-
STUDENT SOLUTIONS MANUAL for STEWART' S
DAN CLEGG • BARBARA FRANK
.Student Solutions Manual
for .
MULTIVARIABLE .CALCULUS
SEVENTH EDITION
DAN CLEGG
Palomar College
BARBARA FRANK ·
Cape Fear Community College
~- . · BROOKS/COLE
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I
·ll
D
PREFACE
This Student Solutions Manual contains detailed solutions to selected exercises in the text
Multivariable Calculus, Seventh Edition (Chapters 10-17 of Calculus, Seventh. Edition, and
Calculus: Early Transcendentals, Sevent,h Edition) by James Stewart. Specifically, it includes solutions to the odd-numbered exercises in each chapter section, review section, True-False Quiz, and
Problems Plus section. Also included are all solutions to the Concept Check questions.
Because of differences between the regular version and the Early Transcendentals version of the
text, some references are given in a dual format. In these cases, readers of the Early Transcendentals
text should use the references denoted by "ET."
Each solution is presented in the context of the corresponding section of the text. In general,
solutions to the initial exercises involving a new concept illustrate that concept in more detail; this
knowledge is then utilized in subsequent solutions. Thus, while the intermediate steps of a solution
are given, you may need to refer back to earlier exercises in the section or prior sections for additional explanation of the concepts involved. Note that, in many cases, different routes to an answer may
exist which are equally valid; also, answers can be expressed in different but equivalent forms. Thus,
the goaJ of this manual is not to give the definitive solution to each exercise, but rather to assist you
as a student in understanc#ng the concepts of the text and learning how to apply them to the. challenge of solving a problem.
We would like to thank James Stewart for entrusting us with the writing of this manual and offering suggestions and Kathi Townes of TECH-arts for typesetting and producing this manual as well as
creati ng the illustrations. We also thank Richard Stratton, Liz Covello, and Elizabeth Neustaetter of
Brooks/Cole, Cengage Learning, for their trust, assistance, and patience.
DAN C LEGG
Palomar College
BARBARA FRANK
Cape Fear Community College
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iii
D
ABBREVIATIONS AND SYMBOLS
CD
concave downward
cu
concave upward
D
the doma in off
PDT
First Derivative Test
HA
horizontal asymptote(s)
I
interval of convergence
1/D
Increasing/Decreasing Test
IP
inflection point(s)
R
radius of convergence
VA
vertical asymptote(s)
CAS
H
· indicates the use of a computer algebra system.
indicates the use of ljHospital's Rule.
' indicates the use ofFormu'Ja j in the Table of Integrals in the back endpapers.
indicates the use of the substitution {u = sin x, if.u = cos x dx}.
indicates the use of the substitution { u = cos x, du = - sin x dx}.
({) 2012 Cengo:~gc LeMning. All RigblS Reserved. May not be scunncd, copied, or duplic:Hed, or posted to a publicly ucce:ssible websire, in whole or jn part.
v
D · CONTENTS
0
10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
10.1
Curves Defined by Parametric Equations
10.2
Calculus with Parametric Curves
0
7
10.3
Polar Coordinates
10.4
10.5
Areas and Lengths in Polar Coordinates
Conic Sections
26
10.6
Conic Sections in Polar Coordinates
Review
35
Problems Plus
13
20
32
43
11 INFINITE SEQUENCES AND SERIES
45
11.1
Sequences
11.2
11.3
11 .4
Series
51
The Integral Test and Estimates of Sums
The Comparison Tests
62
11.5
11 .6
11 .7
Alternating Series
65
Absolute Convergence and the Ratio and Root Tests
Strategy for Testing Series 72
11.8
Power Series
45
59
Problems Plus
0
68
74
11.9 Representations of Functions as Power Series
11.10 Taylor and Maclaurin Series
83
11.1 1 Applications of Taylor Polynomials
90
Review
1
78
97
105
12 VECTQR S AND THE GEOMETRY OF SPACE
12.1
Three-Dimensional Coordinate Systems
12.2
12.3
Vectors
114
The Dot Product
111
111
119
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vii
viii
o
CONTENTS
12.4
The Cross Product
12.5
Equations of Lines and Planes
128
. 12.6
C,ylinders and Quadric Surfaces
135
Review
Problems Plus
0
147
151
13.1
Vector Functions and Space Curves
151
13.2
Derivatives and Integrals of Vector Functions
13.3
Arc Length and Curvature
13.4
Motion in Space: Velocity and Acceleration
Problems Plus
157
161
168
173
179
14 PARTIAL DERIVATIVES
183
14.1
Functions of Several Variables
14.2
Limits and Continuity
183
192
14.3
Partial Derivatives
14.4
Tangent Planes and Linear.Approximations
14.5
The Chain Rule
14.6
Directional Derivatives and the Gradient Vector
195
Maximum and Minimum Values
14.8
Lagrange Multipliers
Problems Plus
203
207
14.7
Review
0
140
13 VECTOR FUNCTIONS
Review
0
123
213
· 220
229
234
245
15 MULTIPLE INTEGRALS
247
15.1
Double Integrals over Rectangles
15.2
Iterated Integrals
247
249
15.3
Double Integrals over General Regions
251
15.4
Double Integrals in Polar Coordinates
258
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CONTENTS
15.5
15.6
15.7
15.8
15.9
15.10
Applications of Double Integrals
Surface Area
267
Triple Integrals
269
Triple Integrals in Cylindrical Coordinates
280
Change ofVariables in Multiple Integrals
285
Problems Plus
289
297
16 . VECTOR CALCULUS
16.1
16.2
16.3
16.4
16.5
16.6
16.7
16.8
16.9
303
Vector Fields
303
Line Integrals
305
The Fundamental Theorem for Line Integrals
Green's Theorem
-316
Parametric Surfaces and Their Areas
Surface Integrals
328
Stokes' Theorem
333
The Divergence Theorem
Problems Plus
321
335 .
337
343
17 SECOND-ORDER DIFFERENTIAL EQUATIONS
.0
17.1
Second-Order Linear Equations
17.2
17.3
17.4
Nonhomogeneous Linear Equations
345
345
·
347
Applications Of Second-Order Differential Equations
Series Solutions
352
Review
354
A.PPENDIX
H
3 10
313
Curl and Divergence
Review
0
276
Triple Integrals in Spherical Coordinates
Review
0
261 '
350
359
Complex Numbers
359
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D
ix
10
D PARAMETRIC EQUATIONS AND POLAR COORDINATES
10.1
Curves Defined by Parametric Equations
y = t2 - t,
1. X= t 2 + t,
y
-2 S t S 2
t=- 2
(2, 6)
t
- 2
- 1
0
l
2
X
2
0
0
2
6
y
6
2
0
0
2
X
3. x = cos2 t,
y
0 S t S 7r /2
y = 1 - sin t,
t=O
(1, 1)
t
0
7r/6
7r/3
7r/ 2
X
1
3/ 4
1/ 4
0
y
1
1/ 2
1 ~ 0.13
0
1-
r=JI.
6
X
y
5. X = 3 - 4t, y = 2 - 3t
(7, S)
t= -1
(a)
t
- 1
0
1
2
X
7
3
-1
-5
y
5
2
-1
-4
(b) X= 3 - 4t
=}
(3, 2)
t= O
X
4t = -X + 3
t = -~ X +~ ' SO
=}
y = 2 - 3t = 2 - 3(- ~x + ~ ) = 2 +~ X - £
1. x = 1 -
=}
(-5, - 4)
1=2
y =~X - ~
e, v = t - 2, - 2 s t s 2
y
(- 3, 0)
t=2
X
(a)
t
- 2
- 1
0
1
2
X
-3
0
1
0
-3
y
-4
-3
-2
- 1
0
(b) y = t - 2
=}
x = - (y + 2)
2
t = y + 2, SO X= 1 - t
(1, - 2)
t= O
2
= 1 - (y + 2?
=}
+ 1, or x =:= - y - 4y -e 3, with - 4 S y S 0
2
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2
D
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
9. X = Vt, y = 1 - t
(a)
t 0 1
y
2
3
4
1.414
1.732
2
X
0
1
y
1
0
=}
t = x2
(b) x = Vt
- 1
- 2
(0, I) ,f = O
X
.:....3
y = 1- t = 1 - x 2 . Since t ~ 0, x ~ 0.
=}
(2, -3) t=4
So the curve is the right half of the parabola y = 1 - x 2 •
11. (a) x = sin ~0, y =cos ~(:1, -1r ~ (:1 ~ 1r.
2
y
(b)
x + y = sin ~8 + cos2 ~8 = 1. For -1r ~ 0 ~ 0, we have
2
2
- 1 ~ x ~ 0 and 0 ~ y ~ 1. For 0 <. 8 ~ 1r, we have 0 < x ~ 1
and 1 > y ~ 0. The graph is a semicircle.
0
-1
13. (a) x = sint, y = csct, 0 < t < ~- y = csct = ~ = !.
.
.
smt
x
I
(b)
'\,,,,
For 0 < t < ~.we haveO < x < 1 andy > 1. Thus, the curve is the
portion of the hyperbola y = 1/x withy> 1.
0
15. (a) x
= e2 t
=}.
2t = ln x
=}
X
t. = ~ ln x.
(b)
X
y
y = t+1 = ~lnx+1.
17. (a) x =sinh t, y = cosh t
' =}
y 2 - x 2 = cosh2 t - sinh2 t = 1. Since
(b)
y
y = cosh t ~ 1, we have the upper branch of the hyperbola y 2 - x 2 = 1.
0
. 19. x = 3 + 2 cost, y = 1 + 2 sin t, 1r/ 2 ~ t ~ 31r/ 2.
X
By Example 4 with r = 2, h = 3, and k = 1, the motion of the particle
takes place on a circle centered at (3, 1) w ith a radius of2. As t goes from -¥- to 3; , the particle starts at the point (3, 3) and
moves counterclockwise along the circle (x - 3) 2 + (y - 1) 2 = 4 to (3, - 1) [one-half of a circle].
21. x = 5sint, y=2 cost
=}
sin t=~, cost=~·
2
2
sin t+cos t =1
=}
(~r + (~r = l.The motionofthe
particle takes place on an ellipse centered at (0, 0). As t goes from - 1r to 511', the particle starts at the point (0, -2) and moves
clockwise around the ellipse 3 times.
23. We must have 1 ~ x ~ 4 and 2 ~ y ~ 3. So the graph of the curve must be contained in the rectangle (1, 4] by [2, 3].
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SECTION 10.1
CURVES DEFINED BY PARAMETRIC EQUATIONS
D
3
25. When t = - 1,-(x,y) = (0, - 1). As t increases to 0, x decreases to - 1 andy
increases to 0. As t increases from 0 to 1, x increases to 0 and y increases. to 1.
As { increases beyo~d 1, both x and y increase. Fort < - 1, x is positive and
decreasing and y is negative and increasing. We could achieve greater accuracy
(-1,0)
t=O
X
t=-1
by estimating x- and y-values for selected values oft from the given graphs and
plotting the correspo~ding points.
27. When t = 0 we see that x = 0 and y = 0, so the curve starts at the origin. As t
y
increases from 0 to~. the graphs show that y increases from 0 to 1 while x
increases from 0 to 1, decreases to 0 and to - 1, then increas.es back to 0, so we
arrive at the point (0, 1). Sinlilarly, as t increases from ~ to 1, y decreases from 1
X
to 0 while x repeats its pattern, and we arrive back at the origin. We could achieve greater accuracy by estimating x- and
y-values for selected values of t from the given graphs and plotting the corresponding points.
29. Use y = t and x = t - 2sin 7ft with at-interval of [-7r, 7r].
31. (a) x = X1 + (x2 - x1)t, y = YI
+ (y2- Y1)t, 0 5 t 5 1. Clearly the curve passes through P1(X1, Yl) when t = 0 and
through P 2(x2 , y2) when t = 1. For 0 <.t < 1, x is strictly between x1 and x2 and y is strictly between y 1 and y 2. For.
2
every value oft, x andy satisfy the relation y- Y1 = y - YI (x - x1), which is the equation of the Line through
X2 - Xl
H (x1, Yl) and P2(;,;2, y2).
Finally, any point (x, y) on that line; satisfies y- Yl = x - x 1 ; if we call that common value t, then the given
.
Y2 - Y1
X2 - X I
,
parametric equations yield the point (x, y); ancf any (x, y) on .th~ line between P1(x 1, y 1) and P2(x2, y 2 ) yields a value of
t in [0, 1]. So the given parametric equations exactly specify the line segment from P1 (x1, y1) to P2(x2, y2).
(b) x = -2 + (3- ( -2)]t = -2 + 5t andy = 7 + (- 1 - 7)t = 7 - Bt for 0 5 t 5 1.
33. The circle x 2 + (y - 1) 2 = 4 has center .(0, 1) and radius 2, so by Example_4 it can be represented by x = 2 cost,
y = 1 + 2 sin t, 0 5 t 5 27r. This representation gives us the circle with a counterclocifwise orientation starting at (2, 1).
(a) To get a clockwise orientation, we could change the equations to x = 2 cost, y =.1- 2 sin t, 0 5 t 5 27r.
(b) To get three times around in the counterclockwise direction, we use the original equations ~ = 2 cost, y = 1 + 2 sin t with
the domain expanded to 0 5 _t 5 67r.
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4
0
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
(c) To start at (0, 3) us ing the original equations, we must have X1 = 0; that is, 2 cos t = 0. Hence, t = ~. So we use
x = 2cost, y = 1 + 2sint, ~ ~ t ~ 3; .
Alternatively, if we want t to start at 0, we could change the equations of the curve. For example, we could use
x = - 2 sin t, y = 1 + 2 cos t , 0 ~ t ~ 1r.
35. Big circle: It's centered at (2, 2) 'with a radius of 2, so by Example 4, parametric equations are
x = 2 + 2cost,
y = 2
+ 2 sin t,
Small circles: They are centered at (1, 3) and (3, 3) with a radius of O.l. By Example 4, parametric equations are
and
(left)
x ;, 1 + O.l cost,
y .= 3 + 0.1 sin t,
0 ~ t ~ 27T
(right)
X =
3 + 0.1 COSt,
y = 3 + 0.1 sin t,·
0 ~ t ~ 27T
Semicircle: It's the lower half of a circle centered at (2, 2) with radius 1. By Example 4, parametric equations are
x = 2 + 1 cost, .
.
y = 2 + 1 sin t,
.
.
To get all four graphs on the same screen with a typical graphing calculator, we need to change the last t-interval to[O, 21r] in
order to match the others." We can do this by changing t to 0.5t. This change gives us the upper half. There are several ways to
get the lower half-one is to change the"+" to a "-" in the y-assigr1ment, giving us
x = 2 + 1 cos(0.5t),
37. (a)x = t 3 :::?
0 ~ t ~ 27T
y = 2 - 1 sin (0.5t ), ·
t = x 1 13 ,so y=t2 = x 213 •
(b) x = t 6
t = x 116 , so y= t 4 = x 4 16 = x 213 •
:::?·.
We get the entire curve y = x 213 traversed in a left to
Since x = t 6 2': 0, we only get the right half of the
right direction.
curve y = x 213 •
.
'
y
y
1
x= t~ y = 1
0
(c) x =e-st = (e- t)3
X
[so e - t = xl/3],
y = e-2t = (e-t )2 = .(xl /3)2 = x2/3.
If t < 0, then x and y are both larger than 1. !f t > 0, then x and y
are between 0 and 1. Since x > 0 and y > 0, the curve never quite
reaches the origin.
39. The case ¥ < (} < 7T is illustrated. 0 has coordinates ( rO, r) as in Example 7,
y
and Q has coordinates (rB, r + r cos(1r- B)) = (r B, 7"(1- cos B))
[sincecos(1r - a) = cos7rc;osa + sin7T sin a· = -cos a ], so.P has
coordinates (r B- rsin(1r - B),r(1- cos·B)) = (r(B :- sin B),r(l - cos B))
[sincesin(1r - a) = sin 1rcosa - cos1r.sin a = sin a ]. Again we have the
parametric equations x = r( (} - sin 8), y = r(l - cos B).
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X
SECTION 10.1
CURVES DEFINED BY PARAMETRIC EQUATIONS
41. It is apparent that x = JOQJ andy= IQPI = JSTJ. From the diagram,
X =
D
y
IOQI = a cos B and y = ISTI = b sine. Thus, the parametric equations are
x = a cos B and y = bsin B. To eliminate B we rearrange: sin B = y / b =>
sin2 B = (y/bl and cosO= x/a
=> cos2 B = (x/a) 2 • Adding the two
X
equations: sin2 B + cos2 B = 1 = x 2 /a 2 + y 2 /b 2 .,Thus, we have an ellipse.
43. C = (2a cot B, 2a), so the x-coordinate of Pis x = 2a cot B. Let B = (0, 2a).
A = ( (2a sin B) cos B, (2a sin B) sin B). Thus, they-coordinate of P
is y = 2asin 2 B.
45. (a)
~
2a
Then L.OAB is a rightangle and L.OBA = B, so JOAI '= 2asinB and
There are 2 points of intersection:
4
( -3, 0) and approximately ( - 2.1, 1.4).
-4
(b) A collision point occurs when x1 =.xz and Yl = Y2 for the same t. So solve the equations:
3sint= - 3+cost
(1)
2 cost = 1 + sin t
(2)
From (2), sin t = 2 cost- 1. Substituting into (1), we get 3(2 cost - 1) = - 3 +cost
cost = 0
=>
t =
=> 5 cost = 0 (*) =>
t or 3; . We check that t = 32..,. satisfies (1) imd (2) butt = t does not. So the on ly collision point
occurs when t = 32", and this gives the point ( - 3, 0). [We could check our work by graphing Xt and x 2 together as
functions oft and, on another plot, y 1 and y2 as functions oft. If we do so, we see that the only value oft for which both
pairs of graphs intersect is t ~ 3; .]
(c) T he circle is centered at (3, 1) instead of ( - 3, 1). Th~re are sti ll 2 intersection points: (3, 0) and (2. 1, 1.4), but there are
no collision points, since(*) in part (b) becomes 5 cost = 6
=> cost = ~ > 1.
47. x = t 2 , y = t 3 - ct. We use a graphing device to produce the graphs for various values of c with -7r ::; t ::; 7r. Note that all
the members of the family are symmetric about the x -axis. For c < 0, the graph does not cross itself, but for c = 0 it has a
cusp at (0, 0) and for c > 0 the graph crosses itself at x = c, so the loop grows larger as c increases.
- I
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5
6
0
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
49. x = t + a cost, y = t + a sin t, a > 0. From the first figure, we see that
curves roughly follow the li ne y = x, and they start having loops when a
is between 1.4 and 1.6. The loops increase in s ize as a increases.
X
While not required, the following is· a solution to determine the exact values for which the curve has a loop,
that is, we seek the values of a for which there exist parameter values t and u such that t < u and
(t+acost,t+asint) = (u + acosu,u+asinu).
y
In the diagram at the left, T denotes the point (t, t), U the point (u, u),
and P the point (t +a cost, t +a sin t) = (u + acosu, u + asinu).
Since PT = PU = a, the triangle PTU is isosceles. Therefore its base
angles, a = LPTU and {3 = LPUT are equal. Since a .= t - i and
(3 = 27r -
3
.;' -
u = 5.;' - u, the relation a = /3.implies that
u + t = s; (1).
'!I
4
X
p
Since TU = distance{{t, t), (u, u)) = J2(u- t) 2 = y'2 (u- t), we see that
cos a= ~TU = (u - t) / v'2, sou - t = v'2acosa, that is,
PT
a
u - t = v'2acos(t- i) (2). Nowcos(t- ~) = sin[f - (t- .;f)] = sinC4,.. - t),
so we can rewrite (2) as u - t = v'2 asine4,. - t) (2'). Subtracting (2') from (1) and
dividing by 2, we obtain t = 34,.. - ~a sine:- t), or 34,..- t =
72 sine
4,.. -
t) (3).
T
u- t
u-t
U
J2
J2
1 -Ji(u -r)--!
Since a> 0 and t < u, it follows from (2') that sin(34" - t) > 0. Thus from (3) we see that t < 34,... [We have
'
implicitly assumed that 0 < t < 1r by the way we drew our diagram, but we lost no generality by doing so since replacing t
by t + 27r merely increases x andy by 21r. The curve's basic shape repeats every time we change t by 21r.) Solving for a in
(3), we get a
v'2(3'~~'-t)
·
v'2z
= sm
. (3! - t ) . Write z = ~,.. - t . Then a = -., where z > 0. Now sin z < z for z > 0, so a > v'2.
sm z
4
[As z --+ o+, that is, as t --+
e
,..)- ,
4
a --+ y'2] .
51. Note that all the Lissajous figures are symmetric about the x -axis. The parameters a and b s imply stretch the graph in the
x- and y-directions respectively. For a = b = n = 1 the graph is simply a circle with radius 1. For n = 2 the graph crosses
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SECTION 10.2
CALCULUS WITH PARAMETRIC-CURVES
0
itself at the origin and there are loops above and b~low the x-axis. In general, the figures have n - 1 points of intersection,
all of which are on the y-axis, and a total of n closed loops.
1.1
\- ---+- (a,b) =( ~.2)'
t1 =3
n =2
n=l
10.2
= (3, 2)
(a, b) = (2, I)
- 2.1
-1.1
(a, b) = (2, 3)
(a, b)
-3.1
- 1.1
- 2.1
-3.1
a=b = l
n=2
n =3
Calculus with Parametric Curves
2
1. x = t sin t, y = t + t
dy
.
dx
.
dy
dy I dt
-d = 2t + 1, -d = t cost+ sm t , and -d = d l d =
::}
z
.
t
t
x
x
t
dy = - 3t2 dx = 4- 2t d dy = dyldt
dt
' dt
, an dx
dx I dt
3
3. x = 1 + 4t - t , y = 2 - t ; t = 1.
2t + 1
. .
t cos t + sm
t
= 4-3tz
-Wh
- 1
- 2t .
en t
'
(x, y) = ( 4, 1) and dy I dx = - ~, so an equa.t ion of the tangent to the curve at the point corresponding to t = 1 is
y- 1 = -~(x- 4), oq/ = - ~x + 7.
5. x = t cos t, y = t sin t; t = 1r.
dy
.
dx
.
' dy
dy I dt
t cos t + sin t
-d = tcost+smt, -d = t(-smt)+cost,and-d = d ld =
.
.
t
t
x
x t
-t s m t+cost
When t = 1r, (x, y) = ( -1r, 0) and dy Idx. = - 1rI( -1) = 1r, so an equation of the tangent to U1e curve at the point
2
corresponding to t = 1r is y - 0 = 1r[x - ( -1r) ) , or y = 1rx + 1r •
7. (a) x = 1 + ln t, y = t 2 + 2; (1, 3).
dy
dx
1
dy
dyldt
2t
2
dt = 2t, dt = t' and dx = dxldt = 1l t = 2t · At( 1 • 3),
ln t = O
=?
t =1 an~ : = 2,so an equationofthetangentis y-3=2(x - l),
ln t=x -1
=?
t=e"'-l,soy=t2.+2=(e"'- 1 ) 2 + 2 = e 2 " - 2 +2,andy'=e 2 x-z . 2.
x=l+ ln t = 1 ::}
ory = 2x + 1.
(b)x = 1 + ln t
=?
At (1, 3), y' = e 2 (l)- z · 2 = 2, so an equation of the tangent is y - 3 = 2(x - 1), or y = 2x + 1.
9. x = 6sin t, y = t2+t;
(0,0).
dy
dy 1dt
2t + 1
.
- = d ld = - -. The potnt (0, 0) corresponds tot = 0, so the
dx
x t
6cost
slope of the tangent at that point is·lJ. An equation of the tangent is therefore
y - 0=
i(x- 0), or y = ix.
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7
8
D
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
2
2
11. x = t + 1• Y = t + t
The curve is CU when
13. x
2
d(~)
'*
dt dx
dx/ dt
~~ > 0, that is, when t < 0.
= et, y = te-t ::::}
d y = dt (h;
dx 2
dx/dt
!!:._ (dy)
dy _ dy I dt _ 2t + 1 _
.!_
dx - dx f dt 2t - 1 + 2t
dy
dyfdt
- te- t + e- t = e-t (1- t) = e- 2t( _ t)
1
dx = dxjdt =
et
et
'*
.
= e- 2 t(- 1) + (1 - t)( - 2e- 2 t) = e - 2 t(- 1 - 2 + 2t) = e-3t(2t- 3). The curve is CU when
et
et
2
d y > 0, that .ts, w hen t > 3 .
dx
2
2
15. x = 2sint, y = 3cost, 0 < t < 21r.
.
i_OO(dy
3
2
dy _ dy/ dt _ -3sint _ -~
d 2 y _ dt dx _ -2sec t _ - ~
3
dx- dxj dt - 2cost 2 tant, so dx2 dxjdt - 2cos t 4 sec t.
The curve is CU when sec3 t < 0
.
17. X = t 3 - 3t, y = t 2 - 3.
(x, y) = (0, - 3).
sect < 0
::::}
d
::::}
cost < 0
¢>
t =0
d
dy = 2t, SO dy = 0
t
'
t
::::}
i < t < 3; .
¢>
dxd = 3t2 - 3 = 3(t + 1)(t - 1), so d:
dx = 0 <=>
t
.
t
t = - 1 or 1 <=> (x, y) = (2, -2) or (-2, -2). The curve has a horizontal
tangent at (0, - 3) and vertical tangents at (2, - 2) and ( -2, -2).
19. x = cos 9, y = cos 39. The whole curve is traced out for 0 ~ 9 ~ 1r.
~~ = - 3'sin 39, so ~~ = 0 <=> sin 39 = 0 <=> 38 = 0, 1r, 21r, or 37r <=>
'e =0, ~ . 23.,.,or1r
2
(-t.1)
(1,1)
9=21T/3
9= 0
<=> (x,y) = (1,1), (~ , - 1), (-~ , 1) , or(-1,- 1 ).
dx
.
dx
d 9 = - sLD 8, so dO = 0 <=> sin 8 = 0 <=> 8 = 0 or 1r <=>
dy
d.x
(x,y_) = (1, 1) or (-1,-1). Both dO and dO equal 0 when 8 = 0 and 1r.
(- 1, -1)
(J= 1T
-2
3
38 11
38
..,
.s the same s Iope when 0 = 1r.
10 fi nd th e s Iope wIten 9 = 0, we fi nd lim dy
dx = lim - sin
. n = lim _g cos ,.. = 9, which 1
o~o
o ~o - sm"
O-•O -cos"
Thus, the curve has horizontal tangents at ( ~ , -1) and (-
t, 1), and there are no vertical tangents.
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SECTION 10.2
CALCULUS WITH PARAMETRIC CURVES
0
9
21. From the graph, it appears that the rightmost point on the curve :J: = t - t 6 , y = et
is about (0.6, 2). To find the exact coordinates, we find the value oft for which the
graph has a vertical tangent, that is, 0 = dxl dt = 1 - 6t5
{::}
t = 11 W.
Hence, the rightmost point is
23. We graph the curve x = t
4
-
2t3 - 2e, y = t 3 - tin ~he viewing rectangle [-2, 1.1] by [- 0.5, 0.5]. This rectangle
corresponds approximately tot E [ -1, 0.8] .
o.s
- 0.5
We estimate that the curve has horizontal tangents at about ( - 1, - 0.4) and ( - 0:11, 0.39) and vertica·Ltangents at
dy
dyl dt·
about (0, 0) and ( -0.19, 0.37). We calculate -d = dxl d =
t
X
dy I dt = 3t2 - 1 = 0
1=}
3t2- 1
.
. The honzontal tangents occur when
·
4t 3 - 6 t2 - 4t
t = ± ~ , so both horizontal tangents are shown in our graph. The vertical tange~ts occur when
dxldt = 2t(2t2 - 3t- 2) = 0 {::} 2t(2t + 1)(t - . 2) = 0 {::} t = 0, - ~or 2. It seems that we have missed one vertical
tangent, and indeed if we plot the curve on the t-interval [-1. 2, 2.2] we see that there is another vertical tangent at ( -8, 6).
25. x = cos t , y = sin tcos t . dxl dt = - sint, dyl dt = - sin2 t + cos 2 t = cos 2t.
(x,y )= (O , O)
{::} cost = .O. {::}
tisanoddmultipleoflf.Whent=~,
dxl dt = - 1 and dyl dt = - 1, so dyl dx = 1. When t = 3; , dxl dt = 1 and
dy I dt = - 1. So dy I dx = - 1. Thus, y = x andy = - x are both tangent to the
curve at (0, 0).
27. x
=rB - d sinB, y = r - dcosB.
dx
dy , .
dy
d sinB
(a) dB = r- dcosll, dB = dsme, so dx = r _ dcosB '
(b) IfO < d < r, then jdcosllj:::; d <. r , so r- dcosfi ;::: r- d > 0. This shows that dxldll never vanishes,
·so the trochoid can have no vertical tangent if d < r .
29. X = 2t3' y = 1 + 4t - t2
6t 2 + 2t - 4 = 0
{::}
---'-
_,..
dy - dy I dt = 4 - 2t N
1. dy = 1
·dx - dxl dt
6t 2 • ow so ve dx
2(3t - 2)(t + 1) = 0
{::}
~
.,_,.
4 - 2t = 1.
6t2
. . . ,._
.....,.
t = ~ or t := - 1. 1ft = ~ . the point is(~~ ' : ) , and ift = -1,
2
the point is (-2, -4).
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10
0
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
31. By symmetry of the ellipse about the x- andy-axes,
A= 4J0a ydx = 4 j~12 bsinB (-a sin B) dB= 4ab I 0" 12 sin2 BdB= 4ab Io" 12 ~(1- cos 28) d8
,
/2
I
= 2ab[8- ~ sin2B]~ = 2ab(~) = 1rab
y
33. The curve x = 1 + et, y = t- t 2 = t(1 - t) intersects the x-axis when y = 0,
that is, when t = 0 and t = 1. The corresponding values of x are 2 and 1 + e.
0
-I
The shaded area is given by
1::1+e
(YT- YB) dx =
=
1::
1
[y(t)- OJ x'(t) dt = I;(t - t2)et dt
J; tet dt- I t et dt = f tet dt- [fet] ~ + 2 j; tet dt
1
0
1 2
0
'
= 3 Io1 tet dt- (e - 0) = 3 [ (t1)et] 01 - e
[Formula 97 or parts]
[Formula 96 or parts]
= 3[0 - (-1)] - e = 3 ~ e
35. x = r8- dsin8, y = r- dcosB.
A= I;~r y dx =
J;rr (r- dcosB)(r - dcos 8) dB= I;rr (r
2
- · 2drcos B
+ d~ cos2 B) dB
= [r 2 B- 2dr sine+ ~d2 (8 + ~ .sin 28) ]~7r = 21rr2 + 1rd2
37. x=t+e~t, y=t-e-t, O~t~· 2.
dx jdt= 1- e- tanddyjdt=1+e-t,so
(dx/dt) 2 + (dyjdt? = (1- e-t) 2 + (1 + e- t? = 1- 2e-t + e- 2 t + 1 + 2e-t + e- 2 t = 2 + 2e- 2t .
Thus, L = I: './(dx/dt) 2 + (dyjdt) 2 dt =I: v2 + 2e- 2 t dt ~ 3.1416.
39. .x = t- 2sint, y = 1 - 2 cos t, 0 ? t ~ 47r.
&cjdt = 1 -2 cost and dyjdt = 2sint, so
(dx/dt) 2 + (dyjdt) 2 = (1- 2 cos t) 2 + (2 sin t) 2 = 1- 4cos t + 4 cos 2 t + 4sin2 t:::;; 5- 4cost.
Thus, L = I: './(dx/dt) 2 + (dyjdt) 2 dt =
41. x = 1 + 3e, y = 4 + 2t~,
o ~ t ~ 1. ~x/dt ~ 6t and dyjdt = 6t2 , so (dxjdt? + (dy / dt) 2 = 36t2 + 36t4
1
Thus,
L =
Io4 " v5- 4cost dt ~ 26.7298.
.
1
Ia './36t + 36t dt = Ia 6t .Jl+i2·dt = 61 JU Gdu)
2
~
4
[u = 1
+ t 2, du = 2tdt]
= 3[~u312 J: = 2(2 3 ; 2 - 1) = 2(2J2 - 1)
43. x ~ t sin t, y =.t cost,. 0 ~ t ~ 1.
(~~ y+ ( ~; y e
=
~~ = t cost + sin t and ~~ = - t sin t + cos t, so
2
2
2
cos t + 2t sin t cost + sin t + t sin t
2
~ 2t sin t cost + cos2 t
= t 2 (cos2 t + sin2 t) + sin2 t + cos2 t = t2 + l.
Thus, L .·= f 0 vt2 + 1 dt ~ [~tvt~ + 1 + ~ ln(t
1
+vt + 1 ) ] ~ = 4J2 + ~ ln(1 + J2).
2
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SECTION 10.2
CALCULUS WITH PARAMETRIC CURVES
0
45.
~~ ) + ( !fJd = [e'(cos t - sin t)f + [et(sin t +cos t )f
2
(
= (e') 2 (cos2 t - 2 cos t sin t + sin 2 t) '
+ (et) 2 (sin2 t+2sint cost+cos2 t
= e 2 t(2 cos2 t + 2 sin2 t ) = 2e 2 t
Thus, L = J~" .J2e2t dt = f 0" -12 et dt =· -12 [ e' ]~ = -/2 (e,.- 1).
The figure shows the curve x = sin t + sin 1. 5t, y = cost for 0 .:::; t 5 47r.
1.4
47.
. dx/dt = cost+ 1.5 cos 1.5t and dy / dt = -sin t, so
-2.1 f->o.o--------,.£-f->'<-----~-12.1
(dx/dt) 2 + (dy / dt? = cos2 t + 3cost cosl.5t + 2.25cos2 1.5t + sin2 t.
Thus, L = J0 " v'1 + 3 cost cos 1.5t + 2..25 cos2 l .5t dt ~ 16.7102.
4
49. x
= t - et, y = t + et, -6 5 t 5 6.
(~~)2 + (!!Jd = (1- et? + (1 + et? = (1 - 2et + e 2 t) + (1.+ 2et + e2 t) = 2 + 2e2 t, soL= f~ 6 v'2 + 2e2t dt.
Set f(t ) = v'2 + 2e2 t. Then by Simpson's Rule with n = .6 and D.t = ·e-~-e) = 2, we get
L ~ j[J( - 6) + 4/( -4) + 2/( -2) + 4/(0) + 2/(2) + 4/(4) + /(6)] ~ 612.3053.
51 . x = sin2 t , y '= cos2 t, 0 5 t 5 37r.
2
2
(dx / dt? + (dy/dt? = (2sin t cos t) 2 + ( - 2 cos tsin t? = 8 sin t cos t = 2 sin 2 2t ==>
~in 2t dt [by symmetry) = -3 -/2 [cos 2t]: = -3 -/2 (- ~ - 1) ~ 6 -/2.
The fu ll curve is traversed as t goes from 0 to ¥, because the curve is.the segment of x + y = 1 that lies in the fi rst quadrant
Distance = J0 " V'i lsin 2tl dt = 6 -/2 J;
3
12
12
(since x , y 2:: 0), and this segment is completely traversed as t goes from 0 to¥ · Thus,·£ ~ f 0" 12 sin 2t dt = -/2, as above.
53. x = asin8, y = bcos O, ·a$ 8 5 21r.
(~7)2 + (!!Jd·= (a cosO?+ ( - bsin8? .,;, a 2 cos2 8 + b2 sin2 8 = a 2 (1- sin2 B)+ b2 sin2 8
= a 2 - (a2 - b2 ) sin2
SoL = 4 j 0"12
..jaz (1- e sin 0) dO
2
2
8= a
[hy '?·mmctty)
.
2
-
c2 sin2
2
8= a (1- ~ sin 8) = a (1 - e sin 8)
2
2
2
2
2
= 4a J;' 12 Vl - e 2 sin2 8 dO.
55. (a) x = 11 cos t - 4 cos(11t/ 2), y = 11 sin t- 4sin(llt/2).
Notice that 0 5 t 5 21r does not give the complete curve because
x(O) f:. x(21r). In fact, we must take t E [0, 47r] in order to obtain the
complete curve, since the first term in each of the parametric equations has
period 27r and the second has period 1ii2 = ~~, and the least common
integer multiple of these two numbers is 47r.
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11
12
0
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
(b) We use the CAS to find the derivatives dx/dt and dy/dt, and then use Theorem 6 to find the arc lengtll. Recent versions
of Maple express the integratJ;" J(dxjdt) 2 + (dyjdt) 2 dt as 88E(2 V2 i), where E (x) is the elliptic i~tegral
x2t2
Jo{1 v/1v'f=t2 dt and i is the imaginary number yCI.
I
Some earlier versions of Maple (a5 well as Mathematica) cannot do the integral exactly, so we use the command
evalf (Int (sqrt (diff (x, t) - 2+diff (y, t) -2), t=O .. 4*Pi)); to estimate the length, and find that the arc
length is approximately 294.03. Derive's Para_a r c_length function in the utility fi le Int_apps simplifies the
integral to 11 }~ ,.
4
J-4 cost cos(l~t ) - 4 sin t sin(l~t) + 5 dt.
57. x=tsint, y=tcost, O~t~Tr/2.
dxjdt=tcost +sintanddyjdt= -tsint +cost,so
e
(dx/dt? + (dyjdt) 2 = t 2 cos2 t + 2tsin t cost+ sin2 t + sin2 t- 2tsin t cos t+ cos2 t
= t 2 (cos2 t + sin2 t) +.sin2 t + cos2 t = t 2 + 1
59. x = 1 + tet, y = (t
2
+ l) et, 0 ~ t ~ 1.
2
(~~ ) + (!fltf = (te 1 + et)2 + [(e + I )et + et(2tW = [et(t + 1}f + [et (e + 2t + 1W
= e 2 t(t + 1? + e2 t(t + 1) 4 = e 21 (t + 1) 2 [1 + (t + 1) 2 ),
3
= 2Tr J(4
-
2!:.
-
81
(uT 4) Ju (fg
2
J
[~u5/2
- §.u3/2] 13 - 2!:. • .L [3u5/2 - 20u3/2] 13
5
3
4 - 81
15
4
= 1;~5 [(3. 13
2
v'13- 20. 13 VI3) - (3 ·.32 - 20. 8)] = 1;~5 (247 VI3 + 64)
63. x = acos 8, y =a sin 8, 0 ~ B ~
3
u = Dt + 4, t 2 = (u- 4)/9,
[ du = 18t dt, so t dt =·f.J tlu
du)
so
3
I·
2
(~~) + (;¥tt) = ( - 3acos2 8 sinG?+ (3a sin 2 8 cos 0) 2 = 9a2 sin2 (I cos 2 0.
2
S = .{0" 12 2Tr · a sin 3 () · 3asin 0 cos() d() = 61ra2 J0rr/ 2 sin~ 0 cos 0 dO = ~1ra2 [sin 5 0] ~ 12 = ~1ra 2
65. X= 3t2, y = 2t 3, 0 ~ t ~ 5 . ::}
C~l + (~/} = (6t? + (6t2) 2 = 36t 2(1 + t2) ::}
s = .r; 21rx J(dx/dt)2 + (dyJdt)2 dt = J; 27r(3e)6t vt + t 2 d.t = l81r J; t2J1 + t 2 2tdt
=
+t2,] =18Trj·2G(u3/2- u1/2) du = 18Tr ['du5/2- ~u3/2] 26
u
1
[ du = 2t dt
1
5
3
= 187r [ ( ~ . 676 J26 - ~ . 26 J26) - ( ~ - ~)] = ¥ 7r (949 J26 + 1)
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1
1
SECTION 10.3
POLAR COORDINATES
0
13
67. If f' is continuous and j' (t) i= 0 for a$ t $ b,.then either!' (t) > 0 for all tin (a, b) or j' (t) < 0 for all t in [a, bj. Thus, f
.
.
'
is monotonic (in fact, strictly increasing or stri<:tly decreasing) on (a, b]. It follows that f has an inver;se. Set F =g o f- 1 ,
that is, define F by F(x) = g(r (x)). Th~n x = f(t)
1
69
· (a)¢> = tan
- 1(
dy)
dx
d<P _ d
~
dt - -dt tan
d(
dy )
d ( if )
fix - xif
dt di = dt ;; =
xz
"a·cttllat
s-- Jrot
!i
-1 (
1
1
dy ) _
1
[ d ( dy ) ]
dy
dy 1dt
i1
dx - 1 + (dyjdx)2 dt dx · But dx = dx/dt = ± ~
d¢>
1
( fl± - xfl )
xy - xif ' .
.
·
dt = 1 + (iJ/x) 2
x2
= xz +iF. Usmg the Cham Rule, and tne
~
(d:r;)2+(!!11..)2dt
dt
dt
r (x) = t , soy= g(t) = g(f- (x)) = F(x).
~
d .• 'Jt
-
~
y /(d:r;)2+(!!11..)2
dt
dt
= ('2+
x
y•2)1/2 , we bave t hat
0
_- I 1- 1(±2xyxy 1- l±ii - xiJI
+ y2)3/ 2 - (±2 + y2)3/2 ·
d¢> _ d¢>/dt _ (xii- xfl )
1
_
xii - xiJ .
d¢>
ds - ds/dt :i;2 + y2 (±2 + y2)1/2 - (~2 + 1;2)3/2 · So"'- - ds (b) x = x_and y = !(x )
~
dy ..
d2y
. 1 .. 0 d .
x = 'x = an y = dx' y = dx2'
- II· (d~y/dx2 ) - 0 . (dyjcb:) i So"' 71. X = 8 - sin8
(1 +
~
:i; = 1----' cosB
-
-~
ld2 y/ dx2 1
(1 + (dyjdx)2] 312 •
X = sin8, and y = 1 - cos8
~
iJ =sine
~
y = cos8. Therefore,
icosB - cos
sin
icosB - (cos 8 + sin 8)1
JcosB - II
.
=
.
=
. The top of the arch 1s
[(1 - cos B) 2 + sin2 B)S/ 2
(1 - 2 cos + cos2 + sm2 8)312
(2 - 2 cos 8)3/2
2
"' =
(dyjdx)2 )31 2
e-
2
el
2
e
2
e
characterized by a horizontal tangent, and from Example 2(b) in Section 10.2, the tangent is horizontal when B = (2n - l )1r,
d b ·
B
· ti
. fi
Jcos 1r -11
1-1- 11
1
_ (- ))3/ 2 =4·
sotaken=1 an su stttute =1rmto 1e ex~resston or ~~;: ~t= ( _ cos1r) 3 / 2 =
2 2
12 2 1
73. The coordinates of T are .(r cosB, r sin B). Since TP was unwound from
arc T A, T P has length rB. Also LPTQ = LPTR - LQTR = t 1r - 8,
soP has coordinates x = r cos B + r8 cos(~1r ~B) = r(cos B + Bsin 8),
y = rsin B - rBsin (~1r - 8) = r(sin8- 8cos9).
X
10.3
Polar Coordinates
1. (a) (2, i)
By adding 21r to i, we obtain the point ( 2, 7; ) . The direction
opposite i is 43.,., so (- 2, ~ ) is a point that satisfies the r < 0
· requirement.
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14
0
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
r > 0: (1, - 3;
(b) (1, - 3; )
+ 21r) = (1, 5; )
r < 0 : (- 1,- 3r + 7r) = (-1,~)
0
_ 311'
4
(c) ( - 1, 2J)
r
r < 0 : (-1 , ~ + 21r) = (- 1, 52,.)
,.
0
> 0: (- (- 1), 1J +1r) = (1, 3; )
2
(-t,f)
3. (a)
X= 1COS7r
= 1(- 1) = -1 and
y = 1 sin 1r = 1(0) = 0 give us
'TT
the Cartesian coordin~tes ( -1, 0).
(l, 'TT)
0
x = 2cos(- 23"' ) = 2(-~) = - 1 and
(b)
y = 2sin( -
2
;)
= 2( - ~) ~ ~VS
give us ( -1, -V3).
x = -2cos 3;
(c)
= -2(-4) = J2 and
y = - 2sin 4" = -~(
3
4) = - J2
gives us ( J2, -J2).
5. (a) x = 2 andy = - 2 => r = J2 2 + (-2)2 = 2 J2 and 8 = tan- 1 (
:n := -~ . Since (2, -2) is in the fourth
quadrant, the polar coordinates are (i) (2 J2, 7; ) and (ii) (- 2 J2, 3; ) .
(b) x = - 1 andy = V3
=> r =
J< -1) + (v'3) = 2 and 8 = tan~ :1)
2
2
1(
=
2
:;. Since_( -1,
VS) is in the second
quadrant, the polar coordinates are (i) (2, ~)and (ii) ( - 2, 6; ) .
'·
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SECTION 10.3
7. r ~ 1.
The curve r = 1 represents a circle with center
POLAR COORDINATES
0
15
9. r ~ 0, 7r/4 S, B S, 37r/ 4 . .
0 and radius 1. So r ~ 1 represents the region on or
8 = IC represents a line through 0.
outside the circle. Note that 8 can take on any value.
0= 37T
4
0='!!..
4
< B < r1r
11· 2 < r < 3 •s.,..
a--a
0 -- 11<
3
r=2
o-- s1f3
13. ·Converting the p~lar coordinates (2, 1r/ 3) and (4, 21r/ 3) to Cartesian coordinates gives us (2 cos f, 2 sin f) = (1 , J3) and
(4 cos 2; , 4 sin 2; ) = (- 2, 2 J3). Now use the distance !ormula.
15. r 2 = 5
<*
17.r =2 cos8
x2 + y 2 = 5, a circle of radius .j5 centered at the origin.
~
r 2 = 2rcos8
<*
2
x 2 + y =2x <* x 2 -2x+ 1 +y2 = 1 <*
radius 1 centered at (1, 0). The first two equations are actually equivalent since r 2 = 2r cos()
(x-1) 2 +y2 = 1,acircleof
~
r(r - 2 cos 8) = 0
r = 0 or r = 2 cos 8. But r = 2 cos 8 gives the point r = 0 (the pole) when 8 = 0. Thus, the equation r = 2 cos 8 is
equivalent to the compound condition ( r = 0 or r = 2 cos 8).
the origin with foci on the x -axis.
21. y = 2
<*
23.y=1 +3x
2
r=--
¢?
r = '2csc8
rsin8 = 1 +3rcos8
¢?
rsin8-3rcosB = 1
r s in B = 2
¢?
¢?
sin8
¢?
r(sin8-3cos8)=1
<*
1
r = -sin::--8:::-- -3-=co_s_e-=
25. x 2 +y2 =2cx
¢?
r = 0 is included in r
r 2 =2crcos8
¢?
r 2 -2crcos8=0 <*
r(r-2ccos8) = 0
¢?
r = Oorr=2ccos8.
= 2c cos 8 when 8 = ~ + n1r, so the curve is represented by the single equation r ~ 2c cors ().
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~
16
D
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
27. (a) The description leads jmmediately to the polar equation 9 = ~.and the Cartesian equation y,; tan (~) x =
7:i xis
slightly more difficult to derive.
(b) The.easier description here is the Cartesian equation x = 3.
29. r = - 2 sin9
r
-2
(2, 3'11'/2)
31. r = 2(1 +cos B)
(4,0)
2'11' 0
33. r = 9, 9;::: 0
r
(J
35. r = 4sin 39
r
./ o=f
4
,..:/ 2 (4.f)
(}
-4
37. r = 2cos49
r
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SECTION 10.3
POLAR COORDINATES
39. r = 1--2sin8
0=!!.
__________..•----- 6
41. r 2 = 9sin28
3
2..-
0
-3
43. r = 2 + sin 38
r
(3, ..-j6)
3
2
2..-
---r--~------~----+- t-~
3..-
6
7..-
6
11..-
0
6
45. r = 1 + 2cos28
IJ
47. For(}=:= 0, 7T, and 27T, r has its minimum ;value of about 0.5. For 0 =~and 32", r attains its maximum va lue of2.
We see that the graph has a similar shape for 0 ~ (} ~ 7T and 7T ~ 8 ~ 27T.
. r
2
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0
17
18
0
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
=>
49. x = r cos 8 = (4 + 2 sec 8) cos 8 = 4 cos 8 + 2. Now, r -+ oo
(4 + 2sec8)-+ 00 => 8-+ (~) - or 8 -+ e2"' )+ [since we need only
consider 0 $ 8 < 271'], so lim x =
.
r -oo
(6, 0)
lim (4 cos 8 + 2) = 2. Also,
8 -+ -rr/2 -
=> (4 + 2 sec B) -+ -oo => 8 -+ ( ~) + or 8 -+ ( 3; ) - , so
T -+ -oo
lim x =
=> x = 2 is a vertical asymptote.
lim ( 4 cos 8 + 2) = 2. Therefore, lim x = 2
r --oo
8--+rr/2+
T"--+±oo
51 . To show that x = 1 is an asymptote we must prove lim x = 1.
r - ±oo
x = (r) cosO= (sin 8 tan 8) cos 8 = sin 2 ().Now; r -+ oo
8 -+ (f) - ,so lim x=
r --+oo
lim
sin2 8 = l.Aiso,r-+-oo
0-+'Tr/2-
.
x= l
=> sinO tan8-+ oo
=>
=> sinBtanB-->-oo =>
e-+
(~) +,so lim X = lim sin2 {;I,;, 1. Therefore, lim X= 1 . =>
.
r ...... - oo
0--+1fj 2+
,·-±oo
X= 1 is
a vertical asymptote. Also notice that X = sin2 e ~ 0 for all e, and X = sin2 {;I $ 1 for all e. And Xi= 1, since the curve is not
defined at odd multiples of ~. Therefore, the curve lies entirely within the vertical strip 0 $ x < 1.
53. (a) We see that the curve r = 1 + csin 8 crosses itself at the origi n, where r = 0 (in fact the inner loop corresponds to
negative r-values,) so we solve the equation of the lima~on for r = 0
<=>
c sin 8 = - 1
<=> sin 8 = - 1/ c. Now if
lei < 1, then this equation has no solution and hence there is no inner loop. But if c < - 1, then on the interval (0, 27T)
the equation has the two solutions 8 = sin- 1 ( -1 /c) and 8 = 1r- sin- 1 ( - 1/c), and if c > 1, the solutions are
(} = 1r + sin - 1 (1/c) and ()= 21r- sin- 1 (1(c). Tn each case, r < 0 for (} between the two so lutions, indicating a loop.
(b) For 0 < c < 1, the dimple (if it exists) is characterized by the fact that y has a local maximum at 8 = 3; . So we
determ.ine for what c-valu~s d ~ is negative at(} = 3 ~, since by the Second Derivative Test tllis indicates a maximum:
~
.
.
2
y = rsin8 =sinO + c sin 2 (}
2
d2 y
dy
=> dB = cos8 + 2csinfJ cos8 = cosfJ + csin 28 =>. -dB~., = - sin8 + 2ccos 28.
At 8 = 32,.., this is equal to - ( - 1) + 2c( -1) = 1 - 2c, which is negative only for c > %. A similar argument shows that
for - 1 < c < 0, y only has a local minimum at e = ~ (indicating a dimple) for c < -~ . .
=>
55. r = 2 sin (}
x = r cos(} = 2 sin fJ cos 8 = sin 28, y = 1· sin 8
= 2 sin 2 B =>
dy = dyjd(J = 2 · 2 sin fJ cosO= sin 28 = tan 2B
dxjdB
cos2fJ·2
cos28
dx
When 8 = 7T , dy
d.'t =tan ( 2 · 7r) =tan 7T = vrr;
3.
6
57. r= l / 8
=>
6
3
· 3.]
[Another method: Use Equation
x=rcos8 = (cosfJ) j B, y=rsin8=(s infJ) / 8
=>
dy
dyjd8
sin.O( - 1/ 82 ) + (1/ 8) cos 8 8 2
- sin (}+(} cosO
dx = dx j dO = cos8(- 1/ fJ 2 ) - (1/ 8) sinB · (} 2 = - cosO - Osin B
.dy
- 0 + 7r(- 1)
- 'Tr
When 8 = 1r, dx = -(-1) -1r(O) = l
= -1r.
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'
SECTION 10 .~
POLAR COORDINATES
0
=> x = r cos() = cos 2() cos(}, y = r sin() = cos 2(} !lin(} =>
59. r = cos 2()
dy
dy / df)
cos 2(} cos(}+ sin(} ( -2 sin 2())
dx = dx / d() = cos 2() (- sin(})+ cos() ( -2 sin 2())
When() =
61 . r = 3cos0
-,0,!-(J2---:2:/---27)+_('-:-v-2
-:::2/,.--!2)~(-_2;,_) - _- J2_2 - 1
O(-J2/ 2) + (J2/ 2}(-2) - -J2 - .
dy
1f
4 , dx
=> x = rcosO = 3cos0 cosO,_ y = rsinO = 3cos0 sinO => ,
2
~ = -3 sin 9 +3cos 2 ,() = 3cos20 = 0 => 2() = ~ or 3; <=> 0 = %or 3; .
So the tangent is horizontal at ( ~, %) and (
~~ = - 6 sin() cos~= -3 sin 20 = 0
63. r = 1 + cos 0 · =>
--32,
3
;)
[same as (
-32,-:a:)] .
=> 29 = 0 or 1r <=> 0 = 0 or~· So the tangent is vertical at (3, 0) and (o, ~).
x = r cos 9 = cos() (1 +cos 9), y = r sin 0 = sin 0 (1 + cos 9)
=>
·.~ = (1 +cosO) cosO- sin2 0 = 2cos 2 0 + cos9 - 1 = (2cos9 - 1) (cos(J + 1) = 0
() = 'i· 1r, or 5;
() = 0, 1r, 2; , or 4;
bx + (~b? + y
=> sin 8 = 0 or cos 9 = - ~ =>
e : j::
= 0.
=> r 2 = ar sin() + br cos 8 '=> x 2 + y 2 = ay + bx =>
65. r = a sin() + b cos 9
-
=>
=> vertical tangent ~t (2, 0), (!, 2; ) , and (!, -t;).
Note that the ~gent is horizontal, not vertical when 8 = 7r, since
2
cos()=~ or - 1
=> horizontal tangent at(~ , 'i), (0, tr), and(~ , 5; ) .
~~ = - ( 1 + cos 8) sin (} - cos ()sin() = - sin (} ( 1 + 2 cos 9) = 0
x
=>
2
2
-
2
ay + ( ~a) = (~b) + (ta?
=>
2
(x - ~b) + (y - ~ a? = Ha2 + b2 ) , and this is a circle
with center (!b, ~a) and radius. ~.Ja 2 + b2 •
67. r = 1 + 2 sin(8 / 2): The parameter intei-val is [0, 47r].
69. r = e•inB - 2cos(40).
The parameter interval is [0, 21l'j.
71. r = 1 + cos999 8. The parameter interval is [0, 2tr).
1.1
/
- 1.1
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19
20
D
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
73. It appears that the graph of r = 1 + sin(~ -
"if) is the same shape as
r = 1 + sin ll
the graph of r = 1 + sine, but ro,tated counterclockwise about the
origin by "if. Similarly, the graph of r = 1 + sin(
r = l +s~(o - ~_)
e. . ,. ~) is rotated by·
r = 1+ sin(ll - ~)
i . In general, the graph of r = f (e - a) is the same shape as that of
r = f (e), but rotated counterclockwise through a about the origin.
That is, for any point (ro, eo) on the curve r = j(e), the point
-0.9
(ro, Bo +a) is on the curve r = j(e- a), since ro = f(Oa) = f((eo +a)- a).
75. Consider curves with polar equation r = 1 + c cos
the pole. For 0 ~ c
e, where c is a real number. If c = 0, we get a circle of radius 1 centered at
s 0.5, the curve gets slightly larger, moves right, and flattens out a bit on the left side. For 0.5 < c < 1,
the left side has a dimple shape. For c = 1, ,the dimple becomes a cusp. For c > 1, there is an internal loop. For c 2: 0, the
rightmost point on the cu;ve is (1 + c, 0). For c < 0, the curves are reflections through the vertical axis of the curves
with c > 0.
c = 0.75
c = 0.25
tan ¢ - tane
77. tan'lj! = tan(¢- e)=
r/>
e
1 + tan tan
_
dy
- - tane
dx
dy
1 + dx ta11e
c= 1
dy/dB
dxjde - tane
dyjde
1 + dx j de tane
~ - ~ tan e ~ (~ ~in e + r cos e) - t a n e ( ~ cos B- r sine)
- : + :~ tane -
(:;.cos
c= 2
e- r sine) + tan B( :; sine + r cos e)
sin2 e
rcosB+r · - case
dr
dr sin2 e
de case+ de. case
_ rcos2 B +rsin2 e
r
2 iJ
dr . 2 e = dr I dO
- dr
de cos . + dB sm
10.4 Areas and Lengths in Polar Coordinates
@ 2012 Censagc Learning. All Rights Reserved. May not be scanned: copied, or duplicated, or posted to 0: publicly ncccssiblc website, in whole or in part.
SECTION 10.4 A~~S AND LENGTHS IN POLAR COORDINATES
3. r 2 = 9sin28, r ~ 0, ·a~ 8::; 7rj2.
A --
1
~/2
0
2
lr
d8 -2
1~/2
.
0
1(9sin2B)d8
-l!.
2
- 2 [- lcos2B)~
2
0
/2
1---l!.(4
1)=l!.2
7. 1' = 4 + 3sin8, -%::; () ::; %·
= ~
~/2
1 (16 + 9sin 8) dO
2
[by Theorem 4.5.6(b) [ET 5.5.7(b)]]
-rr/2
r /2
= ~·2}
[16 + 9 ·~(1 -cos28))d~
0
1 ("; - ~cos28)
rr / 2
=
[by Theorem 4.5.6(a) [ET 5.5.7(a)]] .
'
12
d8 = [~1 8 - tsin2BJ; = ( 4 ~" - O)- (0 _ 0) = 4~"
9. The area is bounded by r = 2 sin 8 for 8 = 0 to 8 = 7r.
A= 1"' ~r d8 := ~ 1 -rr (2 sinfJ) dB= ~
2
2
1"'
(2, 'TT/ 2)
2
4sin Bd8
=21" ~(1 - cos2fJ)dB ~ [e - ~sin2eJ: =1r
Also, note that th is is a circle with radius 1, so its area is 1r(1) 2 = 1r.
1 ~r d8 = l
~1
2
2
11. A =
"
2
fo rr(9+12.cos8+4cos 8)d8
2
2
rr H3 + 2cos8) d8=!
2
2
=
.,.. [9 + 12cos8 + 4 · t(1 .+ cos 28)] dB
= ~
_(_l,_'TT--:)H-=--
-
--'-:..._:..1----+
{2-rr
lo (11 + 12 cos 8 + 2 cos 28) d8 = Hne + 12 sin O+ sin 28) ~71'
= H227r) = 111r
13. A =
r 71' ~r2 d8= r rr t{2 + sin4fJ) 2 dfJ = ~ r"(4+4si~4fJ+sin2 4fJ)d8
Jo
lo
· Jo
·
3
~ fo " I4 + 4sin48 + ~(1- cos88)] dB
2
=
= ~
2"
1
0
(~ + 4 sin48 - ~ cos88) dB=~ [~B- cos4fJ- fs- sin8B] ~"'
.
= ~[(97r- 1) - (- 1)) = ~7('
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0
21
22
D
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
1 .
3
= 21 ( 23 9 + 2o
sm 109] o21r = 21 ( 3'1T) = 27T
-1.4
17. The curve passes through the pole when r = 0
==?-
4 cos 30 = 0
==?-
cos 39 = 0
39 = ~ + 1rn
==?-
=?-
9 = %+·in. The part of the shaded loop above the polar axis is traced out for
9 = 0 to 9 = 7T /6,' so we'll use -7T / 6 and 7T /6 as our limits of integration.
19. r = 0
sin49 = 0
==?-
==?-
48 = 1rn ==?-
r
9 = ~n.
= 4 cos 30
r=sin 40
r'4 Hsin48) dfJ. = t Jor'4sin 48d8 = t Jor '4t(1-cos89)d8
2
2
A = Jo
0
.
.
0
0
· 89 ]"'I 4
.1
= 41 ( 9 - B1 SlD
0
= 41 (,.)
4 = Tii7T
This is a lima~on, with inner loop traced
21.
r = 1 + 2 sin 8 (rect.)
out between 8 = 7; and 1 ~" [found by
3
solving r = 0].
17T ------·
0=6
A = 2[
31r/2
··--••,
ll l'l'
0= 6
.
[37f/2
[ 37r/2
!(1 + 2sin8?
d8 =
(1 + 4sin8 + 4sin2 8) d8 =
[1 +4sin8 + 4· t{1- cos29)] d9
71r /6
77r /6
= [9 - 4cos0 + 20- ~n28J~:~~ = (
23. 2 cos B = 1
.
=> cos 8 = ~
71r /6
9
;)-
C; + 2 v'3 - 4) =
7T -
.
.
¥
=> 8 = :g. or 5; .
A = 2 f 0,. 13 ~[(2 cos8) 2 - 12 ) dB= J0" 13 (4cos2 8
- l) d8
= J;; a { 4 [~ (1 + cos 28)] - 1} d8 = J; /S ( 1 + 2 cos 28) d8
=
[8+ sin 28)~ 13 = i + 4
r = 2cos0
@) 2012 Cengogc Lcaming. All Rights Reserved. Moy not be oamned, copied, or duplicated, or posted to a publicly accessible website, in whole or in prut.
SECTION 10.4
AREAS AND LENGTHS IN POLAR COORDINATES
0
25. To find the area inside the l~miniscate r 2 = 8 cos 29 and outside the circle r = 2,
r=2
we first note that the two curves intersect when r 2 = 8 cos 29 and ~ = 2,
.
I
that is, when cos2B = ~-For -1r < B ::; 1r, cos2B = ~
or ±511' /3
{::} 2B = ±7r/ 3
{::} B = ±1r/ 6 or ±57r/ 6. The figure shows that the desired area is
4 times the area between the curves from 0 to 1r/6. Thus,
A = 4 J0" /. 6 [~(8 cos2B)- ~(2) 2 ] dB = 8 J0" 16 (2 cos 2B - 1) dB
]
. =8 [ sin2B - B
rr~
0
.
.=8(VS/2-7r/6) =4\1'3-411'/3
27. 3cosB=1+cosB {::} cosB=~
=> B=t or-i.
8 =1!.
I
3
,l'
A= 2 f 0" 13 ~ [(3 cos 8?- (1 +cos 0) 2 ) dB
= J0"' 13 (8 cos2 8-'- 2 cos B- 1) dB = f 0" 13 [4(1 +cos 2B) - 2 cos B - 1) dB
= J0.,.13 (3 + 4cos28- 2cosB) dB= [3B + 2sin2B- 2sinf!] ~ 1 3
= 1T +v'3-.J3=1T
29. v'3cosB = sinB
.
=> J3 = sinBB => tanB = J3 => B = i·
cos
2
1'-"' 12 1 (J3 cos B) d()
A -Jo
- f "'13 12 (sin B) 2 dB + hr/
3 .2
3
2
-- Jo
f"' / 1.
1(1
- cos28) dB+ f rr/3
"/ 12 · 3 · 1(1
+ cos2B) dB
2
2
2
12
-- 1
· 28]"o13 +!!4 [B +.!2 sm
· 28]1f7f/3
4 [B - 1
2 sm
=
t [(i - 4) - o] + ~ [ (~ + o)- ( f + 4)]
_
-
.l!.. _
12
31. sin 2B = cos 2B
B=. f
r = J3cos8
+. .!!:8 _ 1fl
_ 5rr _ fl
16
16 - 24
4
_il
=>
2
sin B = 1
cos2B
=> tan 28 = 1 => 28 = ~ =>
=>
A= 8 · 2 f0"/8'~ sin 2 2B dB= 8 f 0"'18 H1- cos 48) dB
= 4[9- t sin48]~
18
= 4(f -
i ·1) = ~ - 1
33. sin2B = cos2B => tan2B = 1 =>
2B = ~
A= 4 f 0"' 18 ~sin 2B dB [since r 2 = sin 2B)
= f 0"'
18
2 sin2B dB= [-;- cos 28] ;
18
=> (} = f
....···B=i
= - ~ v'2- ( - 1) = 1- ~ Y2.
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23
24 • 0
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
35. The darker shaded region (from 0 = 0 to 0 = 27T / 3) represents ~ of the desired area plus ~ of the area of the inner loop.
From this area, we' ll subtract~ of the area of the inner loop (the lighter shaded region from 0 = 27r /3 to 0 = 7r), and then
double that difference to obtain the desired ·area.
A = 2 [J;.,.
= £0 .,.
2
13
.
13
H~ +~os0) 2 dO- g.,.13 H~ +cos 0) 2 dB]
(~+cosO+ cos2 0) dB- f 2: 13 (~ + cos 0"+ cos2 0) dB
.
= J0211'/S [~+cos 0 + ~(1 + cos28)) dO
- I2:/3 a + cosO+ ~(1 + cos28)) d(J
= [~ + sin 8 + ~ + sin 20 ] "
2
4
2
4
13
0
_ [~ + sin 8 + ~ + sin 20
4
2
4
1.,.
211'/ 3
- (1!: + .il
2 + 1!:
3 - .:.8::2) - (!!:
4 + !!:)
2 + (1!:
6 + .il
2 + 1!:
3 - .:.8::2)
-6
= ~ + ~ J3 = H 7T+3v'3)
37. The pole is a point of intersection.
1 + sin 8 = 3 sin 0
=> 1 = 2 sin (J =>
sin 0 = ~
=>
f,
5
6").
(J =~ or ; .
5
The other two ~oints of intersec~ion ~e ( ~, i) and (
r=l + sinO
39. 2 sin 20 = 1
=?
sin 20 = ~
=?
o
&.,. 13.,. or 11"
2 = 1!:
6' 6 ' 6 '
6 °
By symmetry, the eight points of intersection are given by
r = 2sin20
11' 5,. 13rr
1771"
(1, 8) , where (} -_ 12
, an d
, 12 , 12 , and 12
( - 1, 8), where 0 =
i;, 1t;, 1: 2" ; and 2[ ; .
[There are many ways to describe these points.]
41. T he pole is a point of intersection. sin (J = sin 20 = 2 sin(} cos 0 <=>
sinO (1 - 2 cos 9) = 0
8 = 0, 7T,
t. or - t
and ( ~, ; )
2
<=> sinO= 0 or coslJ =
t =>
=> the other intersectio~ points"are ( ~. t )
[by symmen:'].
r =sin20
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SECTION 10.4
. 3.4
43.
3
AREAS AND LENGTHS IN POLAR COORDINATES
0
25
y = 2x
r= 2(}
-3
r = 1 +sinO
-1.4
1.4
-3
- 0.3
From th~ first graph, we see that the pole .is one point of intersection. By zooming in or usin~ the cursor, we find the 0-values
of the intersection points to be a:~ 0.88786 ~ 0.89 and
:r - a ~ 2.25. (The first of these values may be more easily
estimated by plotting y = 1 +sin x andy = 2x in rectangular coordinates; see the second graph.) By symmetry, the total
area contained is twice the area contained in the first quadrant; that is,
A= 21o. ~(20) 2 dO+
=
45. L =
2[1r/Z ~(1 +sin 0.) d8 = 1"' 40 .d8 + l1r/~ [1 + 2sinU + H1- cos 2B)] d8
2
2
12
(~0 3 ] ~ + [8 - 2cosB + (~0 - t sin2B)J:
= ~a + [(~+~)-(a- 2cosa + ~a: - t sin2o:)) ~ 3.4645
3
1b
Jr2+(d7·jd0) 2 dB= fo7r }(2cos0)2 +(-2sinB.)2d8
= 17r
V
4(cos2 B + sin2 B) dB =
1"' ..f4d8 =
[2B]; = 21r
As a check, note that the curve is a circle of radius 1, so its circumference is 27r(1 ) = 21r.
=
1 Je
2
"'
2
(02 + 4)d0 =
1
2
1f
BV02 +4d0
Now let u = 02 + 4, so that du = 20d8 [Ode=
t du] and
49. The curve r = cos 4 (0 / 4) is completely traced with 0 :::; () :::; 47r.
2
r 2 + (drfd8) 2 = [cos4 (B/4)] + [4cos3 (B/ 4) · (-sin(B/4)) · t ]
2
= cos8.(B/4) + cos6 (8/4) sin2 (B/ 4)
= cos6 (8/4)[cos2 (B/4) +sin2 (0/4)] = cos6 (B/4)
L = f 0 r Jcos6 (0/4) dO = J0 "' jcos3 .(0/4) j rj,B .
41
4
= 2 f~" cos8 (B/4)dB [sincecos3 (B/4) ~ 0 for O :S B $ 27r]
= 8 J;12(1 - sin 2 u) cosudu = 8 f 01 (1- x 2 ) dx
= 8J;12 cos3 udu
[u = tB]
x = sinu, ·]
[ dx = cosudu
= 8[x- ~x3) ~ = 8(1- ~) = lf
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26
0
CHAPTER 111 PARAMETRIC EQUATIONS AND POLAR COORDINATES
51. One loop of the curve r = cos 28 is traced with - 1rI 4 ~ 8 ~ 1r I 4.
2
r2 +
(~~) = cos 28 + (- 2 sin 28) = cos 28 + 4sin 28 = 1 +3 sin 28 ::::} L /"' Vl + 3 sin 28 d8:::::: 2.4221.
2
2
2
2
14
2
2
-?r /4
53. The curver = sin(6sin9) is completely traced with 0 ~ 8 ~ 1r.
r2 :
(~;y =s~n2 (6sin8)+36cos2 8cos2 (6sin0) ::::} L
r = sin{6sin9)
L"
::::}
-
~ = cos{6sin9)_- 6cos8, so
Jsin2 (6sin8)+36cos2 8pos2(6sinO)d8:::::8.0091.
55. (a) From (10.2.6),
s =I: 21ryJ(dxld0)2 + (dyld9) 2 dO
[from the derivation of Equation 10.4.5]
(b) The curve r 2 = cos 28 goes through the pole when cos 28 = 0
20 = ~
::::}
::::}
(o. f)
8 = ~. We'll rotate the curve from () = 0 to 8 = ~ and double
? = cos28
(1.0)
this value to obtain the total surface area generated.
r 2 = cos28 ::::}
· dr
2r dO= -2sin28 ::::}
( dr )
dO
2
2
2
sin 28
sin 28
-- ~ - -cos20'
10.5 Conic Sections
1. x 2 = 6y and x 2 = 4py
::::}
4p = 6
::::}
p = ~­
3. 2x = - y 2
::::}
y 2 = -2x. 4p = -2
::::} p = - ~.
'\
The vertex is (0, 0), the focus is (0, ~),and the directrix
The vertex is (0, O),the focus is ( - ~, 0), and the
isy = -~.
directrix is x = ~.
y
x=4
6 X
X
y= -~
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SECTION 10.5
5. (x + 2) 2 = 8 (y- 3). 4p = 8, sop= 2. The vertex is
7. y 2 + 2y + 12x + 25 = 0
0
27
=l>
y2 + 2y + 1 = - 12x- 24
( - 2, 3), the focus is ( -2, 5), and the directrix is y = 1.
CONIC SECTIONS
=;..
(y + 1)2 = - 12(x + 2). 4p = - 12, sop= -3.
y
The vertex is ( - 2, -1), the focus is (-5, - 1), and the
directrix is x = 1.
-------------------------- __ ,y_~_!_
X
0
:
~
X
Ijx=l
9. The equation has the form y 2 = 4px, where p < 0. Since the parabola passes through (- 1, 1), we have 12 = 4p( - 1), so
4p = - 1 and an equation is y 2 = - x or x = - y 2 • 4p = - 1, sop =
-i and the focus is ( -;i,0) while the directrix
.
1
ISX = 4·
y
2
ellipse is centered at {0, 0), with vertices at (0, ±2). The foci are (0, ±v'2).
-..fi
-2
13. x
2
+ 9y2 = 9
<=>
x2
Y2
9 +l
= 1
=l>
a = v'9 = 3,
15. 9x 2 - 18x + 4y2 = 27 <=>
9(x 2 - 2x + 1) + 4y 2 = 27 + 9
b = Vi.= 1, c = ~ = v'9=I = J8 = 2v'2.
<=>
The ellipse is centered at {0, 0), with vertices (± 3, 0).
9(x- 1) 2 + 4y2 = 36 <=>
(x- 1)2
y2
4
+ g- = 1
The foci are (±2v'2, 0).
a = 3, b = 2, c = J5 =:-
center (1, 0),
y
vertices (1 , ± 3}, foci (1, ±J5)
I
-3
0
3
y
X
(1, 3)
- I
3 X
(1,- 3)
2
17. The center is (0, 0), a= 3, and b = 2, so an equation is ~
2
+ Y9 = 1. c = ~ = J5, so the foci are (0, =;I=J5).
© 2012 Ceng3ge LC3JT1ing. All Rights Reser\'ed. Muy not be scuMed. copied, or duplicated, or pos1cd to a publicly access ible wc:bsite, in whole or in pan.
=}
28
CHAPTER 10 PARAMETRIC EQUATIONS AND POLARCOORDINATES
0
yz
19.
xz
9 = 1 ::::? a = 5, b = 3, c = J25 + 9 = J34 ::::?
-
25
center (0, 0), vertices (0, ±5), foci (0, ±J34), asymptotes y = ±~x.
Note: It is helpful to draw a 2a-by-2b rectangle whose center is the center of
the hyperbola. The asymptotes are the extended diagonals of the rectang le.
21 . x 2 - y 2 = 100
x2
<=>
100
y2
-
100
=1
::::?
y
a = b = 10,
c = JlOO + 100 = 10J2 =>
center (0, 0), vertices (±10, 0),
foci (±10 J2, 0), asymptotes y
= .±f§x = ±x
X
,.._--!-----",,
,~_;.......
23. 4x2 - y 2 - 24x - 4y + 28 = 0
4(x
2
-
6x + 9) - (y
2
<=>
+ 4y + 4) = -28 + 36- 4 <=>
4(x '_ 3)2 - (y + 2)2 = 4
<=>
(x - 3)2 - (y + 2)2 = 1 =>
1
4
a = v'f = 1, b = v'4 = 2, c = Vf+4 = J5
::::?
center (3, - 2), vertices (4, - 2) and (2, -2), foci (3 ± J5, -2),
asymptotes y + 2 = ±2(x- 3).
25. x 2 = y + 1 <=> x 2 = 1(y + 1). This is an equation of a parabola with 4p = 1, so p =
i. The vertex is (0, -1) and the
focus is (0, - ~).
27. x 2 = 4y- 2y2
<=> x 2 + 2y2 - 4y = 0 <=> x 2 + 2(y2 - 2y + 1) = 2 <=> x 2 + 2(y - 1) 2 = 2 <=>
x; + (y ~ 1) =
2
l.. This is an equation of an ellipse with vertices
29. y2 + iy = 4x2 + 3
<=> y 2 + 2y + 1 = 4x 2 + 4 <=>
of a hyperbola with vertices (0, - 1 ± 2)
a~ (±J2, 1) . The foci ~eat (±.,J2=1, 1) = (±1, 1) .
(y + 1? - 4x 2 = 4 <=>
1 2
(y: )
-
x 2 = 1. This is an equation
= {0, 1) and (0, -3). The foci are at (0, - 1 ± J4+T) = (0, - 1 ± J5).
31. The parabola wi~ vertex {0, 0) and focus (1 , 0) ?pens to the right and hasp= 1, so its equation is y 2 = 4px, or y 2 = 4x. '
33. The distance from. U1e focus ( -4, 0) to the directrix x = 2,is 2 - ( - 4) = 6, so the distance from the focus to the vertex is
~(6) = 3 and the vertex is ( - 1, 0). Since the focus is to the left of the vertex, p = - 3. An equation is y 2 .= 4p(x + 1)
y
2
= -12(x + 1).
© 2012 CcnGogc: Lcmning. Al l Rjglns Rest.! rvcd. Mny not he scanned. copied, or duplicutccl, or posted to a publicly occcssiblc website. in whole or in p..1rt.
=>
SECTION 10.5
CONIC SECTIONS
0
29
35. A parabola with vertical axis and vertex (2, 3) has equation y- 3 = a(x- 2?. Since it passes through (I; 5), we have
5-3=a(I-2) 2
'* a =2,soanequationisy -3 =2(x-2? .
37. The ellipse with foci (±2, 0) and vertices (± 5, 0) has center (0, 0) and a horizontal major axis, with a = 5 and c = 2,
so b2 = a2 - c 2 = 25- 4 = 21. An equation is
x2
12
+ ; 1 = 1.
25
39. Since the vertices are (0, 0) and (0, 8), the ellipse has center (0, 4) with a vertical axis and a = 4. The foci at (0, 2) and (0, 6)
are 2 units from the center, soc= 2 and b = .,Ja2 - c2 = .,j4 2 - 2 2 = v'f2. An equation is (x - 0)
b2
_41. An equation of an ellipse with center (-1,4) and vertex (- 1,0) is (x
from the center, soc= 2. Thus, b2 + 22 = 4 2
!
2
a
2
2
'* b . = 12, and the equation is (x +121? + (y -164 ) = 1.
2
2
2
2
+ (y-2 4 ) = 1 '*
12
4 2
) + (y ~ ) = 1. The focus (-1,6) is 2 units
2
'* c = 5 and 3 + b =5 '*
43. An equation of a hyperbola with vertices (±3, 0) is ~2 ....:. ; 2 = 1. Foci (±5, 0)
.
2
2
2
2
2
2
b = 25 - 9 = 16, so the equation is ~ - r6 = 1.
45. The center of a hyperbola with vertices ( -3, - 4) and ( -3, 6) is ( -3, 1), so a = 5 and a.n equation is
.
(x + 3) 2
·
·
b2
= l.Focl( - 3, - 7)and(- 3,9)-* c =8,so5 2 +b2 =82
'* b2 =64-25=39andthe
( - 1} 2
{x + 3) 2
equation is y
= I
25
39
.
.
.
2
Asymptotes y = ±2x
b
.
2
•
2
2 t
47. The center of a hyperbola with vertices (±3, 0) is (0, 0), so a = 3 and an equation is ~
-
2
= 1.
2
'* -a = 2 '* b = 2(3) = 6 and the equation is ~9 - JL
= 1.
36
49. In Figure 8, we see that the point on the ellipse closest to a focus is the closer vertex (which is a distance
a - c from it) while the farthest point is the other vertex (at a distance of a+ c). So for this lunar orbit,
(a- c)+ (a+ c) = 2a = (1728 + 110) + (1728 + 314), or a = 1940; and (a+ c)- .(a - c) = 2c = 314 - 110,
or c = 102. Thus, b2 = a
2
-
x2
y2
2
c = 3,753,196, and the equation is ,
+
=I.
,
,
,
3 763 600
3 753 196
51. (a) Set up the coordinate system so that A is ( -200, 0) and B is (200, 0) .
IPAI-\PBI = (1200)(980) = 1,176,000 ft =
b2 -- c2 - a 2 -- 3,339,375
121
121x 2
1,500,625
2
1~ mi = 2a
-*
a= 1 ~ 5 , and c ~ 200 so
2
121y
- 1
3,339,375 - .
© 20 12 Ccngoge Leoming. All Rights Rescn'Cil. May not be scanned, copied. or duplicated. or posted to a publi<:ly uccusible website, in whole or in part.
30
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
0
(121}(200) 2
1,500,625
=> x = 200 =>
(b) Due north of B
2
121y
1
3,339,375 =
=>
y = 133,575 ~ 248 mi
539
53. The function whose graph is the upper branch of this hyperbola is concave upward. The function is
2
2
-
- - 16
55. (a) If k > 16, then k - 16 > 0, and xk + -yk = 1 is an ellipse since it is the sum of two squares on the left side.
2
2
(b) IfO < k < 16, then k - 16 < 0, and xk + k ~
16
= 1 is a hypel"bola since it is the difference of two squares on the
left side.
(c) If k < 0, then k - 16 < 0, and there is no cw11e since the left side is the sum of two negative tenns, which cannot equal I .
2
(d) In case (a), a 2 = k, b2 = k- 16, and c 2 = a2 - b2 = 16, so the foci are at (±4, 0}. Jn case (b), k - 16 < 0, so a = k,
b2 = 16 - k , and c2 = a2 + b2 = 16, and so again the foci are at (±4, 0}.
57. x 2 = 4ptJ =>
2x = 4py'
y
=> y' = ; , so the tangent line at (x0 , y 0 ) is
.
p
2
y- :; = ~; (x - xo). This line passes through the point (a, - p) on the
.
. so - p - x6p = xop (a - xo ) => - 4p 2 - x 2 = 2axo - 2x 2
d1rectnx,
0
0
4
2
x5 - 2axo - 4p2 = 0 <=> x5 - 2axo + a 2 = a2 + 4p2 <=>
<=>
(a,-p)
y= - p
(xo- a) 2 = a 2 + 4p2 <=> x 0 =a± Ja2 + 4p2 • The slopes of the tangent lines at x = a± Ja2 + 4p2
are
a± ..ja2 + 4p2
2
p
, so the product of the two s lopes ts
a+ Ja2 + 4p2 a - Ja2 + 4p2
-----'----'--.
2p
2p
=
a2 _ . (a2 + 4p2)
-4p2.
= - - = -1
4p2
4p2
'
showing that the tangent lines are perpendicular.
59. 9x2 + 4y 2 = 36
2
<=>
2
~ + Y = 1. We use the parametrization x = 2 cost, y = 3 s in t, 0 ~ 't ~ 21r. ·The circumference
9
is given by
2
L =I:" J(dx/dt) 2 + (dyjdt)2 dt = ]~ " J( - 2 s in t) 2 + (3 cost)2 dt =I:" )4sin2 t + 9cos2 t dt
= I:" v4 + 5 cos2 t dt
Now use Simpson's Rule with n = 8, 11t =
2
1T-
8.
0
= ~.and
f(t) = v'4 + 5cos2 t to get
4
L ~ ss = ~ [J(o) + 4f(*) + 2f(~) + 4Jen + 2J(1r) + 4/( 5; ) + 21en + 4tCn + J(21T)J ~ 15.9.
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9
SECTION 10.5
x2
y2
61. ;_2 - - 2 = 1
a
b
A= 2
la a
=?
y2
a:2 _ a2
- 2 = -2
b
a
CONIC SECTIONS
0
31
b
y = ±- Jx2- a2.
a
=?
r !!_ J x 2 ·_ a dx ~ 2ab [~2 J x2 - a2 - 2 In Jx + J x2 - a2J ] a·
42
2
c
b
= -[cvc2 - a 2 -a2 ln jc + vc2 - a 2 j + a 2 ln!al)
a
Since a 2 + b2 = c2 , c2 - a 2 = b2 , and J c2 - a 2 = b.
=!!.a [cb- a 2 1n(c + b)+ a 2 !na) = !!_a [ch + a2 (!na - !n(b +c)))
= b2 c/a + abln[a/( b +c)), where c2 = a 2 + b2 .
x2
63. 9x2 + 4y2 = 36 <::>
Y2
'4 + g = 1
top half of the ellipse is
H1rab) = 37T. Solve 9x2 + 4y2 = 36 for y to get an equation for the top half of the ell ipse:
9x2 +4y 2 =36 <::> 4y 2 =36-9x 2
y = -1
A
1b
a
= - 3 · '2
87T
-1 [f(x))2 dx = 12
~
y 2 = ~(4-x 2 )
1
2
2
-1 ( -3 V 4- x 2 )
37T - 2 2 2
2
1
a= 3, b = 2. By symmetry, x = 0. By Example 2 in Section 7.3, the area o_f the
=?
2
( 4 - x ) dx = -3 [4x - -1 x
0
47T
3
dx= - 3
87T
y = ~v'4-x 2 .Now
=?
12
(4 -x 2 )dx
y
- 2
2
3] = -3 ( -16) = -4
0
47T
3
7T
so the centroid is (0, 4/ rr).
2
2
. . .tmp I'tctt
. Iy, •x
65. Dt' ffierenttatmg
a 2 + yb2 = 1 =?
line at Pis -
[y =f. 0]. Thus, the slope ofthe tangent
b:x1 . The slope of F1P is ___)f];_
and of F2P is ____lt!__ By the formula from Problems Plus, we have
+
a Y1
X1
c
X1 -
c
2
Yl
b Xt
-Xt_+_c + -a 2_y_1 _ a 2 + b2 x 1(x1 +c) _ a 2 b2 + b2 cx1
tan a = _.:........;.---:-:.-----'-'-=-1_
b2 x1Y1
- a 2 y1 (x1 + c) - b2X tY1 - C2 XtY1 + a 2CIJ1
2
a yt (Xt +c)
yt
b2 (cxt+a2 ) · b2
= CIJt(CXt + a 2 ) = cy1
and
2
2 2
y'f. - b2 x1 ( x1 -c)
- a b + b cx1
b2 (ext - a 2)
b2
= a 2y1 (x1- c) - b2X 1Y 1 = c2x1y1 - a 2C1Jt = CIJ1 (cx1 - a 2) = CIJ1
- a
2
Thus, a = {3.
© 2012 Cc:ngoge Lco.ming. All Rights Reserved: Muy not be scanned, copied, or duplicaled, or posted to a publicly ncccs.•db lc wcbsiu~. in whole or in part.
32
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
0
10.6 Conic Sections in Polar Coordinates
1. The.directrix x = 4 is to the right of the focus at the origin, so we use the form with "+ e cos(}" in the denominator.
ed
l2 . 4
4
(See Theorem 6 and Figure 2.) An equation is r =
(} =
(} =
(}.
1
2 +cos
'
1+ ecos
1+ 2 cos
3. The directrix y = 2 is above the focus at the origin, so Wf! use the form w ith "+ e sin(}" in the d enominator. An equation is
ed
_
1.5(2)
_
6
1 + e sin () - 1 + 1.5 sin(} - 2 + 3 sin(}·
_
r -
5. The vertex (4, 3Jr/2) is 4 units below the focus at the origin, so the directrix is 8 units below the focus ( d = 8), and we
use the form with "- e sin(}" in the denominator. e = 1 for a parabola, so an equation is
ed
1(8)
8
r = 1 - e sin () = 1 - 1 sin (} = 1 - sin (} ·
7. The directrix r = 4 sec (} (equ ivalent to r cos(} = 4 or x· = 4) is to the right of the focus at the origin, so we w ill use the form
with"+ e cos()" in the denominator. The distance from the focus to the directrix is d = 4, so an equation is
~{4)
ed
2
4
r = 1 + ecosB = 1 + ~cos(}· 2 = 2 +cos((
4
9. r =
5 - 4sinB
5
4 5
where e = .1 and ed = .1
=
/
1/5 . 1 - ~sinB'
5
&
. l/
=> d = 1.
y
(4 , 7r/2)
(a) Eccentricity = e = ~
(b) Since e = ~ < 1, the conic is an ellipse.
(c) Since"- e sin (}" appears in the denominator, the directrix is below the focus
at the origin, d = IFll = 1, so an equat'ion of the directrix is y = - 1.
F2
(d) The vertices are (4, ¥) and(~ , 3; ) .
11. r =
2
1/3
2/ 3
X
-------1
---------·
y=-1
( 4 3'1T)
'2
3 + 3 sinB · . 1 /~ = l + 1 sinB ' wheree=1and ed= 3 => d= ~.
y
y = 2/ 3
(a) Eccentricity= e = 1
(b) Since e = .1, Lhe conic is a parabola.
(c) Since"+ e sin(}" appears in the denom inator, the directrix is above the focus
at the origin. d = IFll =
j , so an equation o flhe directrix is y = i·
(d) T~e vertex is at ( ~, ¥), midway between .the focus and directrix.
@) 2012 CcngJige Leon~ing. All Rights Rc.c:n-..1. May not be scanned. copied, or duplieoted. or posted to a publicly accessible website, in whole or in P"fL
SECTION 10.6
13. r =
:
6+
3 2
1 6
· 1 =
(
, where e = ~ and e d = ~
cos 9 116
1+ 3 cos0
(a) Eccentricity= e =
:::}
CONIC SECTIONS IN POLAR COORDINATES
d = ~-
t
t < 1, the conic is an ellipse.
(c) Since " + e cos e" appears in the denominator, the directrix is to the right of
(b) Since e =
the focus at the origin. d = !Fll = ~.so an equation of the directrix is
I
9:
x=21
:
(f,1T)
!
x = ~(d) The vertices are ( ~, 0) and (
that is, (
t, 1r), so the center is midway between them~
fn, 1r) .
3
1/ 4
3/ 4
3
15. r = 4 - 8 cos 9 . 1I 4 = 1 - 2 cos 9' where e = 2 and ed = 4
d -- 1s ·
(a) Eccentricity = e = 2
(b) Since e = 2 > 1, the conic is a hyperbola.
(c) Since "- e cos 0" appears in the denominator, the directrix is to the left of
the focus at the origin. d'= IFll = ~.so an equation of the directrix is
3
.X -- - s·
(d) The vertices are ( - ~. 0) and (i, 1r), so the center is midway between them,
that is, (~ , 1r).
17. (a) r =
1-
; .
sm 9
, where e = 2 and ed = 1 :::} d = 12 . The eccentricity
e = 2 > 1, so the conic is a hyperbola. Since" -e sin B" appears in the
denominator, the directrix is below the focus at the origin. d = IFll = ~.
so an equation of the directrix is y = - ~ . The vertices are ( - 1, ~) and
1 3
.s, ( 32 , 23 " ) .
( 3 , 2'~~" ) , sothe center .IS m1"dway between th l'!m, t hat 1
-3
(b) By the discussion that precedes Example 4, the equation
.
1
IS T =
0
1 - 2sin ( e- 3; ) •
-2
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33
34
0
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
19. Fore < 1 the curve is an ellipse. It is nearly circular when e is close to 0. As e
increases, the graph is stretched out to the right, and grows larger (that is, its
right-hand focus moves to the right while its left-hand focus remains at the
origin.) At e = 1, the curve becomes a parabola with focus at the origin.
21.I PFI= e iPLI
=> r =e[d-rcos(7r-9)J = e(d+rcos8)
=*
ed
=> r=---....,.
r(1 - ecos~) = ed
1- ecosO
X
x=-d
23. IPFI = ei P ll
y
=> r = e[d- rsin(9 -1r)] = e(d+r sinO) =>
ed
=> r = ---""7
r(1 - esin 9) = ed
.. 1- esin9
X
l
•
y= - d
25. We are given e = 0.093 and a = 2.28 x 10 8 . By (7), we have
r=
a(1 - e 2 )
2.28 x 108 (1- (0.093) 2 ] . 2.26 x 108
=
~ -::-----::-=::------::
1 + ecos 8
1 + 0.093 cosO
1 + 0.093cos8
=> a = 18.09 AU and e = 0.97. By (7), the equation of the orbit is
27. Here 2a = length of major axis = 36.18 AU
2
r = 18.09(1 - (0.97) ] ~
1 + 0.97 cos 9
1.07
1 + 0.97 cos 8
·
d"1stance from the comet to t he sun 1s
·
. By (&), the maxtmum
18.09(1 + 0.97) ~ 35.64 AU or about 3.314 billion miles.
29. The minimum distance is at perihelion, where 4.6 x 107 = r = a(1 - e) = a(1 - 0.206) = a(0.794}
=>
a = 4.6 x 107/ 0.794. So the maximum distance, which is at aphelion, is
r = a(1 + e) = (4.6 x 107/ 0.794) (1.206} ~ 7.0 x 107 km.
31. From Exercise 29, we have e =
.
0.206 and a(l- e)= 4.6 x 10 7 km. Thus, a= 4.6 x 10 7/ 0.794. From (7), we can write the
1- e
2
.
equation of Mercury's orbit as r =a
1 + ecos 8
•
. So smce
2
dr
a(1- e )esin8
-=
d9
(1 + ecos8) 2
=>
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CHAPTER 10 REVIEW
0
35
the length of the orbit is
1
2
L=
,. Jr 2
+ (dr/d0)2 d(J = a(l- e2 )
1
2
,.
.Jl(t e + ~)~sO dO:::::: 3.6 x 108 km
0
0
2
2
+ecos
This seems reasonable, since ,Mercury's orbit is nearly circular, and the circumference of a circle of radius a
is 21ra :::::: 3.6 x 108 km.·
10
Review
CONCEPT CHECK
1. (a) A parametric curve is a set of points of the form (x, y) = (J(t), g(t)), where f and g are continuous funct ions of a
variable t.
(b) Sketching a parametric curve, like sketching the.graph of a function, is difficult to do in general. We can plot points on the
curve by finding f(t) and g(t) for various values oft, either by hand or with a calculator or computer. Sometimes, when
f and g are given by formulas, we can eliminate t from the equations x = j(t) andy = g(t) to get a Cartesian equation
relating x andy. It may be easier to graph that equation than to work with the original formulas for x andy in terms oft.
2. (a) You can find :
as a function oft by
calculating~~ = ~~J~~ [if dxjdt f: 0].
(b) Calculate the area as I: y dx = I: g(t) j'(t)dt [or I; g(t) !' (t)dt ifthe leftmost point is (!({3), g({J)) ~ther
than (f(a),g(a))].
+[.g'(t))2 dt
3. (a) L = I: J(dxjdt)2 + (dy j dt)2 dt =I: vlf'(t))2
(b)
s =:: I: 27ryJ(dxfdt)2 + (dyfdt) 2 dt =I: 21rg(t) v lf'(t)F + [g'(t)J2 dt .
4. (a) See Figure 5 in Section 10.3.
(b) x = rcosO, y = r sinO
(c) To find a polar representation (r, 0) with r ;?. 0 and 0
s 0 < 27r, firs t calculate r =·Jx2 + y2. TI1en 0 is specified by
cosO= xjr and sinO= yjr.
d
5. (a) Calculate
!:.._ (y)
dy
Jx = ~ = ~
dO
dO (x)
!:.._ (r ::;in 8)
=
. : : l J:;:-- -
d() (rcosB)
(~) sin(} + r cos(}
dr)
(dO
. ,
wherer = f(O).
cos () - r sm (}
2
(b) calculate A= I: !r dO= I: tlfCOW d8
(c) L =I: J(dx/d8.)2 + (dy/d0) 2 dO= I: Jr 2 + (dr/d0) 2 dO=
.r: J[f(0))2 + [!'(0)]2 dO
6. (a) A parabola is a set of points in a plane whose distances from a fixed point F (t11e focus) and a fixed line l (the 'd irectrix)
are equal.
(b) x 2 = 4py; y 2 = 4px
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36
CHAPTER 10
D
PARAMETRIC EQUATIONS AND POLAR COORDINATES
7. (a) An ellipse is a set of points in a plane the sum of whose distances from two fixed points (the foci ) is a constant.
2
2
+ _Y_
= 1.
a2
a2 - c2
(b) X
8. (a) A hyp~rbola is a set of points in a plane the difference of whose distances from two fixed points (the foci) is a constant.
This difference should be interpreted as the larger distance minus the smaller distance.
2
2
(b) ::__ - _ Y_
c2- a2
a2
(c) y
=±
=1
Jc2 _ o,2
a
x
9. (a) Jfa conic section has focus F and corresponding directrix l, then the eccentricity e is the fixed ratio /PFI / /Pl / for points
P of the conic section.
(b) e < 1 for an ellipse; e > 1 for a hyperbola; e = 1 for a parabola.
(c) x = d: r =
ed
. ed
·
ed
ed
. x = - d:.r =
. y = d: 1' =
. . y = - d: r = -:----:--::
1 + e cos (}
1 - e cos (}
1 + e sm 8
1 - e sin 8 ·
TRUE-FALSEQUIZ
1. False.
Consider the curve defined by x = f(t) = (t - 1) 3 and y= g(t) = (t- 1) 2 : Then g'(t) = 2(t - 1), so g'(1) = 0,
but its graph bas a vertical tangent when t = 1.
3. False.
Note: The statement is true iff' (1) =I= 0 when l (1) = 0.
For example, iff (t) = cost and g( t) = sin t for 0 ::; t :S 4rr, then the curve is a circle of radius I, hence its length
is 2rr, but f 0 " J [!'(t)]2 + [g'(t))2 dt = .{0 " J(- sin t) 2 +(cos t) 2 dt = f0 " 1 dt = 4rr, since as t increases
4
4
4
from 0 to 4rr, the circle is traversed twice.
. 5. True.
The curve r = 1 - sin 28 is unchanged if we rotate it through 180° about 0 because
1 - sin 2(8 + rr) = 1 - sin(28 + 2rr) = 1 - sin 28. So it's unchanged if we replacer by -r. (See the discussion
after Example 8 in Section 10.3.) In other words, it's the same curve as r = -(1 -sin 20) = sin 28 - 1.
7. False.
The first pair of equations gives the portion of the parabola y = x 2 with x ;::: 0, whereas the second pair of equations
traces out the whole parabola y = x2 •
9. True.
By rotating and translating ~he parabola, we can assume it has an equation of the form y = cx2 , where c > 0.
Tile tangent at the point (a, ca2 ) is the line y - ca2 = 2ca(x - a); i.e., y = 2cax - ca2 • This tangent meets
the parabola at the points (x, cx2 ) where cx 2 = 2cax - ca2 • This equation is equivalent to x2 = 2a~ - a 2
[since c > 0]. But x 2 = 2ax - a 2
2
<=> x 2 - 2ax + a 2 = 0 <=>
(x- a) 2 = 0 <=>
x =a <=>
2
(x, cx ) = (a, ca ). This shows that eacb tangent meets the parabola at exactly one point.
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CHAPTER 10 REVIEW
0
37
EXERCISES
1. X = t 2 + 4t, y = 2 - t, - 4 ~ t ~ 1. ~ = 2 - y, SO
X =
y
{2 - y) 2 + 4{2- y) = 4- 4y + y 2 + B - 4y = y 2 - By+ 12 {::}
x + 4 = y 2 - By+ 16 = (y - 4) 2 . This is part of a parabola with vertex
( -4, 4), opening to the right.
X
3. y = sec() = ~
() = .! . Since 0 ~ () ~ 1r /2, 0 < x ~ 1 and y ;::: 1.
COS
y
X
This is part of the hyperbola y = 1/x.
\ ,1,1),0-0
0
5. Three different sets of parametric equations for the curve y = -IX are
(i) X = t, y = Vt
(ii) X = t 4 , y = t 2
(iii) x = tan2 t, y = tan t, 0 ~ t < 1r/2
There are many other sets of equations that also give this curve.
The Cartesian coordinates are x = 4 cos 2; = 4 (- ~) = -2 and
7. (a)
y = 4sin ;,. = 4(
(b) Give~ x = - 3 and y= 3, we haver =
4) = 2 J3, that is, the point ( - 2,2 J3).
J( -3) 2 + 32 = JIB = 3 v'2. Also, tan() = ~ '* tan() = ~3 , and since
( -3, 3) is in the second quadrant,(}= ' 34". Thus, one set of polar coordinates for ( -3, 3) is (3 V2, 3,;'"), and two others are
1 111") and (- 3 '2 7 ,.)
( 3 y '2
~,
4
v. .L.i, 4 .
9. r = 1 - cosO. This cardioid is
symmetric about the polar axis.
r
2
(2, 7T)
7T
27T 0
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38
0
CHAPTER 10
PARAMETRIC EQUATIONS AND POLAR COORDINATES
r
11. r = cos 30. This is a
three-leaved rose. The curve is
traced twice.
13. r = 1 +cos 2e. The curve is
sym·metric about the pole and
axes.
-1
1T
2
15. r =
(2,0)
(2, 1r)
both the horizontal and vertical
3
t + 2sin£1
=}
1T
31T
2
e = 2 > 1, so the conic is a hyperbola. ·de = 3
=>
d = ~ and the form "+2 s in(}" imply that the directrix is above the focus at
the origin and bas equation y = ~-The vertices are (1, ~) and ( - 3, 3; ) .
17. x+y=2
¢:>
rcosll+rsinll = 2
¢:>
r(cos0+sin 0)=2
2 .
¢:>
r= - - -
--
cos(J + s in£1
19. r = (sin £1)/0. As B-. ±oo, r __, 0.
As (} __, 0, r --.. 1. In the first figure,
there are an infinite number of
x -intercepts at x = 1rn, n a nonzero
integer. These correspond to pole
points in the second figure.
-0.25
- 0.75
dy
dx
1
dy
dy / dt
2t
2
2
21. x = lnt, y = 1 + t ; t = 1. dt = 2t and dt = t' so dx = dxjdt = /t = 2t ·
1
When t = 1, (x , y) = (0, 2) and dyjdx = 2.
23. r = e- 0
=> y = rsin£1 = e- 0 s in(} and x = T"cos O = e - 0 cosO =>
dy
dyjd£1
*sin(} +rcosO _ -e- 0 sin(}+ e- 0 cos(} - e 0
sine- cosO
dx = dx j d(} = ~;cos(} - rsin(} - -e- 0 cosB - e- 0 sinO · - e 0 = cos B+sine ·
dy
0 - (- 1)
1
=
= = - 1.
dx
-1 + 0
-1
When (} = 1r, -
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CHAPTER 10 REVIEW
25. x = t + sin t, y = t - cost
d (dy)
dt dx
dxjdt
0
39
=}
dy
dyjdt
1 + sint
dx = d.xjdt = 1 +cos t
~
{1+cost) cost- {1 +sint)(- sint)
2
2
_ __ _ ___:(,_1,...-+.:..__c_os_ -t'-)-2 _ _ _ __ = cost+ cos t +sin t + sin t = 1 + cost + sin t
l +cost
(l+cost)3
(1 + cost)3
3
2 .
27. We graph the curve x = t - 3t, y = t + t + 1 for -2.2 :=::; t :=::; 1.2.
By zooming in or using a cursor, we find that the lowest point is about
(1.4, 0. 75). To find the exact values, we find the t-value at which
dyjdt=2t+1 = 0
t=~~ <=> , (x,y),;,(lj,~).
¢=>
dx = -2asin t + 2asin2t = 2asint(2cost - 1) = 0
dt
29. x ·= 2acost- acos2t
~
sint = 0 or cost = ~
~
t = O,·t, 1r, orr;.
y = 2asin t - a sin 2t
~
dy = 2a cost - 2a cos 2t = 2a(l +cost- 2 cos2 t) = 2a(1 -cos t)(1 + 2 cost) = 0
dt
¢:>
271"
4"
t = 0'3,or3.
Thus the graph has vertical tangents where t =
t. and ~">3" , and horizontal tangents where t =
1r
2
;
and 4; . To determine
what the slope is where t = 0, we use !'Hospital's Ru;e to evaluate lim ddy/jddt = 0, so there is a. horizontal tangent there.
t->0
t
X
0
a
0
y
3"
~a
2rr
3
- ~a
~a
2
7r
- 3a
0
4 7r
-~a
-~a
2
51r
~a
_ :.::]:a
3
t
y
:.::]:a
3
X
2
(-3a,O)
(a,O)
)C
2
31. The curve r 2 = 9 cos 50 has 10 " petals." For instance, for - -fu :=::; 8 :=::; {'0 , there are two petals, one with r > 0 and one
with r < 0.
33. The curves intersect when 4 cos 8
=2 ~
cos 8 = ~
~
8 = ±i
r = 4cos 0
for -1r :=::; 8 :=::; 1r. The points of intersection are (2 , i) a nd (2,- ~).
(f) 2012 Ce:ngoge Learning. All Rights Reserved. M.o.y nol be scanm-d. copiL-d, or duplica1ed, or posted toll publicly accessible: website; in whole or In part.
40
D
CHAPTER 10
PARAMETRIC EQUATIONS AND POLAR COORDINATES
35. The curves intersect where 2 sin () = sin() + cos ()
sin 0 = cos 0
=>
=> 0 = "i, and also at the origin (at which 0 = 34rr
8=f
on the second curve).
A= I 0" 14 ~(2 sin0) 2 dO+ I:;f ~(sinO+ cos 0) 2 dO
4
= Io"
14
(1- cos 20) dO+~ I:;t (1 + sin2fJ) dO
r = 2 sin 8
= [0 ~ ~ sin20]~ 14 + [~0-:- t cos2BJ!~~ = ~(1r -1)
4
'
r = sin 0 + cos 8
L =I~ .J(dxjdt)2 + (dyjdt) 2 dt = 1~ .J(6t)2 + {6t2)2 dt =I: V36t_2 + 36t4 dt = I~ y'36t2v'1 + t2 dt
=I~ 6\t! v1 + t 2 dt = 6 I~ t v'f+t2 dt = 6 I 1 u 1 ; 2 (~du)
5
J:
= 6. ~ · ~ [u312 = 2(53 12 - 1) = 2(5VS -1)
39. L =I:" Jr2 + (drjdB)2 dB = I:" }(1/0)2 + (-1/0 2 )2 dB=
L7
2
"'
dB
~ [- #+! +In (B + J B2 + 1 )] 2,. = #+I - v'471"2 + 1 + In ( 271" + ~ )
()
7r:
271"
11"
71" + v' 71"2 + 1
s = I 14 27ry.J(dxfdt)2 -t. (dyfdt)2 dt = I 14 21r{~t3 + ~c 2 ) J(2/0) 2 + (t2- t-3)2 dt
4
3 + .!c 2) . l(t2 + t - 3) 2 dt = 21r f 4 (.!t 5 + !! + l t- 5) dt - 21r [...!..t6 + l!t - 1c4) 4 - 471 ·295 1r
= 21r J1 (.!t
3
2
v
1 3
6
2
18
6
8
1 1024
43. For all c except - 1, the curve ·is asymptotic to the line x = 1. For
c < -1, the curve bulges to the right near y = 0. As c increases, the
bulge becomes smaller, until at c = - 1 the curve is the straight line x = 1.
As c continues to increase, the curve bulges to the left, until at c = 0 there
is a cusp at the origin. For c > 0, there is a loop to the left of the origin,
whose size and roundness increase as c increases. Note that the x-intercept
of the curve is always -c.
© 2012 Ccngag~ Ltaming. All Rights R.cscrvcd. Muy not be scanned, copied, or duplicated, or po!!1etl lo n publicly accessible website, in whole or in part.
CHAPTER 10 REVIEW
2
2
9
8
.
47. 6y2 + X - 36y + 55 = 0
45. ~ + 'lL = 1 is an ellipse with center (0, 0).
a = 3, b = 2 J2, c = 1
~
2
(y - 3) = --[;(x + 1), aparabola with vertex (-1, 3),
foci (±1, 0), vertices (±3, 0).
y
41
~
6(y2 - 6y + 9) = - (x + 1)
=?
0
opening to the left, p
J-
2 2
= - f.t
focus (- ;~, 3) and
=?
directrix x = -~.
y
3
-3
X
(- 1,3)
0
X
49. The ellipse with foci (±4, 0) and vertices (±5, 0) has center (0, 0) and a horizontal major axis, with a = 5 and c
2
2
2
so b2 = a 2 - c2 ·= 5 - 4
= 9. An equation is ~5 + Y9 = 1.
2
xb: = 1.
?
51 . The center of a hyperbola with foci {0, ±4) is (0, 0), so c= 4 and an equation is y 2 -
.
a
a
3
The asymptote y
= 3x has slope 3, sob = l - ::::} a= 3b and a 2 + b2 = c2
10b2 = 16
b2 =
::::}
*
"
= 4,
2
and so a
2
=?
·
{3b? + b2 = 4 2
y2
x2
1
1
5y2
=?
5x2
= 16 - ~" = L52 • Thus, an equation is -72 -5 - -8 5 = 1, or -72 - - 8 = 1.
53. x 2 = -(y - 100) has its vertex at (0,100), so one of the vertices ofthe ellipse is (0, 100). Another form of the equation of a
parabola is x 2
found at .(0,
b2
__
= 4p(y - 100) so 4p(y - 100) = - (y- 100)
3 9
3 9
=?
4p
= -1
or 25 +
p = -~.Therefore the shared focus is
3 9
.
x2
(
----,=--= 25. So the equation of th e e ll ipse is /1 +
2
8
x2
::::}
~ ) so 2c = ~ -0 ::::} c = ~ and the center of the ellipse is (0, ~ ). So a= 100- 3~ 9 = 4 ~ 1 and
a 2 _ c2 __ 401 2 - 399 2
3 9
y - 839!))2
a2
=1
=?
(8y - 399? =
1
160,801
·
55. Directrix x = 4
=?
d = 4, so e = ~
=?
ed
4
- ..,..--- -.,
- 1 +ecosfJ- 3+cosfJ'
r -
3
57. (a) Jf (a, b) lies on the curve, then there is some parameter value t1 such that~ = a and
tr ~ =b. l ft 1 = 0,
1 + tl
1 +tl
the point is (0, 0), which lies on the line y = x. lft1 =I 0, then the point corresponding tot=
3(1/tl )
3ti
3(1/td
x = 1 + ( 1 /t ) 3 = t~ + 1 = b, Y = 1 + ( 1 /h) 3 =
1
..!.
is given by
tl
tt3tl+ 1 = a. So (b, a) also lies on the curve. [Another way to see
2
3
t
this is to do part (e) first; the result is immediate.] The curve intersects the line y = x when ~ =
1+ t
1 +t3
3
t = t2
::::}
t = 0 or. 1, so the points are (0, 0) and ( ~, ~).
<D 2012 Cengogc Leamins. ;\II RightS Rescr.,.cd. May noc be scanned, COJ')ict.l. or duplicated, or postt.'tlto u publicly llCCessiblc website, in whole or in (mrt.
::::}
42 . D
.CHAPTER 10
PARAMETRIC EQUATIONS AND POLAR COORDINATES
3
1
34
3 3 2
(b) dy = ( +t )(Gt) - t2( t ) = Gt - t =Dwhen6t - 3t4 =3t(.2-t3 )=D ·=>
dt
(1+t3)2
(1+t3)2
horizontal tangents at (0, D) and (
t=Dort= 3'2,so thereare
y.r.
V2', ?-'4). Using the symmetry from part (a), we see that there are vertical tangents at
(0, D) and ( ?-'4, .v2).
(c) Notice that as t --> -1 +,we have x--> -oo andy --> oo. As t--> -1-, we have x --. oo andy --> - oo. Also
.
3t+3t 2 +{1+t3)
{t+1) 3
{t+1) 2
y - {-x - 1) = y + x + 1 =
=
=
-> 0 as t -> -1. Soy = -x - 1 is a
1+ t 3
1 +t3
t 2 -t+1
slant asymptote.
.d) dx = (I-+ t 3 )(3)- 3t(3t 2 ) = 3 - 6t3
dy
6t- 3t4
dy
dyjdt
t(2 - t 3)
( dt
(1 + t3)2
(1 + t3)2 and from part (b) we have dt = (1 + t3)2. So dx = dxjdt = 1- 2t3 .
y
So the curve is concave upward there and has a minimum point at (D, D)
and a maximum point at ( .v2, W). Using this together with the
X
information from parts (a), (b), and (c), we sketch the curve.
3
3
3
3t )
(e) x? + y = ( 1 + t 3
27t3 + 27t6
.(1 + t3)3
+ ( 1 3t2
+ t3 )
.
2
'y =-x - l
27t3 (1 +t3 )
(1 + t3)3
3
3xy = 3C!\3 )C~t3 ) = ( 1 ~~3 ) 2 ,sox 3 +y3 =3xy.
(f) We start with the equation from part (e) and substitute x = r cos 0, y = r sin 0. Then x 3 + y 3 = 3xy
r 3 cos3 0 + 1'3 s in3
r
r
e ~ 3r2 cos e sin e. For =1- 0, this gives = 3 ~OS e s~:
)
3(_1
cosO · cosO
cos O+ sm 0
=>
.
Dividing numerator and denominator
sinO
.
by cos3 0, we obtam r =
3sec0 tan B
1 + tan 3
.
sm3 e
1 + -cos3 0
e.
(g) The loop corresponds to 0 E (D, ~).so its area is
-
{"/
2
r
A-}0
2
-.! ("/ 2 (3sec0 tan0) 2
1+tan3 0
2d0-2}0
r / (l+
sec e tan e 0 - ~ {""' u du
tan 0)2d -2}
{1+u 2
~
d0-2}0
-
2
2
2
2
3
0
3)
[let u = tan OJ
= lim l!.[-~(l+u3)-l]b =;!
b--+oo 2
3
a
2
(h) By symmetry, the area between the foUum and the line y = -x - 1 is equal to the enclosed area in the third quadrant,
plus twice the enclosed area in the fourth quadrant. The area in the third quadrant is~ . and since y = -x - 1
r sine = - r cos 0- 1
-1 /_-"'/
2 - rr/2
4
[(
-
=> r = - . e
. () 1
() )
sm + cos
!;JU
2
1
e' the area in the fourth quadrant is
+cos
2
3
]
()
- (3sec0 tan0)
d() CAS
= -21.. Therefore, the total area IS. 21 + 2 (1)
2 = 2.
1 +tan3
© 2012 Ccngage Learning. Al l Rights Reserved. M1ty not be scanned, copied. ~r duplicated, or posted too. publicly accessible wcb~itc, in whole or in part.
=>
0
PROBLEMS PLUS
1. x =
l
t cos u
- - du, y =
1
tL
dx = 0
dt
{:}
lt
sin u
dx
cost
dy
sin t
. - - du, so by FTCl, we have dt = -t- and dt = - t - . Vertical tangent lines occur when
1
tL
cost = 0. The parameter value corresponding to (x, y) = {0, 0) is t = 1, so tl1e nearest vertical tangent
.
occurs when t = ~. Therefore, tl1e arc length between these points is
7r/2
L=
1
1
(dx) + (dy)
2
-dt
dt
2
. t = j '1f/Z-dt ~ [ lnt]7f 12 = ln :!!.
11r/Z cost2 t + ~dt
t2
1
t
1
2
2
dt=
2
1
3. ln. terms ofx andy, we have x =?'cosO= (1 + csin B)cosB =cos O+ csinBcosB =cosO + ~csin2B and
2
y = rsinB = (1 + csinO)sinB =sinO+ csin B. Now - 1 $sinO $ 1
=> -1 $sinO -t csin2 0 $ 1 + c $2, so
-1 $ y $ 2. Furthermore, y = 2 when c = 1 and B = ~ , while y = -1 for c = 0 and 0 = 3; . Therefore, we need a viewing.
rectangle with - 1 $ y $ 2.
To find the x-values, iook at the equation x =cos B + ~ csin 20 and use the fact that sin 20 ~ 0 for 0 $ B $ ~ and
sin 28 $ 0 for - ~ $ B $ 0. [Because r =::, 1 + csin 0 is symmetric about the y-axis, we only need to consider
- ~ S 8. $ ~ .) So for - ~ $ B .$ 0, x has a maximum value when c = 0 and then x = cos 0 has a maximum value
of 1 at B = 0. Thus, the maximum value of x must occur on [0, ~] with c = 1. Then x = cos 0 + ~ sin 20
~~ = -sinB +cos 2B=- sinB + 1-2 sin2 9
=?
[but sin B # - 1 for 0 $ 9 $ H If sin 9 = ~, then B =
=>
j~ = -(2sin9-1)(sin9+1) = 0when sin9 = - 1or~
*
and
2.1
c=t
x = cos {f + ~sin 3 = ~V3. Thus, the maximum value of xis ~V3, and,
by symmetry, the minimum value is - ~ V3. Therefore, the smallest
viewing rectangle that contains every member of the famil y of polar curves
r = 1 + csinB, where 0 $ c $ 1, is [-~V3, ~V3] x [-1 , 2].
5. Without Joss of generality, assume the hyperbola has equation :: -
~: = 1. Use implicit differentiation to get
2
2
: - Yb2y' = 0, soy' = b:x. The tangent line at the point (c, d) on the hyperbola has equation y- d = b:d
c (x - c).
a
a y
.
a
' 2
The tangent line intersects the asymptote y = !!. x when .!!.x - d = bzdc (x- c)
a
a
a
=> abdx- a 2 d 2 = b2 cx - b2 c 2 =>
© 2012 Cengnge le>ming. All Rigllls Reserved. Moy not be: seonncd, copied, or duplicated, or posted to n publicly a=ssiblc website. in who le or in pan.
43
44
D
CHAPTER 10 PROBLEMS PLUS
2 2
=>
X =
2 2
a d -b c
b(ad _be)
ad + bc
= - -b-
d h
. bad +be
ad '-t-be
an t e y-va 1Ue IS - b- = - a- ·
a
. .1ar1y, t he tangent 1.me mtersects
.
b at (be
-bad, --aad - be ) . Th e m1.d pomt
. oftl1ese mtersect1
.
.on pomts
. 1
.s
S1m1
y = -ax
--
be-ad) (ad+be + -ad-b
e)) = (12bc
12ad)
.
a2b , 2--;;:= ( c, d), the pomt of tangency.
1 (ad+bc
1
a( 2 - -b- + --b- • 2 - -
Note: Jf y = ·o, then at (±a, 0), the tangent line is x = ± a, and the points of intetsection are clearly equidistant from the point
of tangency.
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INFINITE SEQUENCES AND SERIES
11 . D
11.1
Sequences
1. (a) A sequence is an ordered list of numbers. It can also be defined as a function whose domain is the set of positive integers,
(b) The terms a,. approach 8 as n becomes large. In fact, we can make an as close to 8 as we like by taking n sufficiently
large.
(c) The terms an become large as n becomes large. In fact, we can make an as large as we like by taking n sufficiently large.
2n
. { 2
4
6
8
10
}
{ 4 3 8 5
}
3. an = n2 + 1, so the sequence IS 1 + 1' 4 + 1 ' 9 + 1' 16 + 1' 25 + 1' . . . = 1, 5' 5' 17' 13 ' . . . .
5. an =
(-1)n- l .
. { 1 -1 1 -1 1
}
{1
1 1
1
1
} ·
5n
, so the sequence IS 51' 52' 53 ' 54' 55 '... = 5' - 25' 125 ' - 625' 3125'... .
. {1
1
7. ~ = (n + 1)!, so the sequence IS
9. a 1 = 1, a,,+ 1 = 5an- 3.
1 1 1 1
}
{11 1
1
1
}
2!'"3!' 4f•5!' 6!'... = 2' 6' 24 , 120, 720 '... .
Each term is defined in terms of the preceding term.
a 2 = 5a1 - 3 = 5(1)- 3 = 2.
a 3 = 5a2 - 3 = 5(2) - 3 = 7. a4 = 5a3 - 3 = 5(7) - 3 = 32. as = 5a4 - 3 = 5(32) - 3 = 157.
The sequence is {1, 2, 7, 32, 157, ... }.
Q.4
a5 =
4
1
: a
4
a3
2
=- = 1 +2/52/5 = 7•
1 + a3
=:' 1 ~;/7 = ~- ,The sequence is {2, ~. ~. ~' ~ •... }.
1
The denominator of the nth term is the nth positive odd integer, so an= - - - .
2n-1
15. { - 3, 2, -1, ~,- ;~, ... }. The first term is -3 and each term is - ~ times the preceding one, so a,. = -3 ( -~
17. { ~, -~,%,- 1: , 25~, ... }.
f -1 .
The numerator of the nth term is n 2 and its denominator is n + 1. Including the alternating signs,
n2
wegetan = (-1)n+l _ _ _
n+1
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45
46
D
INFINITE SEQUENCES AND SERIES
CHAPTER 11
19.
n
3n
an= 1+ 6n
1
0.4286
2
0.4615
3
4
0.4737
5
0.4839
0.4865
'
21 .
10
5
n
It appears that lim an = 0.5.
n-+oo
3
lim ~ = lim
(3n)jn = lim
= ~ = .!
+ 6n n-+oo (1 + 6n)jn n-+oo 1/n + 6 6 2
0.4884
0.4898
8
9
10
.
0
0.4800
6
7
.........
0.5
0.4
n-+oo 1
0.4909
0.4918
n
an= 1 + (-U~
1
2
0.5000
3
1.2500
0.8750
4
1.0625
5
6
7
0.9688
1.0156 I
8
9
1.0039
0.9980
1.0010
.....
0
n
It appears that lim a,. = 1.
n-co
0.9922
10
10
5
I
lim (1+(- ~f) =lim 1+ lim (- ~f=1+0 = 1 since
n -oo
n-too
n-oo
""
23. an = 1 - (0.2)", so lim a,. = 1 - 0 = 1 by (9) . Converges
n - oo
3 + 5n2
(3 + 5n2)jn2
5 + 3/n2
5+ 0
25. a,, = n + n 2 = (n + n 2)/n2 = 1 + l /n, so a,, -+ i + 0 = 5 as n -+ oo.
Converges
27.. Because the natural exponential function is continuous at 0, Theorem 7 enables us to write
lim an= lim e 1 fn = e1imn-oo( l /nl = e0 = 1. Converges
,.,_ex:>
n-oo
2mr h lim' b
1'
(2mr)/n
lim
271'
271'
71' s·
.
.
.,. b
29. Ifb n = - --, t en
n =
1m (
)/ =
/
= - = - . mce tan IS contmuous at 4 , y
1 + 8n
n-+oo
n-+oo 1 + 8n
n
n->oo 1 n + 8
8
4
Theorem 7, lim tan (
n --->oo
31. an =
2
2
n1r ) =tan~ = 1.
n 71' ) = tan ( lim
1 + 8n
n-+oo 1 + 8n
4
n2 j,;r;J
n2
vn3 + 4n = ../n3 + 4n/v'n3 =
Converges
.,jii
Jl + 4jn2 , so an -+ oo as n--+ oo since lim .,jii = oo and
n--->OO
lim j l + 4/ n 2 = 1. Diverges
1\-+00
33. lim Jan! = lim
n--+oo
n-+oo
I(-%"I
= -21
2vn
lim
1
n-+oo n 112
1
= - (0) = 0, so lim an = 0 by (6).
n-oo
2
Converges
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SECTION 11.1
35. a,. = cos( n /2).
SEQUENCES
0
47
This sequence diverges s ince the tenns don 't approach any particular real number as n---> oo.
The terms take on val4es between - 1 and I.
1
_ (2n - 1)! =
(2n - 1}!
· =
· ___. 0 as n---> oo.
37
· an - (2n + 1)!
(2n + 1}(2n}(2n- 1)!
(2n + 1}(2n)
C
onverges
1 + e - 2n
--'------ ---> 0 as n---> oo because 1 + e- 2 n ---> 1 and e" -
en- e n
41. an = n 2 e
-n
2
2
e-n -+ oo. Converges
n
x
2x
2
= -.
Smce lim - = lim - = lim - = 0, it follows from Theorem 3 that lim an= 0 . Converges
en
e:t
n-oo
•
•
X--+00 e X
11
H
X-tCO eX
2
43. 0 < cos n < _..!:.._ · [since 0 :s; cos2 n :s; 1],
2n - 2"
. ( 1/ n ) = sin(1/
/ n n) .
45. an = n sm
1
X--tOO
82
1
so since lim
= 0, { 'c 02n n } converges
to 0 by the Squeeze Theorem.
n->oe> 2"
·
s·mce lim sin(1/x)
/
= 1.tm sin t [w here t = 1/ x ] = 1, .tt .,•OIIows ·from Theorem 3
x ..... oo
1 X
t-o+
t
that {an } converges to I .
= lim
x -> oo 1
lim
!t-oo
( 2)"' =
1+ -
X
.
lim e 1n Y = e 2 , so by Theorem 3, lim
:r-oo
n -+oo
2
( 2)" =
1+ n
2
+ 2/x = 2
::::}
e 2 • Converges
2
2
2
+ 1/n
49. an = In(2n + 1)- ln(n + 1) =In ( 2n
n 2 ++ 1) = In .(2 +
/ n ) -+ In 2 as n ---> oo.
1
1 1 2
Converges
51. a.,, = arct an(ln n). Let f(x) = arctan(ln x). Then lim f(x) = ~since In x -+ oo as x---> oo and arctan is continuous.
x - oo
Thus, lim a ,= lim f(n) = ~n-+oo
n -+oo
Converges
53. {0, 1, 0 , 0, 1, 0, 0, 0, 1, ... } diverges since the sequence takes on only two values, 0 and I , and never stays arbitrarily close to
either one (or any other value) for n sufficiently large.
n!
1 2 3
(n- 1) n
1 n
55. an= 2 n = 2 · 2 · 2 · · · · · - -2- · 2 2': 2 · 2
57.
2~------------~
n
(for n > 1] = 4--> oo as n--+ oo, so {an} diverges.
From the graph, it appears.that the sequence converges to 1.
{ ( -2/e)n} converges to 0 by (7), and hence {1 + (- 2/e)" }
.. . . .. . .
converges to 1 + 0 = 1.
0 '---~--~--~---~ 21
® 2012 Ccng;,gc Lc011T1ing. AURighiS R.escn·l.-d. May not be scanned, copied. or duplicated. or posted to apublicly accessible website, in whole or in part.
I
48
D
CHAPTER 11
'
INFINITE SEQUENCES AND SERIES
From the graph, it appears that the sequence converges to t.
59.
As n--+ oo,
3/ n 2 +2
.. ...
8 + 1/ n
=>
so lim an =~ ·
n -+oo
0 '----~--~--"--'----./ 21
2
61.
2.~-------------..
..
From the. graph, it appears 'that the sequence {an} ,;_ { ~ ~o~: } is
divergent, since it osci llates between 1 and -1 (approximately). To
n2
, then
prove this, suppose that {a,.} converges to L. If bn = - 2
1+n
..
{bn} converges to 1, and lim abn =!::. = L. But an = cosn, so
n-+oo n
1
bn
lim abn does not exist. This contradiction shows that {an} diverges.
-2
-oon
•
From the graph, it appears that the sequence approaches 0.
63.
O < a,.=
:<::;
0~-~-~~-----~-J IO
65. (a) a,. = 1000(1.06)."
1 · 3 · 5 · · · · · (2n- 1)
1 3
5
2n- 1
n
= - · - · - · · .. · -(2n)
2n 2n 2n
2n
1
1
n · (1) · (1) · · · · · (1) = n --+ 0 as n --+ oo
2
2
So by the Squeeze Theorem, {
1 · 3 · 5 · · · · · (2n - 1)}
( n)"
converges to 0.
2
=> at = 1060, a2 = 1123.60, a3 = 1191.02, a4 = 1262.48, and as = 1338.23.
(b) lim a,. = 1000 lim (1.06)", so the sequence diverges by (9) with r = 1.06 > 1.
n - oo
n-+oo
67. (a) We are given that the init!al population is 5000, so Po = 5000. The number of catfish increases by 8% per month and is
decreased by 300 per month, so Pt = Po + 8%Po - 300 = 1.08Po - 300, H = l.08P1 - 300, and so on. Thus,
P,.
= 1.08Pn-t - 300.
(b) Using the recursive formula with Po = 5000, we get P 1 = 5100, P 2
= 520.8, P 3 = 5325 (rounding any portion of a
catfish), P,. = 5451, P 5 = 5587, and P6 = 5734, which is lhe number of catfish in the pond after six monlhs.
69.
If lrl ~ 1, then {r"} diverges by (9), so {nrn} diverges also, since lnr''l = n lr"l ~ lr''l· If lrl < 1 then
lim xr"' = lim -=..._ :lb ~ (
:C - t OO
:c-oo r-:Z:
X-+00
-
In\
T T-%
= lim ___!__I
."' :::: 0, so lim nr"
%-+00 nT
n-oo
= 0, and. hence {nr"} converges.
whenever lrl < 1.
® 2012 Ccngogc Lcoming. All Rights Rc:sen'Ctl. May not be S<llllllCd. copied, orduplic:oted, or posted ton publicly accessible website, in whole or in p.1rt.
SECTION 11.1
SEQUENCES
D
49
71. Since {a,.} is a decreasing sequence, an > an+l for all n ~ 1. Because all of its terms lie between 5 and 8, {an} is a
bounded sequence. By the Monotonic Sequence Theorem,'{an} is convergent; that is, {an} has a limit L. L must be less than
8 since {an} is decreasing, so 5 $ L < 8.
73. a,. =
1
' d
.
.
1
1
1
"
n + IS ecreasmg smce an+1 = (n + ) + = n + < n + = a 1, tOr each n ~ 1. The sequence is
2
1
3
2
5
2
3
2
3
bounded since 0 <an $
i fo r all n ~ 1. Note that a1 = t·
75. The tenns of an = n( -1)n alternate in sign, so the sequen~e is not monotonic. The first five terms are - 1, 2, - 3, 4, and -5.
Since lim Ia.. I = lim n = oo, the sequence is not bounded.
n. -+oo
n-+oo
n
.
.
( )
x
77. an= - 2 - - defines a decreasmg sequence smce fo r f x = - 2 -
n
+1
.
x
-
+.1
'( )
(x
,f x =
2
+( 1)(1)-)2x(2x) = 1 - x 2 < 0
x2 + 1
(x2 + 1)2 -
for x ~ 1. The sequence is bounded since 0 < a" $ ~ for all n ~ 1.
79. For { v tn
2, y rn-Fn
2 y 2, y. I 2 y rn-Fn
2 v 2, .. . } , a1· -_ 2 112 , a2 -_ 2 3/4 , aa -_ 2 7/B , ... , so an -_ 2 (2"-1)/2" = 2 1-(1/2">.
lim a, = lim 21 - <112 ") = 2 1 = 2.
n-oo
n-+oo
Alternate solution: Let L = lim an. (We could show the limit exists by showing that {an} is bounded and increasing.)
n--+oo
Then L must satisfy L = ~
.
.
2
=> L = 2L => L(L - 2) = 0. L =/= 0 since the sequence increases, soL = 2.
1
81 . a 1 = 1, an.+l = 3 - - . We show by induction that {an} is increasing and bounded above by 3. Let Pn be the proposition
an
that an+l > an and 0 < an < 3. Clearly P1 is true. Assume that Pn is true. Then an+l > an
1
1
1
1
- - ·- > - -.Now an+2 = 3 - - - > 3 - - = an+l
an+l
a,.,
an+l
a.,
~
=> -1- < -1
an+l
an
=>
Pn+l· This proves that {an} is increasing and bounded
above by 3, so 1 = a 1 < an < 3, that is, {an} is bound~d , and hence convergent by the Monotonic Seque11ce Theorem.
If L = lim an, then lim an+l = L also, soL must satisfy L = 3- 1/ L
n---+oo
n---+oo
=> £ 2 - 3L + 1 = 0 => £ -_ 3±.,/5
2
•
But L > 1, soL=¥.
83. (a) Let a,. be the number of rabbit pairs in the nth montl1. Clearly a1 = 1 = a2. In the nth month, each pair that is
2 or more months old (that is, a,.-2 pairs) will produce a new pair to add to the an-1 pairs already present. Thus,
an= qn- 1 + a n-2, so that {an}= {fn}, the Fibonacci sequence.
( b) an= f,.+ 1
f ,.
fn
fn-1 + fn - 2 = 1 + .f,.- 2 = 1 +
1
1
an-1 = - - =
/
·= 1 + --. If L = lim an,
fn - 1
fn- 1
f n-1
f n- 1 fn-2
an-2
n-oo
1
then L = . lim an- 1 and L = lim a,.- 2, soL must satisfy L = 1 + -L
n-+oo
n -+oo
·
=> L 2 - L - l = 0 => L = ¥
[s ince L must be positive].
® 2012 Ccngagc Lcoming. /\11 Riglns Rcsc~rvcd. May no1 be scnnncd. copied, or duplicated, or poi!.cd loa publicly occ<..-ssiblc wcbsilc, in whole or in parL
50
D
CHAPTER 11
IN_FINITE SEQUENCES AND SERIES
From the graph, it appears that the sequence { ~~ }
50
85. (a)
5
converges to 0, that is, lim n 1 = 0.
n- c::o n.
0 '-~~----~----~---L----~ 10
O.Or3_ _ _ _ _ _ _ _ _ _ _ _ _"'
(b)
y= 0.1
y = 0.001
7.5 "'--'~--~-----'-----~-------' 12.5
0
9.5 0'<:::::============~ l5.5
From the first graph, it seems that the smallest possible value of N corresponding to c = 0.1 is 9, since n 6 / n! < 0.1
whenever n ~ 10, but 96 / 9! > 0.1. From the second graph, it seems that for c = 0.001, the smallest pos~ible value for N
is 11 sin~e n 5 fn ! < 0.001 whenever n ~ 12.
87. Theorem 6: If lim lanl = 0 then lim -lanl = 0, and since -ian I ~an ~ Ia,. I, we have that lim an= 0 by the
n -+oo
n-+oo
n-oo
Squeeze Theorem.
89. To Prove: If lim an = 0 and {bn} is bounded, then lim (a.,b,.) = 0.
n -+ oo
tl. -tOO
Proof: Since {bn} is bounded, there is a posi!ive number M such that lb,.l ~ M and hence, lanl lbnl ~ lanl M for
all n ~ 1. Let f: > 0 be given. Since lim an = 0, there is an integer N such that ian - 01 < Me if n > N . Then.
n~oo
la,.bn - Ol = lanbnl = lanl lbnl ~ lanl M = Ia,.- OlM < ~ · M = f: for all n > N . Since f: was arbitrary,
lim (anbn) = 0.
n -t oo
91 . (a) First we show that a > a1 > b1 > b.
a 1 - b1 =
ai -.Jab = ~ ( ~- 2Vllb +b) = ~ ( y'a - -/bf > 0 [since a > b] => a1 > b1 . Also
b
a- a1 = a-Ha+ b)= Ha - ~) > 0 and b- b1 = b- .Jab=
Vii( Vb - 01) < 0, so a >.a1 > b1 >b. In the same
way we can show that a1 > a2 > b2 > b1 and so .the given assertion is true for n = 1. Suppose it is true for n = k, that is,
ak > ak+l > bk+ l > bk. Then
® 20J2 Cengoge Learning. All Rights Rescr...cd. Mny not be scanned, copied, or dupl icated. or posted to t1 publicly accessible wcbsilc, in whole or in putt.
SECTION 11.2
SERIES
0
51
ak+l - a1.:+2 = ak+l- Hak+l + bk+l) = ~(ak+l - bk+l) > 0, and
bk+l - bk+ 2 = bk+ 1 - Jak+lbk+l = ~ ( ~- Jak+l ) < 0
=*
a 1.:+1 > ak+2 > bk+2 > bk+l•
so the assertion is true for n = k + 1. Thus, it is true for a ll n by mathematical induction.
(b) From part (a) we have a > a,. > an+ 1 > bn+1 > b.. > b, which shows that both sequences, {an } and { bn}, are
monotonic and bounded. So they are both convergent by the Monotonic Sequence Theorem.
. b,. = {3 . Then lim an+l = lim -a,. -+b.,
(c) Let lim an =a and Itm
-n--+oo
n--+oo
n - oo
n--+oo
2
2a: = Q + {3
93. (a) Suppose
Q = {3.
{Pn} converges top. T hen Pn+l =. ~
=>
a+p...
p 2 + ap = bp
(b)
=>
a: +f3
- =>
=> a : = 2
"*
p(p + a - b) = 0
=>
=>
bp
a+p
p=--
p = 0 or p = b - a.
Pn+l = ~
= (~);n < (!!.)Pn since 1 + Pn > 1.
a + pn
1 + ....2::
a
a
r
a
(c) By part (b), P1 <
(~)Po. P2 < ( ~ )P1 _< ( ~
so lim Pn $ lim
n---..oo
n--+00
=>
y
P<l. P3 < (
y
~ )P2 < ( ~ Po, etc. In general, Pn < ( ~ Po.
(!!.)n
·po = 0 since b < a. [sy (7), lim rn = 0 if -;- 1 < r < 1. Here r = !!.a E (0, 1) .J
a
11.--+00
(d) Let a < b. We first show, by induction, that ifpo < b - a, then Pn < b - a arid Pn+l > Pn ·
bpo
a + Po
For n = 0, we have Pl - po = - -- - po =
Po(b - a a+po
Po) > 0 since Po < b- a. So Pl > po.
·
Now we suppose the assertion is true for n = k, that is, Pk < b- a and Pk+l > Pk · Then
a(ba - Pk)
- b _ a_ ~ __ a(b - a)+ bp~.;- ap~.; - bpk __ __.!..
_ ___...!~ > 0 because p~; < b a. So
b- a - Pk+l a+pk
a+pk
a + pk
Pk+l < b - a. And Pk+2 - Pk+l =
bpk+l
Pk+l(b- a - Pk+l)
.
- Pk+l =
> 0 smce Pk+l < b - a. Therefore,
a + Pk+l
a+ Pk+l
Pk+2 > ·Pk+l· Thus, the assertion is true for n = k + 1. It is therefore true for all n by mathematical induction.
A similar proof by induction shows that ifpo > b- a, then Pn > b- a and {Pn} is decreasing.
In either case the sequence {Pn } is bounded and monotonic, so it is convergent by the Monotonic Sequence Theorem.
rt then follows from part (a) that lim Pn = b- a.
n-oo
11.2 Series
1. (a) A sequence is an ordered list of numbers whereas a series is the sum of a list of numbers.
(b) A series is convergent if the sequence of partial s ums is a convergent sequence. A series is divergent if it is not convergent.
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52
0
CHAPTER 11
INFINITESEQUENCES AND SERIES
00
3. I: an = lim s, = lim [2- 3(0.8)n) = lim 2 - 3 lim (0.8)" = 2 - 3(0) = 2
n-oo
n=l
00
1
n-.oo
1
5. For I: 3 • a,, = 3·
n=l n
n
n--too
n-+oo
1
1
81 = a1 = 13 = 1, 82 = s1 + a2 = 1 +
= 1.125, sa = 82 + aa ~ 1.1620,
23
s~ = 83 +a~ ~ 1.1777, ss = 84 +as~ 1.1857, so = ss + ao ~ 1.1903, s 7 = 8G + a 7 ~ 1.1932, and
ss = sr + as ~ 1.1952. It appears that the series is convergent.
'
n
oo
n
7. For I: ~·a,, = ----r.:;·
n=l 1 + yn
1 + yn
St = a1 i;= -
1
-- =
1 + J1
2
0.5 82 = St + a2 = 0.5 + ~ ~ 1.3284,
'
1 + v2
s3 = s2 +as~ 2.4265, S4 = sa+ a4 :::::: 3.7598, 8::, = 84 + a 5 :::::: 5.3049,
8 6 = s 5 + a 6 :::::: 7.0443,
sr = so+ ar:::::: 8.9644, ss = 87 +as~ 11.0540. It appears that the series is divergent.
9.
n
8n
1
- 2.40000
2
- 1.92000
3
- 2.01600
4
- 1.99680
5
- 2.00064
6
- 1.99987
7
-2.00003
8
-1.99999
9
- 2.00000
10
-2.00000
Ot - ---+--;---+---+----..----+--i ll
-3
From the graph and the table, it seems that the series converges to - 2. In fact, it is a geometric
·
·th
2
d
I
·
· . ~
12
-2.4
-2.4
sen es w1 a= - .4 an r =-&·so tis sum IS 6 ( - )n =
( 1) = - = - 2.
5
.
n=l
1- - &
1.2
Note that the dot corresponding to n = 1 is part of both {an} and {Sn}.
TI-86 Note: To graph {an} and { sn }, set your calculator to Param mode and DrawDot mode. (Draw Dot is under
GRAPH, MORE, FORMT (F3).) Now under E (t) = make the assignments: xt1=t, yt1=12/ ( -5) -t, xt 2=t ,
yt2=surn seq ( ytl, t, 1, t , 1) . (sum and seq are under LIST, OPS (F5), MORE.) Under WIND use
1 , 10, 1, 0, 1 0, 1, - 3, 1, 1 toobtain a graph s im ilar to the one above. Then use TRACE (F4) to see the values.
10/------------~
11.
n
Sn.
1
0.44721
2
1.15432
3
1.98637
4
2.88080
5
6
7
3.80927
4.75796
5.71948
8
6.68962
9
7.66581
10
8.64639
0....___....___
The series
__.._ _..__
__..__ _..__, 11
f: ~
diverges, since its terms do not approach 0.
n +4
n=l
2
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SECTION 11.2
SERIES
0
13.
n
s.,
1
0.29289
2
0.42265
3
0.50000
0.55279
4
0.59175
5
6
0.62204
0.64645
7
o'-~~~~--~~~~~L-~~
u
From the graph and the table, it seems that the series converges.
0.66667
8
9
0.68377
0.69849
10
00 (1-Vn - -v'n-1)
- =
+
so I:
1
n=l
lim
k -oo
( y'kTI
1) = ·
1 - - --
1
15. (a) lim an = lim ~ = ~.so the seqitence {an} is convergent by (11.1.1).
n-oo 3n
n-oo
+1
3
(b) Since lim an = ~ =I 0, the series
n -+oo
17. 3 - 4 + 1;
64
9
-
f: an is divergent by the Test for Divergence.
n= l
+ ·· · is a geometric series with ratio r = -l Since lrl = ~ > 1, the series diverges.
19. 10 - 2 + 0.4 - 0.08 + · · · is a geometric series with ratio - 120 =
21.
-i· Since lr\ = k < 1, the series converges to
f: 6(0.9)n-l is a geometric series with first term a = 6 and ratio r = 0.9. Since lrl = 0.9 < 1, the series converges to
n=l
a
6
6
- - = - - = - = 60.
1- 7·
1-0.9
0.1
f: (~ 3~n-l ~ f:· (-~) n- .
1
23.
n=l
=
4
converges to
oo
25. I:
?Tn
n+l
n=O 3
1-
4 n= l
(
1
4
/ ) = ~. Thus, tbe given series converges to (-!) (~) =
- 3 4
1 oo (?T) n·
= -
The latter series is geometric w.ith a= 1 and ratio r
I: -
3 n= O 3
IS
=-~.Since lrl = ~ < 1, it
t.
.
.
.
?T .
.
.
ageometnc. senes
w1th rat1o r = - . Smce \rl > 1, the senes dtverges.
3
1
1 ~ 1 Th'IS ts
. a constant mu It1p
. Ie o f the d'tvergent harmoruc
. senes,
. so
27. -1 + -1 + -1 + -1 + -1 + · · · = ~
L- - - = - L- - .
3 n= l n
3
6
9
12
15
n = l 3n
it diverges.
1 .
by the "'.est ".or o·1vergence smce
. lim. an = lim -n - 1 = -1 ....£
29. ~
L., -n- - - - dtverges
r 0.
3
n=l 3n - 1
n-oo
n-oo 3n - 1
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53
54
0
CHAPTER 11
INFINITE SEQUENCES AND SERIES
31. Converges.
00
00
00
1 + 2"
( 2) "]
L:--=L:
-1 + -2") =L: -1 ) " +n=1 3"
n=1 3n 3" . n=1 3
3
(
[sum of two convergent geometric series]
[ (
-~
~-!
- 1 - 1/ 3 + 1 - 2/3 - 2 +
2-~
- 2
00
33.
L: o/2 = 2 + v'2 + ?12 + ~ + · · · diverges by the Test for Divergence since
n=l
o/2 = n-oo
lim 21 / n = 2° = 1 =/= 0.
lim a,. = lim
n.-oo
35.
37.
n-oo
E In ( 2nn ++11 diverges by the Test for Divergence since
2
2
n=l
)
.
E (~)" is a geometric series with ratio r = -I ~ 1.047. It diverges because lrl ~ 1.
k=O
00
39.
L: arctan n diverges by the Test for Divergence sincen.-+oo
lim an = lim arctan n = %=/= 0.
n=l
n-+oo
1
1
1
1
1
41 . ~
( - )" 1·s a geometric
· senes
· w1t
· h fi rst term a = - and ratio
· r = - . s·mce Ir I = - < 1, the senes
· converges
6 ~ = ~
6
n= l e
n=1
e
e
e
e
1/e
1 - 11e
1/e
1 - 11e
(1
1
e
e
1
e- 1
( 1
oo
to - -- = - -- · - = --.By Example 7, L:
oo
00
·)
1
·
) = 1. Thus, by Theorem 8(ii),
n=1 n n + 1
""
· 1
·
1
1
e
e- 1
n~1 en + n(n + 1) = n~l en + n~l n(n + 1) = e- 1 + 1 = e - 1 + e- 1 = e - 1 ·
43. Using partial fractions, the partial sums of the series
'
n
s,. = L:
i= 2
E + a-1r e
n =2 n
n
(
. 2.
= L:
-. 1- - -.1- )
(t - 1)(t + 1) i=2 t - 1 t + 1
!) (!2 -!)
+ (! -!) + ... + ( '- 1 - _ 1 ) + ( 1 - .!.)
4
3 5
n-3 n- 1
n-2
n
= (1- 3 +
. senes
. an d sn = 1 + -1 ~ -1- - -1 ·
. sum 1.s a te Iescopmg
Th 1s
2
n- 1 n
Thus,
E --i:-= l~
- 1
n=2 n
n-+oo
45. For the series :E
oo
Sn
3
n=1 n(n+3)
.!.)
= lim (1 +!2 - -n--1 - n = ~2
1
n-+00
n
3
n (1
, s,. = :E
-. -.- = L:
-:- - -. -1 )
i =1 t(1 + 3)
i=1
\
+ (l2 ....: l)
+ (13 - l)
+ (l4 - l)
+ ... +
(1 - l)
4
5
6
7
[using partial fractions). The latter sum is
t+3
(-1 - l) + (-1 - _1 ) + (-1 - _1 ) + (.!. - --4)
n-3
n
n -2
1 n+1
1 = 1 + 21 + 3TIms, 6~
n + 1
1
n+ 2 -
n-1
1
n+3
n+2
[e t
l . senes
' ]
escopmg
l'tm Bn = l'1m (1 + 21 + 31 - - 1
1 - ..,----+
1 ) = 1 + 21 + 31 = 11 . converges
+
- - +
2
6
+ 3) = n-oo
n -oo
n 1
n 3
( 3
n= l n n
1l
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n
n+3
SECTION 11.2
SERIES
0
55
1
47. F~r the series ..~ ( e1 1" - e 1/ (n+l)) ,
Sn =
i~ ( el/i _ el/(i+I)) = (el _ el / 2) + (el / 2 _ el/ 3) + ... + ( el /n _ el / (n+l)) = e _ el /(n+l)
[telescoping series]
Thus,
f: ( el/n - el/ (n+l)) = lim
n =l
n-oo
Sn
= lim ( e - e1 / (n+l) ) = e - e 0 = e- 1.
Converges
n--+oo
49. (a) Many people would guess that x < 1, but note that x consists of an infinite number of9s.
(b) x = 0.99999 .. . =
~ 9
l"h"
.
.
.
9
9
9
+
+
+ 10,9000 + · · · = .~
10
100
1000
1 10.. , w 11c 1s a geo~etnc senes w1th a 1 = 0.9 and
0.9
0.9
1 h .
1
.
r = 0.1. Its sum IS _ 0. = 0. = , t at 1s, x = .
1
1
9
(c) The number 1 has two decimal representations, 1.00000 . .. and 0.99999 . . . .
(d) Except for 0, all rational nurnbe.rs that have a terminating decimal representation can be written in more than one way. For
example, 0.5 can be written as 0.49999 . . . as well as 0.50000 ... .
-
51. 0.8 =
8
8
.
.tl
8
d
1 l
a
8/10
+
+ · ·· .IS a geometriC. senes
WI 1 a =
an r =
. t converges to - - =
/ = -8
102
10
10
10
1 - 1"
1 - 1 10
9:
516 516
"516
516
.
.
.
.
516
1
53. 2.516 = 2 +
+
+
+ · · · IS a geometnc senes With a=
and r =
. It converges to
+ · · · . Now
106
103
106
103
103
103
3
3
_a_ = 516/ 10 = 516/ 10 = 516 . Thus. 2 .516 = 2 516 = 2514 = 838
1 .:... r
1 - 1/ 103
999/ 103
999
'
+ 999
999
333 ·
-
42
42
42
42
.
.
.
.
42
1
104 + 106 + · · · . Now 104 + 106 + · · · 1s a geometnc senes w1th a = 104 and r = 102 .
55. 1.5342 = 1.53 +
a
42/ 104
42/ 104
42
It converges to 1 - 1· = 1 - 1/ 102 = 99/ 102 = 9900 ·
42
153
42
15,147
42
15,189
5063
Thus, 1.5342 = 1.53 + 9900 = 100 + 9900 = 9900 + 9900 = 9900 ~r 3300 ·
57.
00
00
n=l
n=l
2..: (-5)"xn = 2..: (- 5x )" is a geometric series with r = - 5x, so ilie series converges
~
J- 5xJ < 1
JxJ <
00
59.
00
'E (x - ..2)" = E
n=O
3
n=O
(X
- - 2 ) " is a geometric series with r = -X-2, so the series converges
3
3
- 1 < - 3- < 1
- 3 < x- 2 < 3
a
1
1
3
1- r =
_ x -2
3 - (x - 2)
5- x ·
3
3
1
~ ---;;
2" =
61. ' L..n=O X
2 < Jxl
~
X
~
_-
X
~
Jrl < 1
~
- 1 < x < 5. In that case, the sum of the series is
~
(2)".
.
. senes
. Wlw
.... r = :-•
2 sothe senes
. converges
L..IS a geometric
n=O
lrl < 1
t, that is, -t < X< r. In that case, the sum of the series is _1 -a_r = 1 - ( 5-X5x ) = 1-+55x
x
·
x- 2
¢:?
¢:?
~
Jrl < 1
2
x > 2 or x < -2. In that case, the sum of the series is - a = __!__
/ = ~.
1
1- 2 X
X 1- r
© 2012Ccngage !.earning. All RighiS Rcscn "Cd. May nol be scaru>Cd. copic:d. or duplicated. or posted lo a publicly aecC>Sible wcbsilc. in whole or in part.
56
0
INFINITE SEQUENCES AND SERIES
63.
:L: en:z' = :L: (e"T is a geometric series with r = e"', so the series converges
CHAPTER 11
00
00
n =O
n=O
- 1 < e"' < 1
lrl < 1 {::} lex! < 1 {::}
{::}
1
l -ex
<=> 0 < e"' < 1 {::} x < 0. In that case, the sum of the series is _a_ = - - .
1- r
65. After defining J, We use convert (f , parfrac ); in Maple, Apart in Mathematica, or Expand Rational and
.
. 0 .
fi d th h
I
. 3n2 + 3n + 1
enve to n
at t e genera term IS (n 2 + n)3
.
s~mpl~fy 10
Sn =
= n13
1
(
) . So the nth partial sum is
. n+1 3
n(1
~.:'f1
1
1
)
(
1)
(1
1)
(1
1
)
k3 - (k + 1)3 = 1 - 23 + 23 - 33 + " . + n3 - (n + 1)3 = . 1 - (n + 1)3
The series converges to lim s,.
n-oo
= 1. This can be confirmed by directly computing the sum using
s um (f , n=l.. i nfinity) ; (in Maple), Sum [f , {n , 1 , In f inity}] (in Mathematica), or Calculus Sum
(from 1 to oo) and Simplify (in Derive).
67. For n
= 1, a1 = 0 since s1 = 0. For n > 1,
.
an = Sn -
Sn- 1 =
n- 1
(n - 1) - 1
n_ _+ _l - (n- 1) + 1 =
(n - l )n- (n + l)(n- 2)
2
(n + l)n
= -n '(
n- +:-1=-)
~
I'
1'
1 - 1/n 1
A Iso, nL;:l a,.,. = n~~ Sn = n~ 1 + 1/n = .
69. (a) The quantity of the drug in the body after the first tablet is 150 m g. After the second tablet, there is 150 mg plus 5%
of the first 150-mg tablet, that is, (150 + 150(0.05)] mg. After the third tablet, the quantity is
(150 + 150(0.05) + 150(0.05) 2 ]
= 157.875 mg. After n tablets, the quantity (in mg) is
n
1
.
.
150(1 - 0 .05")
3000 (
")
_ 0.0
=W
1 - 0.05 ..
1
5
.
150 + 150 ( 0.05) + ... + 150 ( 0.05 ) - . ·We can use Formula 3 to wnte this as
_(b) The number of milligrams remaining in the body in the long run is lim ( 3~go (1 - 0.05")] = 3 ~go (1- 0) ~ 157.895,
n.-.oo
.
only 0.02 mg more than the amount after 3 tablets.
71. (a) The first step in the chain occurs when the local government spends D dollars. The people who receive it spend a
fraction e of those D dollars, that is, D e dollars. Those who receive the De dollars spend a fraction c of it, that is,
D c 2 dollars. Continuing in U1is way, we see U1at the total spending after n transactions is
Sn = D + De + De2 + · · · + Dc" - 1 =
D(1 en)
by (3).
1 -c
1
(b) lim Sn = lim D( - c" ) = __..!2_ lim (1 - c") = __..!2_
n.-oo
n.-+oo
1- C
1 - C n -oo
1- C
= D
8
[since c + s = 1] = kD
[s ince 0 < c < 1
=>
lim c" =
n--.oo
.
o]
[since k = 1/ s]
If c = 0.8, then s = 1 - c = 0.2 and the multiplier is k = 1/s = 5.
73.
f (1 + c)- n is a geometric series with a ,;, (1 + c)- and r = (1 + c)- t, so the series converges when
2
n=2
1
IC1+ c)- 1 < 1
{::}
l1 +el > 1
{::}
1 +e>1or1+e< - 1
{::}
c > Oore < 7 2.Wecalculatethesumofthe
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SECTION 11.2
.
2c2 + 2c - 1 = 0
1 = 2
)
1- (1 + c
c = - 2~ m =
<:::}
D
57
2
(1+ c) - 2
.
scn es and set 1t equal to 2:
SERIES
<:::}
1- )
(1+ c
-2 - 2 ( -1- )
1 +c
<:::}
1 = 2(1 + c) 2 - 2(1 +c)
<:::}
±'1- However, the negative root is inadmissible because -2 < --q- 1 < 0.
1
.
Soc= :02- 1 .
1
1
1
.
.
75. e"" = e1+2+3+··-+;; = e 1e 1 12e 113 · · · e 1/ n > (1 + 1) (1 + ~) (1 + ~) · · · (1
+ ~)
[e"' > 1 + x]
= ~~~ ... n + 1 =n + 1
123
n
Thus, e·'" > n + 1 and lim e"" = oo. Since { sn} is increasing, lim s,. = oo, implying that the harmonic series is
n -+ oo
n-+ oo
divergent.
77. Let dn be the diameter of Cn. We draw lines from the centers of the C; to
the center of D (or C), and using the Pythagorean Theorem, we can write
2
1 2 + (1 - ~d1) = (1 + ~d1)
2
¢:}
1 = (1 + ~d 1 ) - (1- ~d1 ) = 2d 1 [differenceofsquares] ==>
2
2
d1
= ~·
Sim ilarly,
1 = (l + ~d2) - (1 - d1 - ~d2) = 2d2 + 2d1- di - d1d2
2
2
= (2 - dl)(d1 + d2)
1
d2 = 2 _ d 1
1
d,.+1 = ( -=2
1
12
d1 =
¢:}
(1-
dd , 1 = (1 + 2d3
1 )2 - (1 - d1 - d2 - 2ds
1 )2
2_ d
¥ ~) .
1
2
1
i=1
T
lfwe actually calculate d2 and
'
<:::}
- [1-_ (dl
+ d2)) 2
.
(dt + d ) , and m general,
d3 -
2
2
d~ from the formulas above, we find that they are.!.6 = -2 1. -3 and
1
3·4
- = - - respect1·veIy, so we suspect that .m general , d n = (
1
. duct1on:
.
• 0 prove th'IS, we use m
Assume that for all
) . .,.,
n n+ 1
1
1
n
1
n
ly < n dk =
= -k1 - -.Then 2:= d; = 1 - - - = - - '
k (k + 1)
k+ 1
. ;= 1
n+ 1
n+ 1
_ _ n ]
[ 1 n+1
.
formula for dn+l. we get dn+l =
(
2-
n
-
)
-
n+1
2
[telescoping sum). Substituting this into o ur
1
2
(n+1)
1
n + 2 = (n + 1)(n + 2 ), and the induction is complete.
n+1
Now, we observe that the partial s ums 2:=~= 1 d; of the diameters of the circles approach 1 as n--+ oo; that is,
00
00
:2:= an = :2:= (
n=1
n =1 n n
1
) = 1, which is what we wanted to PJOVe.
+1
79. The series 1 - 1 + 1- 1 + 1 - 1 + · · · diverges (geometric series with r = -1) so we cannot say that
0=1- 1 +1 - 1 + 1-1+· · · .
81. 2:=""=
can = n-+
limoo l:=~=l ca; = nlim
c 2:='~
a ; = c nlim
2:=~~ 1 a; = c 2:::,'= 1 a,., which exists by hypothesis.\
n 1
-oo
' 1
-+ oo
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58
0
CHAPTER 11
INFINITE SEQUENCES AND SERIES
83. Suppose on the contrary that l:(an + bn) converges. Then l:(an + bn) and I: an are convergent series. So by
Theorem 8(iii), I: [(a,. + bn) - a,.] would also b~ convergent. But I: [(an + bn) -an] = I: bn, a contradiction, since
I: bn is given to be divergent.
85. The partial sums {sn} form an increasing sequence, since s,. - Sn- 1 =an > 0 for all n. Also, the sequence {sn} is bounded
since Sn ::::; 1000 for all n. So by the Monotonic Sequence Theorem, the sequence of partial sums converges, that is, the series
I: an is convergent.
87. (a) At the first step, only the interval ( ~ , ~) (length ~) is removed. At the second step, we remove the intervals (
( ~ ' ~).which have a total length of2 ·
i, ~) and
Ut At the third step we remove 2 intervals, each of length (~ ) In general,
3
2
1
•
at the nth step we remove 2n- 1 intervals, each oflength ( ~) n, for a length of 2"- 1 · ( ~)"' = ~ ( ~) n- • Thus, the total
1
length of all removed intervals is
f ~ (t)
n-
= 1 :./;13 = 1 [geometric se~ies with a = t and 7' = t] . Notice. that at
1
n =l
the n th step, the leftmost interval that is removed is ( (
the rightmost interval removed is (1 -
1)" , (j )"),so we never remov~ 0, and 0 is in the Cantor set. Also,
(j)", 1 - (tf'), so 1 is never removed. Some other numbers in the Cantor set
dB
12127
are 3• 3•·9• 9• 9• an 9·
(b) The area removed at the first step is ~; at the second step, 8 · ( ~) ; at tbe third step, {8)
2
removed at t.he nth step is {8)"00
n>;1
1(8)"- = 1 - 8/ 9 =
9 9
1
1ar H~r-l.
=
2
i) 3 • In general, the area
so the totat area of all removed squares is
1/ 9
1.
oo
n
1
1
1
2
5
5
3
89. (a) For n>;1 (n + 1)!' 81 = 1. 2 = 2' S2 = 2 + ~ = 6' sa = 6 + 1 . 2. 3. 4
23
· (
4
119
.
23
24,
s4 = - +
= -.The denommators are (n + 1}!, so a guess would be sn =
24 1 · 2 · 3 · 4 · 5
120
(b) For n = 1, 8 1 =
Sk+l =
1
21 - 1
2 = ~·so the formula holds for n = 1. Assume 8k =
(n + 1}!-1
)
(
n + 1 .1
(k + 1)! - 1
(k + )! . Then
1
(k + 1}! - 1
k+1
(k + 1}! - 1
k +1
(k + 2) ! - (k + 2) + k + 1
(k + 1)! + (k + 2)! = (k + 1)! + (k + l)!(k + 2) =
(k + 2)!
- (k + 2)! -1
-
(It+ 2)!
Thus, the formula is true for n = k + 1. So by induction, the guess is correct.
.
·
.
(c) lim 8n = lim
n-oo
,._00
(n + 1)! - 1
.
[
1
]
oo
n
(n + 1)I. = nhm 1 - (
L: (n + 1)! = 1.
oo
n + 1)'. = 1 and so "=1
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SECTION 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS
0
59
11.3 The Integral :rest and Estimates of Sums
1. The picture sh?WS that a 2 =
1
a 3 = "1.3 <
.3
131
1
2
1. 3
r · 1 dx,
<1
2
y
x 1.3
1
00 1
[ 00 1
"1.3 dx, and so on, so 2:: 1':3 <
"1.3 dx. The
. 2
n= 2 n .
X
X
1
integral converges by (7.8.2) withp = 1.3 > 1, so the series converges.
0
2
3
X
4
3. The function f(x) = 1/ ~ = x- 1/ 5 is continuous, positive, and decreasing on [1 , oo ), so the Integral Test applies.
4 15 ] t = lim (~t4 1 5 - .§.) = oo, so E
= t-.oo
lim J' x- 1 / 5 dx = lim [2x
.1/ ifrf diverges.
1
t-.oo 4
1
t--.oo
·. 4
n= 1
1
f
oo x - 1 15 dx
1
. .
5. The function f(x) = ( x ~ ) 3 is continuous, positi~e, and decreasing on [1, oo), so the lf1tegral Test applies .
2
1
.1roo (2x +1 1)3 dx ~ lim 1r (2x +1 1) dx = lim [-.!4 (2x +1 1)2 ] = t-limoo ( - 4(2t 1+ 1)2 + _.!_)
= _.!__
36
36
t-+oo
1
t
3
1
t-.oo.
Since this improper integral is-convergent, the series
.
7. The function f(x)
1
E (2n +1 1) is also convergent by the Integral Test.
3
n=l
=-/-positive, and decreasing on [1, oo), so the futegral Test applies.
+ 1 is continuous,
.
.
X
1
1
roo ~
+ 1 'dx = lim r -/-+ 1 dx = lim [-2 ln(x + 1)] = -2 lim [ln(e + 1) - ln 2] = oo. Sine~ this improper
2
}
1
t -.oo } 1
X
X
t--> oo
t
.
.
1
t--+oo
·
E~
is also divergent by the Integral Test.
+1
integral is divergent, the series
n=1 . n
9.
E ~ is a p-series with
p = -/2 > 1, so it converges by (1 ).
.
.
n=1 n
8
27
1 Thi .
1
1
'th p = 3 > 1,so1tconverges
.
by ( I.
)
..
++ ···= ~
s tsap-seneswL
6 3 ·
64 125
n=l n
1
3
1
5
1
7
1
1
11.1 +-+-+
-
1
9
00
13. 1 +- +- + - + - + · · · =
1
1
L: - -.
The function f( x) = x _- · is
2
1
1
n =l 2n -
continuous, positive, and decreasing on [1, oo), so the· Integral Test applies.
1
-·- dx = lim
2x- 1
t -+oo
r oo -
}1
ft-2 1 dx = lim a1n 12x - 111 ~ = ~ lim (1n(2t - 1) - 0) = oo: so the series r: -2n 11
X- 1
t -+ oo
t-->oo
n=l
diverges.
oo
vfn+ 4
n
'oo
15. L: - - 2 - = L:
n =1
n= l
( vfn
4)
oo
1
oo 4
- 2 +2
= nL:
3 / 2 + L: 2•
n
n
=l n
n =1 n
~
6
n= l
1 .IS a convergent p-senes
. w1'th p = 3 > 1.
.
3!2
2
n
E n4 = 4. E ~n is a constant multiple of a convergent p-series with = 2 > 1, so it converges. The sum of two
n=l
2
p
n= l
convergent series_is-convergent, so the original series is conv~rgent.
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1
60
0
CI:IAPTER 11
INFINITE SEQUENCES AND SERIES
+4
17. The function f(x) = ~ is continuous, positive, and decreasing on [1, oo), so·we can apply the Integral Test.
X
.
l 1 ·= . ~t 1 · . [1 _ = ·-1. [· _(t) _(1)]
oo
1
- - dx
x2 + 4
- 2 - dx = lim -tan 1 -x]t
x + 4
t --+ oo 2
2 1
lim
1
t --+00
·
hm tan 1 2 t--+ oa
2
- tan 1 2
=.!. [~ - tan(.!.)]
.
2
1
. 2 2
Therefore, the series
f: ~
converges.
n +
n=l
ill n =
. ~
ln n smce
.
. f(. x ) = -ln x 1s
. contmuous
.
· · on [2, oo ) .
19 . ~
~ ~ -ln 1 = 0 . Tbe funchon
and postttve
3
3
3
n= l n
n =t n
1
X
I
3
2
2
2
f'( x) = x (1/x)-(ln x)(3x )=x -3x lnx=1 - 3lnx <O {::}
.
(x3)2
x6
x4
1 _ 3 lnx < O {::}
ln x> -1
{::}
3
x > e113 ~ 1.4,' so f is decre~ing on [2, oo), and the Integral Test applies.
1t
00
1
2
Inx dx = lim
x3
t -oo
2
1
ln x dx ~ lim [- lnx2 - - -]
t = lim [-_!._ (2 ln t + 1) +
x3
t-+oo
2x
4x2 1 t-+oo
4t2
4
.!.] (~) -41 , so the series f: In~
n= 2
n
converges.
(*): u = hl x, dv = x - 3 dx
I
=> du = (1 /x) dx, v = -~x- 2 , so
ln x d x = - 1 x -2 1n x - .;· - 1 x -2 ( 1/ x ) dx = - 1 x -21n x· + 1 ; · x -3dx = - 1 x -2 1nx- 1 x -2 +
. C
.
2
2
2
2
2
4
~
(**):lim (~2lnt+ 1 ) J:!:_lim 2/ t =- l4 lim ]:_ =0.
t--+oo
4t2
t --+ oa 8t
t--+oo t2
21. f (x) = + is continuous and positive on [2, oo), and also decreasing since J'(x) =-
x= x
.
use the lntegra1Test.1
.""
.
2
X
1
x2
Tlnln)2 < 0 for x > 2, so we can'
x
1
ln dx= lim [ln(lnx)J; = lim [ln(lnt)~ln(ln2)]= oo,sotheseries
X
t--+oo
t--+oo
.
f -n 11n-diverges.
n
n=2
23. The function f(x )'= e1 1"jx 2 is continuous, positive, and decreasing on [1, oo), so the Integral Test applies.
[g(x) = e 1 1~ is decreasing and dividing by x 2 does~'t change that fact.]
l
~t e1frz;
oo
f(x) dx = lim
t-oo
1
1
·
x] = - lim (e1/ t - e) = -(1 - e) = e- 1, so the series 2:: -eln/n2-
- 2- dx = lim [-e 11
X
t --t-oo
.
t
·
1
.
00
n=1
t --+CXJ
converges.
.
1
1
1
1
25. The funct10n f(x) = - 2- - 3 = - 2 - - + - - [by partial fractions] is continuous, positive and decreasing on [1, oo),
q; +x
x
x
x+ l
so the Integral Test applies.
00
1
1
f(x)dx = lim
t --+ oo
It(
J
1 + -1-) dx = lim [ - -1 - ln x +ln(x +:t)
. t
21 - -
1
X
X
X
+1
t--+oo
X
l
. [ -1 + ln t +1 + 1-ln2] =0 + 0 + 1-ln 2
= hm
t--+oo
t
t
' I converges, so the senes
. ~
1
The .mtegra
~ - - converges. .
2
3
I
n=l n
+n
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,
SECTION 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS
0
61
27. T he function f(x) = coFxx is neither positive nor decreasing on (1, oo), so the hypotheses of the Integral Test are not
.
· oo COS7Tn
satisfied for the senes L: ~ .
n=l
y n
1
00
29. We have already shown (in Exercise 2 1) that when p = 1 the series E
.
f (x) =
(
X
n=2
(l
) diverges, so assume that p =1- L
n nn p
1
) is continuous and positive on [2, oo), and f' (x) = - ~~ ~ :
ln x P
·
X
X 'P
1 < 0 ifx > e - P, so that f is eventually
decreasing and we can use the Integral Test.
roo ---:-:--1 -:-- d l' [ (ln X)1-p] t
} 2 x (ln x)P x=t~
1- p
2
This limit exists whenever 1 - p < 0
[forp :f- 1] = lim [ (ln t) 1- p - (ln 2)1 - p]
h oo · 1- p
1- p
p > 1, so the series converges for p > L
¢:?
31 . Clearly the series cannot converge if p 2': -~. because then lim n(1 + n 2)P =1- 0. So assume p. < -~ . Then ·
n--<oo
f(x) = x(1 + x 2)P is continuous, positive, and eventually decreasing on [1 , oo), and we can use the Integral Test.
This limit exists and is finite
¢:?
p+1 <b
¢:?
p < - 1, ~o the series converges whenever p < - 1.
33. Since th is is a p-series with p = x, ( (x) is defined when x > 1. Unless specified otherwise, the domain of a function f is the
set of real numbers x such that the expression for f (x) makes sense and defines a real number. So, in the case of a series, it's
the set of real numbers x such that the series is convergent.
(3)
4
00
35. (a) n~l
00
81
1
00
(7T4)
97T4
E 4n = 81 nI:
4 = 81 90 = -1o
= 1 n
;;:
n=l
4
00
00
1
1
1
1
1 . 7T
(b) k~6 (k - 2)" = 34 + 44 + 54 + ... = k~3 k4 = 90 -
(
1
4
7r
17
[subtract a I and a 2] = 90 - 16
1)
14 + 24
37. (a) j (x ) = ...;- is positive and continuous and f'( x ) =- ~ is negative for x > 0, and so the Integral Test applies.
X
00
X
.1
1
1
1
-
1
n~l n2 ~ sw = I2 + 22 + 32 + .. : + 102 ~ 1.549768.
Rw :::;
(b) 8 10 +
1
roo ...;dx = tlim
[- ]
X
--.oo
X
} 10
00
1
11
= lim
t
t --.oo
10
1
(-.!.t + 10 )
1
00
21 dx :::; S :::; SlO +
X
10
21 dx
X
=?
SlQ
=
1
, so the error is at most 0.1.
10
+ 1\ :::; S :::; 8 10 + fa
1.549768 + 0 .090909 = 1.640677 :::; s :::; 1.549768 + 0.1 = 1.649768, so we gets ~ 1.64522 (the average of 1.640677
and 1.649768) with error :::; 0.005 (the maximum of 1.649768 - 1.64522 and 1.64522- 1.640677, rounded up).
® 20 12 Cengage Lc:unint;. All RighiS Reserved. May riot be scanned, copied. or duplicntcd, or posted to n publicly accessible \\"<bsitc, in wbolc or in port.
62
0
CHAPTER 11
INFINITE SEQUENCES 1,\ND SERIES
(c) The estimate in part (b) iss ~ 1.64522 with error ~ 0.005. The exact value given in Exercise ~4 is 1r2 / 6 ~ 1.644934.
The difference is less than 0.0003.
1
(d) Rn ~ /.~ --;. dx = .!.. So Rn < 0.001 if.!. < - ,. x
n
n
1000
{::}
n > 1000.
39. f(x) = 1/{2x + 1) 6 is continuous, pos itive, and decreasing on [1, oo), so the Integral Test applies. Using (3),
R,.
~ 1 oo (2x + 1) - 6 dx = t~ [ 10(2; ~ 1) 5
1
10( n + ) 5 ~
2
1
4
S4
41.
1~ 6
1
(2n + 1)5 ~ 20,000
¢:}
1
1
1
r
l0( n\
2
) . To be correct to five decimal places, we want
15
n ~ ~ ( .e-'20,000- 1) ~ 3.12, ;muse n = 4.
{::}
1
= n~1 (2n + 1)6 = 36 + S6 + 76 + ge ~ 0.001446 ~ 0.00145.
f: n-1.oo1 = f: n 1 ~01 is a convergent p -series with p = 1.001 > 1. Using (2), we get
n=1
n=1
Rn < !.oo x-l.001 dx = lim [ x - o.oo1 ] t = - 1000 lim [-1-]t = - 1000(-- 1 -)
- n
t->oo x0.001
t->oo - 0.001 "
We want Rn < 0.000 000 005
{::}
1000
n 0.001 < 5 x 10 9
{::}
n0.001
n
n°·001 >
5
1000
10 9
x
= 1000.
n0.001
{::}
n > (2 X 10ll) 1000 = 21000 X 1011,000 ~ l.07 X 10301 X 1011.000 = l.07 X 1011.301.
43. (a) From the figure, a2 +as+···+ an ~
1 1
1
1
f(x)=-,-+-+-+
.. ·+ -1 ~
x2
3 4
n
Thus, Bn = 1 +
Jt f(x) dx, so with
y
1"
1
-dx=
ln n.
x
1
z1 + '31 + 41 + .. · + ;:;:1 1 + ln n.
~
(b) By part (a), s10o ~ 1 + ln 106 ~ 14.82 < 15 and
0
2
3
4 ·"
n
8109 ~ 1 + ln 109 ~ 21.72 < 22.
1
45. bin" = (e1" b) Inn = (e1" ") In b = nln b = n _ 1n b. This is a p-series, which converges for all b such that - ln b > 1
ln b < - 1
¢:}
b < e- 1
{::}
¢:}
b < 1/e [with b > 0].
11.4 The Comparison Tests
1. (a) We cannot say anything about I: an. If an > bn for all nand :L b,. is convergent, then :L an could be convergent or
divergent. (See the note after Example 2.)
(b) If an < bn for all n, then I; an is convergent. [This is part (i) of the Comparison.Test.)
3.
" a II n ~ 1, so L..~ - -n
b
.
'th ~ 1 , wh'1cI1 converges
n n+ < - n 3 = - 1 2 < 21 .or
2
3 - converges y companson WI nL..2n
2 3 1
2n
n
n=l 2n + 1
=l n
because it is a p-series with p = 2 > 1.
® 2012 Cengage Learning. A ll Ri£bts Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
SECTION 11.4
= -
nVn
p-series with p =
63
1
! ~ 1.
00
9"
< -9" = ( -9 )" for all n > 1. L::
3 + 10n
10"
10
- . n=l
7.
D
~ n + 1 d'1verges by companson
·
··h L~ 1 , wh'1ch d'Jverges because 1t· 1s
· a
for all n 2: 1, soLw1t
1
1
Jn
n=l n vn
n=l vn
n +1 > - n5. nVn
THE COMPARISON TESTS
oo
'9n
(for is a convergent geometric series (lrl'= fo < 1), so L::
3 + 10
n
n=l
converges by the Comparison Test.
·
9. -lnk > - for all k 2: 3 ['
smce ln k• > 1 "k
.or .>
_ 3] , so ~
LJ -lnkd.
k
1verges by compan·son w1t'h~
LJ -k , wh'h
1c d'1verges because 1t
k
k
k =3
k=3 •
1
1
is a p-series with p
=:1 ~ 1 (the harmonic series). Thus, f lnkk diverges since a finite number of terms doesn't affect the
k=l
I
convergence or divergence of a series.
11 .
{fk
Jk3 + 4k + 3
<
ifk
kl/3
1
00 •
{fk
.
. 00 1
~ = k
= k7 16 for all k 2: 1, so
converges by compar1son w1th
,
312
3
716
.v k3
k= l ,jk + 4k + 3
k=l k
L::
L::
· which converges because it is a p-series with p = ~ > 1.
13.
arctann
7r/2 fior aII n 2: 1, so ~
arctann
.
1
hi h
.
1.
< ""1.2
LJ
~,
converges by comparison
w1t. h -1r ~
LJ ""1.2• w c converges because 1t is a
2 n=l n
n 2 . n
n =l
n 2
constant times ap-series with p = _1 .2 > 1.
15.
4''
+
n _
1
00
00
(4)" a
3
4 . 4" = 4 (4)"
> 3n""
3 for all n 2: 1. n~l 4 (4)"
3 = 4 n~l 3
2
00
4n+l
n=l
3"- 2
is divergent geometric series (lrl = ~ > 1), so
L:: - -- diverges by the Comparison Test.
n:
1
1
17. Use the Lirni~ Comparison Test with an = JTi2'+T and bn =
lim an = lim
n
n-oo Jn 2
n-oo bn
+1
1
= lim '
n-<oo )1
+ (1 fn2)
= 1 > 0. Since the harmonic series f .!. diverges, so does
n=l
n
1
00
L::l Jn2+1'
n=
n +1
4
1 4"
. . Companson
.
TiestwJtlan·
.I
19. Use the L1m1t
= - +
andbn = - "n :
.
1 + 3n
3
1 +4"
. lim-=
.
a..
.
1 +,-,3"
. 1 +4" ·3"
. 1+
.
( -1 +1 ) · -1 -=1 > 0
lim-- = hm
- = hm
-4"
· - -3"
- = hm
n__.ob bn n__.oo !._ · n-oo 1 + 3" 4" n__.oo 4"
1 + 3"
n-.oo 4"
.I_ +
~
~
'
1
1r diverges, so does n=l
f 11 ++ 34: . Alternatively, use the Comparison Test with
Since the geometric series 2::: bn = 2::: (
1+4"
.
-1+4"
- - >- - > - (4"- ) = -1(4)"
or use the Test for Divergence.
1 + 3"
3n + 3"
2 3"
2
3
© 201 2 Cengoge Le:uning. All R;ghL< Resel'\'ed, May not be SCIUUled, copied, or duplicated, or posted to a publicly acc'cssihlc website, in whole or in pM.
64
D
CHAPTER 11
INFINITE SEQUENCES AND SERIES
21 . Use the Limit Comparison Test with an =
and bn =
; :
n 12
.
n 1 Jn+2
. (n 1 Jn+2)l(n 1 vn)
. ) 1 +2ln = -/1 = -1 > 0
= n--.oo
hm
= lim
= lim
2n2 + n + 1
n--.oo
(2n2 + n + 1)ln 2
n--.oo 2 + 1ln + 1l n 2
2
2
·
3 2
.
an
n--.oo bn
hm -
Since
f+"2
2n +n+ 1
3 2
3 2
f: ; is a convergent p-series [P = ~ > 1], the series. n=f:l 2nf+"2
also converges.
+ n+ 1
n=l n 12
5+ 2n
1
23. Use the Limit Compa'rison Test with a,. = ( + n 2 )2 and bn = n 3 :
1
*
li
n (5 + 2n)
l'
5n + 2n
1ln
l'
+ 2 . 2 0 s·
~ 1 .
lim an
n --.oo bn. = n--.~ ( 1 + n2)2 = n~ (1 + n2)2 · l l (n2 )2 = n!...~ (.l;,. + l )2 =
> . mce n'S::l n 3 IS a convergent
3
3
4
4
...
. [p
p-senes
. L.J
~ ( 5 + 2n) a lso converges.
= 3 > 1], t he senes
n.=l 1 + n 2 2
n2
1
oo ../n4 + 1
diverges by comparison with
) = - - for all n ~ 1, so I:
n n+ 1
n+ 1
·
n=l n 3 + n 2
2(
f: -
f:
1
- =
~.which diverges because it is a p-series with p = 1 ~ 1.
n.=ln+1 n=2n
2
27. Use the Limit Comparison Test with an = (1 +
2
~)
e- n and bn = e-": lim abn = lim .(1 + ~) = 1 > 0. Since
n
n
n
n-oo
n-oo
2
f: e-n = n=l
f: ·e:. is a convergent geometric series [lrl = ~ < 1], the series n=l
f: (1 + ~)
e-n also converges.
n=l
n
E-
1
1
- is a convergent geometric
29. Clearly n! = n(n - 1)(n - 2) · · · (3)(2) > 2 · 2 · 2 · · .. · 2 · 2 = 2''- 1 so..!_ < - -.
.
'
n! - 2n-l n =1 2 n -1
series [lr I
= ~ < 1], so n=l
E --\n . converges by the Comparison Test.
31. Use the Limit Comparison Test with an == sin ( ; ) and bn = ; . Then I: an and 2: bn are series with positive terms and
.
an
.
I1m - = 1rm
n--.oo bn
n--.oo
sin(1ln)
1ln
. sinO
.
~
.
.
.
.
_= hm - - = 1 > 0. Smce L.J bn IS the divergent harmontc senes,
0 --+D
0
n=l
00
2: sin (1 l n) also diverges.
n= 1
[Note that we could also use !' Hospital's Rule to ev~luate the limit:
·
. sin(ll x) H •
cos(l/x) · ( -l/x 2 )
1
.
hm
I
= %-oo
lim
I
= x_____.
hmoo cos -X =cos O= 1.)
1 X
-1 X 2
x - oo
lO
1
1
1
1
1
1
33. n~l ~ = .J2 + ..ff7 + v'82 + · · · + .JI0,05I ~ 1.24856. Now ~ <
00
Rw ~ Tw ~
1
10
1
1
.
..JTiJ = n 2 , so the error IS
1 ] t = lim ( - 1 +-1 ) = -1 = 0.1.
21 dx = lim [- X
t --.oo
X 10
t --.oo
t 10
10
© . 20 12 Ccngage Leaming . All Ri(;hts Reserved. Mny no t be scanned, copied, or duplicated, or posted ton publicly accessible website, in who le or ln pan.
SECTION 11.5
35.
10
cos2 1
cos2 2
cos2 3
cos2 10
cos2 n
n =l
5
5
5
5
5"
ALTERNATING SERIES
0
65
1
.
so the error IS
- 5n'
I; s- ncos2 n = ~ + - 2- + - 3- + .. · + - -10 - ~ 0.07393. Now - - < -
R 10 ~ T10 ~
l
oo
10
. j·t 5
1
-dx= lim
t~oo
5"'
10
-x
-1o) = 5101ln < 6.4 x 10_8.
5
5
.
[ 5- "' ] '
. ( 5- t
dx= lim - ln
= lim - + ln S
5 10 t~ oo
1n 5
t~oo
.
~
.s a convergent
.
. senes
. (I r I = 1 < 1) , o.d 1 d 2 d 3 . .. = ~
< - 9 for each n, and smce
L.., - 9 1
geometnc
L.., - dn
10
10" - 10"
n=1 10n
n = l 10"
37. Since -d,.
will always converge by the Comparison Test.
39. Since I: a.. converges, lim an = 0, so there exists N such that ian - 01 < 1 for all n > N
n->oo
=> . 0 ~ ar,. < 1 for
=> 0 ~ a~, ~ an. Since I; an converges, so does I; a~ by the Comparison Test.
all n > N
41 . (a) S ince lim an = oo, there is an integer N such that abn > 1 whenever n > N. (Take M = 1 in Definition 11.1.5 .)
-oo~
n
.
Then an > bn whenever n > Nand since I: b,. is divergent, I; a" is also divergent by the Comparison Test.
. If an = - 1 and bn = -1 fior n >
X
(b) (1)
_ .2, t hen I'1m -an
b = lim. -. n- = lim -ln
=II lim - 1
n
Inn
n- oo
n ........ oo 1n n
n
X
X--+ 00
1
x-oo 1 X
= lim x = oo,
x-oo
~ 1 . .
so by part (a), L.., . -ln IS d1vergent.
n=2
n
.. If an = -ln n and bn = -, then ~
·s the d'1vergent h armomc
· senes
· an d I'1m -b
an
(11)
L.., bn 1
1
n
n
n=l
n--+oo
n.
lim Inn = l im lnx = oo,
,,_ oo
x-oo
00
so I: an. diverges by part (a).
n=l
.! .
43. lim nan = lim a/,. , so we apply the Limit Comparison Test w ith bn =
Since lim nan > 0 we know that either both
n--+oo
n - oo 1 n
n
,,_00
series converge or both series diverge, and we also know that
.
f .! diverges [p-serics with p = 1]. Therefore, I; an must be
n =l n
divergent.
45. Yes. Since
I: an is a convergent series with positive terms, lim an= 0 by Theorem 11.2.6, and I: bn =I: sin( an) is a
n ~oo
series with positive terms (for large enough n). We have lim bn = lim sin(an) = 1 > 0 by Tbeor~m 2.4.2
n-oo an
n --+oo
an
[ET Theorem 3.3.2]. Thus, l::bn is also convergent by the Limit Comparison Test.
11.5
Alternating Series
1. (a) An alternating series is a series whose terms are alternately positive and negative.
(b) An alternating series
f; an= n=l
f; (-l)"- bn, where bn =ian I, converges if O < b,.+l ~ b,. for all nand lim b,, = 0.
1
n=l
n-+oo
(This is the Alternating Series Test.)
(c) The error involved in using the partial sum s.,. as an approximation to the total sum s is the remainder R ,. = s - sn and the
size of the error is smaller than bn+l; that is, IR~,I ~ bn+l· (This is the Alternating Series Estimation Theor~m.)
© 2012 Cc ngace Learning. AU Rights Reserved. Mny not be scanned, copied, orduplicntcd, o r posted loa publicly i1Cccssiblc website, in whole or In p:.rt
66
0
CHAPTER 11
INFINITE SEQUENCES AND SERIES
00
2 4 6 8 10
n 2n
2n
.
2
2
. 3. - - +- - - +- - - +···= :E{-1) - - .Now limb,. = l i m - - = lun
/ = - =fO.Smce
8
7
5
6
9
n=l
n+ 4
n-oo
n-oo n + 4
n-too 1 + 4 n
1
lim an =I 0 (in fact the limit does not exist), the series diverges by the Test for Divergence.
n-+oo
00
00
n= l
n=l
5. :E a,. = :E (-1f- 1
2n
1
+
1
00
1
n=l
2n + 1
= :E (-1)"- 1 b,.. Now b,. = -
> 0, {bn} is decreasing, and lim b,. = 0, so the
n-oo
series converges by the Alternating Series Test.
7.
E a,.= E(- 1)" 32nn +- 11 = f (-1)"b,.. Now lim b,. = lim ~-+ 11jnn = -32 =f 0. Since lim a,. =f 0
n=l
n=l
n=l
n-+oo
n-+00
n-+oo
(in fact the limit does not exist), the series diverges by the Test for Divergence.
00
00
1
00
9. :E a,.= :E ( - 1)"e-" = :E ( -1)"bn. Now b,. = n
n= l
n=l
n =l
e
> 0, {bn} is decreasing, and lim b,. = 0, so the series converges
n-+oo
by the Alternating Series Test.
n2
11. b,. = n 3 +
4
> 0 for n 2:: 1.
{bn} is decreasing for n 2:: 2 since
+ 4)(2x) - x 2 (3x 2 ) _ x(2x3 + 8- 3x3 ) _ x(8- x3 )
(x 3
(x3 + 4 )2
lim b,. = lim
n-op 1
n -+00
-
(x3 + 4 ) 2
- (x 3 + 4 )2 < 0 for x > 2. Also,
El (-1)"+1 ~converges
by the Alternating Series Test.
+ /4n/n 3 = 0. Thus, the series n=
n3 + 4
1
13. lim b~ = lim e2 fn = e0 = 1, so lim ( -l)n- 1 e21" does not exist. Thus, the series
n-+oo
ra.-oo
n-+oo
f: (-1)"- e 1" diverges by the
1 2
n=l
Test for Divergence.
15. a,. =
oo
:E
1
+ .,fti
=
(-1)"
1
1
+ .,fii," Now b,. = 1 + Vn > 0 for n 2:: q, {b,.} is decreasing, and ..~~~ b,. = 0, so the series
fon converges by the Alternating Series Test.
sin(n +.! )1r
n=O
17.
sin(n + ~)1r
1+
n~l ( - 1)" sin(~J bn =sin(~) > 0 for n 2:: 2 and sin(~) 2:: sin( n : 1 ). and ,.~ sin(~) = sinO = 0, so the
series converges by the AJternating Series Test.
n"
n·n···· ·n
19. - =
>n
n!
1· 2 · ·· · · n -
=>
n"
n!
l i m - =oo
n.-+oo
=>
lim
n ..... oo
( 1)" "
-
n.1
n
·
oo
by the Test for Divergence.
21.
The graph gives us an estimate for the sum of the series
E (-O.~)"
of -0.55.
n.
n=l
bs
n
does not exist. So the series :E ( -1}" ~, diverges
n=l
n.
(0 8)"
=~
: : : 0.000 004, so
-1
© 20U ~ngoge Learning. All Rights Reserved. Muy no1 bo scanned, copied, or duplicated. or po!lttd too publicly accessible website, In \\tlole or in port.
SECTION 11.5
ALTERNATING SERIES
D
67
~ ( -0.8t :.._
- ,;.., ( - 0.8t
'
~57- LJ
'
LJ
n=1
n.
n.
n=1
~ -0.8 + o.32 - o.oss3 + o.oi 706- o.oo2 731 + o.ooo 364- o.ooo 042 ~ -o.sso7
Adding b8 to s 7 does t:~ot change the fourth decimal place of 57, so the sum_of the series, correct to four decimal places,
is - 0.5507.
1
f: (- 1n);+satisfies (i) of the Alternating
Series Test because (
) < -i and (ii)' lim -i = 0, so the
·
n +1
n
n
1
23. The series
6
n=l
series is convergent. Now bs =
n -+ oo
1
= 0.000064 > 0.00005 and b6' = 661 ~ 0.00002 < 0.00005, so by the Alternating Series
56
Estimation Theorem, n = 5. (That is, since the 6th term is less than the desired error, we need to afld the first 5 terms to get the
sum to the desired accuracy.)
25. The series
f: (- 1)",·n. satisfies (i) of the Alternating Series Test because 10."+ltn + 1 )'. < -10"1 n 1 and (ii) lim -10"1- n ! = 0,
n =O 10"
n-ioo
! ~
so the series !s convergent. Now bs =
10 31
0.000 167 > 0.000 005 and b4 =
! =
10 41
0.000 004 < 0.000 005, so by
the Alternating Series Estimation Theorem, n = 4 (since the series starts with n = 0, -not n = 1). (That is, since the 5th tenn
is less than the desired error, we need to add the first 4 terms to get the sum to the desired accuracy.)
1
1
27. b4 = I = - - :::; o.ooo 025, so
8.
40,320
00
(- 1)"
3
(-1)" '
1
1
1
n~1 (2n)! ~ 53 = n~l (2n)! = -2 + 24 - . 720 ~ ~0 .459722
Adding b4 to S3 does not change the fourth decinlal place of 5s , so by th~ Alternating Series Estimation Theorem, the sum of
. the series, correct to four decimal places, is - 0.4597.
72
29. b7 = 107 = 0.000 004 9, so
00
"'
n~
1
(-
6 (-1)n- ln2
1)n- ln2
"'
10"
:::; 56 = n-7;:
10"
=
1
1
10 -
4
9
100 + 1000 -
16
2s
10,000 + 100,000 -
36
·
1,000,000 = 0 ·067 614
Adding b7 to 56 does not change the fourth decimal place of 56, so by the Alternating Series Estimation Theorem, the sum of
the series, correct to four decimal places, is 0.0676.
I
00
31.
I:
n= 1
(
-1)n-l
1
1
1
1
1
1
1
= 1- - + - - - + · · · + - - - + - - - + · · · . The 50th partial sum of this series is an
n
2
3
4
49
50
51
52
underestimate, since "~ ( - ~n-l = 5so + (
1
1
1
1
1
) + ( ; ) + ; · ·, and the te~s in parentheses are all positive. '
52
54
51
3
The result can be seen geometrically in Figure 1.
1
33. Clearly bn = - - is decreasing and eventually positive and lim bn = 0 f~r any p. So the Series converges (by the
n+p
n -+ oo
Alternating Series Test) for any P. for which every bn is defined, that is, n + p :/:- 0 for n 2 1, or p is not a negative integer.
® 2012 Ccngnge Learning. All Rights Reserved. May not be scanned, copied, or dupJicatcd, or posted to a publicly actcss ible website, in whole or in part.
68
CHAPTER 11
D
INFINITE SEQUENCES AND SERIES
35. 2: b2 n = 2: 1/(2n)2 clearly converges (by comparison with ~e p-series for p = 2). So suppose that 2: (-1r-I bn
.
.
1
converges. Then by Th~orem 11.2.8(ii), so does 2: [(-1 1 bn + bn] ::: 2 ( 1 + + + · · · ) = 22: . But thi~
2n -1
.
l i
t-
diverges by comparison with the harmonic series, a contradiction. Therefore, 2: (....:1)"- 1 b,. must diverge. The Alternating
Series Test does not apply since {bn} is not.decreasing.
11.6
Absolute Convergence and the Ratio and Root Tests
1. (a) Siilce lim Ian+l I = 8 > 1, part (b)
n-oo
an.
(b) Since lim I an+l I
n.-CX>
an
.
.
of the Ratio Test tells us that the series 2: an is divergent.
= 0.8 < 1: part (a) of the Ratio Test tells us that the series 2: an is absolutely convergent (and
therefore convergent).
(c) Since lim I an+l I = 1, the Ratio Test fails and the series 2: an might converge or it might diverge.
-~
3. lim
n-too
an
.
.
1 5
1
1 1
lan+l
l = n-oo
lim In+ · " 1 = lim 1.!. · n+ 1 =.!.lim + /n = .!. (1) = .!. < 1,sotheseries E ~is
an
5n+l n
n-too 5
n
5 n-too
1
5
5
n=l 5n
absolutely co.nvergent by the Ratio Test.
E (- t
1
1
5. bn = > 0 for n 2::' 0, {bn} is decreasing for n ~ 0, and lim bn = 0, so
converges by the Alternating
5n + 1
n-->oo
. n=O 5n + 1
Series Test. To determine absolute convergence, choose an = ~ to get
n
an = 1'tm / ( 1/n ) = l'rm' 5n +
1 = 5 > 0 , so ~
1 .d .
b .h L ' . C
.
,..
. h the
lim -b
~ - - 1verges y t e mut ompar1son . est w1t
n n-~ 1 5n + 1
n-+oo n
n=l 5n + 1
'
.
n~~
hannonic series. Thus, the series
E (- 1)" is conditionally convergent.
n=O 5n+ 1
(~_)'"+ ] = lim k +k 1(~)
= ~3 k~oo
lim (1+ .!..)
= ~'(1) = 1 < ·1,sotheseries
k(~)k
3
k
1
1
7. lim lak+Il = lim [(k-+: )
.
k~oo
ak
k-oo
1
k -too
3
'
'
3
E k ( ~) is apsolutely convergent by the Ratio Test. since the terms of this series are positive, absolute convergence is the
k
n=1
same as convergence.
4
1
9. lim 1an+ I I = lim [ (l.l)n+l4 . ~J = lim (1.1)n 4 ·= (1.1) lim
· = (1.1) lim (
~~ )4
n --<oo
an
n-too (n + 1)
(1.1)n
n-+oo (n + 1)
~-oo (n + 1)4
n -o oo 1 +
n
= (1.1)(1) = 1.1 > 1,
so the series
f: (- 1)n (l.~)n
diverges by the Ratio Test.
n
n= l
el/n
e
3- ~ 3
11 . Since 0 ~ -
n
oo
(- 1)"el/n.
n=l
n3
2::
n
(
1)
= e 3
n
oo
1
and 2: 3
n=l n
·
'
00
e l /n
n= l
n
is a convergent p-series [p = 3 > 1], L: - 3- converges, and so
is absolutely convergent.
® 2012 Cengagc: Learning. All JUghts Reserved. May not be scan~. copied. or duplicated, or posled 10 a publicly accessible website, in whole or in p3l1.
SECTION 11.6
2
ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
D
69
1
.
. lan+l l
.
[
w n+ l
{n+1)4 "+ ]
(10 n + 1)
5
.
. 00
10"
13. Ji_~ ~ = J2...~ (n + 2) 42n+3 . . 10"
= n~ 42 . n + 2 = 8 < 1, so the senes n~l (n + 1)42~+1
,
is absolutely convergent by the Ratio Test. Since the.terms of this series are positive, absolute convergence is the same as
convergence.
15.
\
(-1)" arctann\
rr/ 2
.
oo rr/ 2
rr 00 1
.
. oo (-1)" arctann
=
<
,
so
smce
2:::
2::: 2 converges (p = 2 > 1), the giVen
senes 2:::
2
2
2 n =l n
n2
n
n= l n
n=l
n2
converges absolutely by the Comparison Test.
17.
1
1
f: (-Inn1)" converges by the Alte~nating Series Test s.ince lim -1nn
= 0 and {-ln } is decreasing. Now Inn < n, so
n
..= 2
- 1-
Inn
n -oo
> .!, and since
n
f:
.! is the divergent (partial) harmonic series, f: -1 1 diverges by the Comparison Test. Thus,
n=2.n
nn
n= 2
~
(-I1n
)".IS condi'
w twnaIIy convergent.
n=2
n
19.
00
00
icos{nrr/ 3)1
.
.
cos{nrr/ 3)
:5 11 and 2::: 11 converges (use the Ratto
Test), so the senes 2:::
converges absolutely by the
1
n.1
n.
n.
n= l
n=l
n.
Comparison Test.
.
21 . lim
n-oo
2
. -n + 1 =
y/i(lJ = lim
2 2 1
n-+oo
n
+
2
2
00
. 2::: ( n + 1 ) " IS
.
1+
+ 1/n
. absolutely convergent by the
/ 2 = -1 < 1, so the senes
hm
1 n
2
2n 2 + 1
n -oo 2
n =l
Root Test.
2
y'jaJ = n-...
lim " (1 + .!)" = lim (1 + .!)" = e > 1 [by Equation 7.4.9 (or 7.4*.9) [ ET 3.6.6] ],
n--..oo
oo
n
n
23. lim
n~ oo
1)"
2
00 ( 1 + so the series L;:
n =l
n
diverges by the Root Test.
.
.
100
10
. I-an+ll
. I(n + 1) C!1QO"+l ·
n! . I- lim
.
100
1)
. 100
25. hm
- - Inn
- (n
-+- lim
- ( 1 + -1 )100
10
a,. -n- co
(n + 1)!
n 0100" .-n-oo n+ 1
n
- n-co n + 1
n
n-oo
= 0 ·1 = 0 < 1
oo
nl00100"
n=l
n1
so the series 2:::
.
is absolutely convergent by the Ratio Test.
27. Use the Ratio Test with the series
1·3 ·5 1 ·3 ·5 ·7
n - 1 1 · 3 · 5 · · · · · (2n - 1)
_ co
,._ 1 1 · 3 · 5 · · · · · (2n - 1)
1 ·3
1 - 3 ! + - 5-!- 7!
+ .. · +(- 1)
{2n - 1)!
+···-n~l( - 1 ) ·
(2n - 1)!
(2n-1)!
I
lim l an+ll = lim ~ ( - 1)"~1·3·5 .... ·(2n -1)[2(n +1)-l]_
a,.
n-oo
(2(n+1) - 1]!
(- 1)"- 1 ·1·3·5·····(2n-1)
n - oo
=lim ~ (-1)(2n+l)(2n- 1)! 1 = lim _.!_=0 < 1,
n -oo (2n + 1)(2n)(2n - 1)!
n-oo 2n..
so the given series is ahsolutely convergent and therefore convergent.
® 2012 Ccngagc Lc.lm..ing. All Rights Resen·cd. May not be scanned. copiL-d. or dupllcaled. or posted to a publicly ncccssiblc: \\'Cbsilc. in whole or in pan.
70
INFINITE SEQUENCES AND SERIES
D
CHAPTER 11
00
00
2 · 4 · 6 · · · · · (2n)
(2 · 1) · (2 · 2) · (2 · 3) · · · ·. (2. n)
oo .2nn!
oo
.
.
n!
= n~l
nl
= .~ -;;;[ = n~l 2n, wh1ch diverges by the Test for
29 · n~l
1
Divergence since lim 2n = oo.
n -oo
I a,. I = lim 15n
+ 11 = -5 > .1, so the series diverges by the Ratio Test.
-4n + 3
4
..
lim an+l
. de fi mt1on,
31 . B y the recurs1ve
n-oo
00
b" cosn1r
n=l
n
•
33. The series I:
"
a n+l l
.
lirn - = 11m
n-+oo
I
an
n-oo
n--400
00
bn
n=l
n
1
2
= I: (-1 )" 2':., where bn > 0 for n ~ 1 and lim bn = - .
n -oo
I(- l)n+lb~+l
n
I . b · n = -1 (1) = -1. < 1, so the senes
. ~ b~ cos n 1r .
+
· ( ) b = 1
n
- 1 n ~
1
1m
n -~o oo
n --
n +1
2
2
LJ
n= l
n
iS
absolutely convergent by the Ratio Test.
3
1
1 3
1
35. (a) lim 1 / (n + ) 1 = lim
n 3 = lim
= 1. Inconclusive
n _,oo
1/n3
n -+oo (n + 1)
n ->oo (1 + 1/ n) 3
(b) lim
I(n2 ++ 1) · 2n" I= lim n 2+n 1 = lim ( -12 + -21n ) = -21 . Conclusive (convergent)
(c) lim
Iv~
·(
n+1 - 3
n - oo
n .... oo
(d) lim
n .... oo
1
n
n -t oo
V:
n -
1
n --+oo
1 = 3 lim ·;
n->oo
n = 3 lim
n +1
n -+ oo
v~/
I+Vn
= 3. Conclusive (divergent)
I
.Jn + 1 · 1 +-n21
1/n2 + 1
] = 1. rnconc lus1ve
.
- = lim. [ R
+1
- ·
1 + (n + 1) 2
Vn
n->oo
n 1jn2 + (1 + 1/n)2
37. (a) lim \ an+l
a,.
n ->oo
00
1
I= lim I(nxn+~)'
x--1 = lx l lim - - = lxl · 0 = 0 < 1, so by the Ratio Test the
+ . · n! I = lim 1
n +1
n +1
xn
series I; I
n = O n.
X"
n ->oo
n->oo
n -> oo
converges for all x .
(b) Since the series of part (a) always convergt<s, we must have lim xn, = 0 by Theorem 11.2.6.
n.- oo n.
0
1
1
1 . 1
1
1
661
.
39. (a) Ss = n~l n2" = 2 + 8 + 24 + 64 + 160 = 960 :::::: 0.68854. Now the ratiOS
= 2 (n: 1) form an increasing sequence, since
n+1
1'n+l -
rn =
n
2 (n+ 2) - 2 (n+ 1) =
in using s s is Rs ~
a6
1
_ lim T n =
n->oo
(n + 1) 2 - n(n + 2)
2 (n + 1)(n + 2 )
1/(6 · 2 6 )
_
1 112
1
= 192 :::::: 0.00521.
(b) The error in using s ,. as an approximation to the sum is Rn =
1
(n + 1) n < 0.00005
2
¢?
1
2(n + 1)(n+ 2) > 0. SobyExercise34(b), theerror
an+~ = (
1- 2
2
) n+I . We want Rn < 0.00005
n+12
¢?
(n + 1)2n > 20,000. To find such an n we can use trial and error or a graph. We calculate
(11 + 1 )2 11 = 24,576, so su =
£; n 12" :::::: 0.693109 is within 0.00005 ofthe actual sum.
n= l
© 20l2 Ccngo.gc Learning. All Rights Reserved. May nol be scanned, copied, or duplicttted, or posted to tt publicly accessible website, in whole or in p.1r1.
SECTION 11.6
ABSOLUTE CONVERGENCE AND THE RATIO AND ROOTTESTS
0
71
41. (i) Following the hint, we get that Iani < rn for n 2: N, and ~o since the geometric series 2:::'= 1 r" converges [0 < r < 1],
the series 2::=N lanl converges as well by the Comparison Test, and hence so does 2:::'= 1 lanl, so 2::_:'=1 an is absolutely
convergent.
(ii) If lim . y'j'aJ = L > 1, then there is an integer N such that y'j'aJ >.1 for all n 2: N, so lanl > 1 for n 2: N. Thus,
n-too
-
lim an "# 0, so 2:::'=1 an diverges by the Test for Divergence:
n~oo
(iii) Consider
f ~ [diverges] and f ~
[converges]. For each sum, lim y'j'aJ = 1, so the Root Test is inconclusive·.
n
n =l n
n=l
n--+oo
43. (a) Since 2:: an is absolutely convergent, and since la;t I ~ !ani and Ia~ I ~. I ani (because a;t' and a~ each equal
either an or 0), we conclude by the Comparison Test that both 2:: a;t and 2:: a;;- must be absolutely convergent.
Or: Use Theorem 11 .2.8.
(b) We will show by contradiction that both 2:: a;t and 2:: a~ must diverge. For suppose t hat 2:: a;t converged. Then so
'
would l::(a;t'- ~an) by Theorem 11.2.8. But l::(a;t - ~an) = 2:: [ ~(an + !ani)- ~an] = ~ 2:: Ian i, which
diverges because 2: a,. is only conditionally convergent. Hence, 2: a;t can't converge. Similarly, neither can 2:: a~.
45. Suppose that 2:: an is conditionally convergent.
(a) 2:: n 2 an is divergent: Suppose 2:: n a,. converges. Then lim n an = 0 by Theorem 6 in Section 11.2, so there is an
2
2
•
'
'1. ----+ 00
integer N > 0 such that n > N
=}
n
comparison with the convergent p -series
.
2
;
1
lanl < 1. For n > N, we have !ani < 2 , so 2: !ani converges by
.
n
n>N
I;: -.;.. In other words, 2:: an converges absoluiely, contradicting the
n>N n
assumption that 2: an is conditionally convergent. This contradiction shows that 2:: n 2 an diverges .
Remark: The same argument shows that 2: n'~' a .. diverges for any p > 1.
(b)
f (- lnl)nn is conditionally convergent. It converges by the Alternating
Series Test, but does not converge absolutely
.
n =2 n
1
·[by the Integral Test, since the function f (x) = - ln
.
00
1
dx
- = tlim
x 1n x
~oo
00
= n~
2
n~ nan
2
00
(
1t
X
X
is c~ntinuous, positive, and decreasing on [2, oo) and
dx- = lim [ln{ln x )] t = oo] . Setting an = (-l}n
I n for n ~ 2, we find that
x 1n x
t ~ oo
2
n n
-
2
-1)"
2 ln n converges by the Alternating Series Test.
00
(-1t-1
n= l
n
It is easy to find conditionally convergent series 2:: an such·that ~na n diverges. Tw~ examples are 2::
00
"~
1
(
-1)"-1
and
fo , both of which converge by the Alternating Series Test and fai l to converge absolutely because 2:: Ian i is a
p-series with p ~ 1. In both cases, 2:: nan diverges by the Test for Divergence.
© 20l2 Ccngage Le(1111 ing. All Rights Reserved. May not be scanned, copied, or duplicated. or posted to a publ ic ly occcssiblc website, in whole or in part.
72
D
INFINITE SEQUENCES AND SERIES
CHAPTER 11
11.7 Strategy for Testing Series
f (~)n is a convergent geometric series [lrl = t < 1), so f ~
n +3
1. -1- < -1 = (1)n
for all n > 1.
n+3n
3n
3
-
n=l
.
n=l
converges by the Comparison Test.
f
3. lim la..l = lim _2:_ = 1, so lim a n= lim (-1}" _2:_ does not exist. Thus, the series
(- 1)n __.!:_diverges by
n - oo
n-oo n + 2
n--<00
n -<oo
n+ 2
n=l
n + .2
the Test for Divergence.
5. lim Jan+II = lim
n-oo
00
an
.
1 2
I(n -5)n+l
+ ?
n
n-<oo
•
(-
(
.!)
5)" I = lim 2(n+ 1) 2 = ~
(1 +
n - oo
5n2
5 n-oo
n
llm
n 2 2 n-l
2
= ~(1) = ~ < 1, so the series
5 · 5
n2 2n-l
n~l ( _ )" converges by the Ratio Test.
5
7. Let f( x) =
Since
~ -Then f is positive, continuous, and decreasing on [2, oo), so we can apply the In..tegral Test.
xvlnx
J~
1°/x,] = j u- 112 du = 2u112+ C = 2 JinX+ C, we find
[d..,:'
1 x
u - ex
dx
xvlnx
r
f oo
~ = lim
d~ = lim [2J!nX] = lim ·(2 villt- 2 v1Ii2) = oo. Since the integral diverges, the
J2
X y lnx
oo J2 X y lnx
t_.oo
2 t-oo
t ....
given series
00
f
t
~ diverges.
n = 2 n vlnn
•
9. 2: k 2 e-k =
k=l
00
k2
.
.
2: ""'"'k· Usmg the Rat1o Test, we get
k=I e
= 12 .
11.
.e! .e!
=
< 1, so the series converges.
f: (~
+ ~, ) ~ E ~ + E (.!)n
·The first series converges since it is a p-series with p = 3 > 1 and the second
3
n
n
3
n=l
n= l
n=l
series converges since it is geometric with lrl = ~ < 1. The sum of two convergent series is convergent.
1
3
13. lim la"+ 1 1= lim J n+l (n+ )
n-oo
an . .
n--<00
(n + 1)!
2
-~1= lim 3 (n + 1) 2 =3lim n~ 1 =0 < 1,sotheseries f 3n~
2
3nn2
n--<00
(n + 1)n
n --<00
n.
'
n=l
2
n.
converges by the Ratio Test.
00
k~l
(6)k
2k-13k+l = k~l 23(6)k
k converges. It follows from Theorem 8(i) in Section 11.2 that the given series, k~l
k ,
00
00
kk
also converges.
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SECTION 11.7 STRATEGY FOR TESTING SERIES
17
lim
· n-oo
I I=
0
73
I
lim 11 · 3 · 5 .. · · · (2n- 1)(2n + 1) . 2 · 5 · 8 · .... (3n - 1) = lim 2n + 1
2 · 5 · 8 · · · · · (3n- 1)(3n + 2) 1 · 3 · 5 · · · · · (2n- 1)
n--+oo 3n + 2
an+ l
an
n--+oo
= lim 2 + 1/ n = ~ < 1 ,
n--+oo 3 + 2/ n
3
oo
1 . 3. 5 .. · .. (2n - 1)
n= l
2 · 5 · 8 · · · · · 3n - 1
so the series 2:::
19. Let f(x) =
(
) converges by the Ratio Test.
.
2 - ln x
?
lrin.
.
ln x
r . Then f 1 (x) = x 312 < 0 when lnx > 2 or x > e-, so r ts decreasmg for n > e2 .
2 .
vx
.
vn
=
. I's RuIe, lim Inrn
By I'Hosptta
n--+oo vn
I'tm
n--+oo 1/
(1/n )
2 Vn
=
1'tm
2r
n - oo vn
=,
0 sothe senes
. n~
( - 1)"Inn
r;;, converges by the
6-1
vn
Alternating Series Test.
2
21 . lim lan. l = lim IC- l )ncos(l/n2 )1= lim icos(1 /n ).1=cos O= 1, so the series
n-oo
n -'oo
n-oo
.
E(-l)'' cos(l/ n diverges by the
2
)
n =l
Test for Divergence.
23. Using the Limit Comparison Test with a n = tan(.!.) and b,. = .!., we have
n
.
n
lim an = lim tan(1/ n) = lim tan(l/ x) M: lim sec2(1 /x). (- 1/x2) = lim sec2(1/ x) = 12 = 1 > 0. Since
n --+oo
1/ n
:z:--+oo
1/x
:r -oo
-1/x 2
:z:--+oo
n-oo bn
00
00
n= l
n.=l
2::: bn is the divergent harmonic series, 2::: a,.. is also divergent.
· ~
li
an+l
1'
I (n + 1)!
c" 1
oo n!
. (n + 1)n! · e"
. n+ 1
25. Use t I1e Ratto . est. f"L-H>O
m - an- = .,._.00
un
< +t)~ · - 1 = 1rm
= 11m ~+t = 0 < 1, so :L ---:r
2
en
n.
ft. - 00
en + 2n+ln!
r&.-oc e n
n = l en
I
I.
2
2
converges.
27.
roo ln: dx = lim [- ln
./2
t - oo
X
k ln k
k ln k
(k + l)
..
- ----,3 < - k3
29.
.!.] t . [using integration by parts] M: 1. So
X -
X
X
ln k
.
. oo
= -k? , the gtven senes 2:::
·-
E ln:
converges by the Integral Test, and since
n
n= l
1
k In A:
k=t(k+ l)
•
3 converges by the Companson Test.
1
f a n = f (-1)" ___!_h
= f (- 1)" b,,. Now b,. = - - - > 0, {b,..} is decreasing, and lim b, = 0, so the series
cos n
cos1 n
n =l
n=l
"·=1
1
n -oo
converges by the Alternating Series Test.
.
1
0 r: W rtte
co~lt n =
~
Comparison Test. So
2
e" + e-"
1 '.
.
1
< -2 and ~
ts a convergent geometrt.c senes,
so 6~ - - .ts convergent by t he
6
e"
n = l cos11 n
n = l e"
E(- 1) " cos
___!_h is absolutely convergent and therefore convergent.
n
n=l
31. lim a~.; = lim
k-oo 3
k--+oo
oo
s"
k = 1, 3
+4
k
5k
+ 4 k = (divide by 4")
Thus, 2::: ~ diverges by the T<;st for Divergence.
'
<D 2012 Ccn~:tgc Lc:tmin{l- All Rights Rescn·cd. May no1 be scanned, copied, or dup1icau:d, or ('K)stl-d h) n publicly ncccs..'lihl~ wcbsih!, in whole or in rart.
74
CHAPTER 11 INFINITE SEQUENCES AND SERIES
0
33. n-+oo
lim
~ = n-+00
lim (~ )
n + 1
1
"2 jn
= lim [(
n-+00
1
)I
n +1 n
1
n2
t = li m (1· + 11n r = -e < 1,. so the series n=1
I: (·-n
-)
n + 1
oo
n-->oo
converges by the Root Test.
35. an = n 1; 1/n = n . ~ /n, so let bn = ; and use the Limit Comparison Test.
1
1
l .un -a, = lim
.
--= 1 > 0
n-+oo bn
n-+oo n1/n
f: 1
[see Exercise 4.4.6 1], so th~ series
,. diverges by comparison with the divergent harmonic series.
.
n=l n 1 11
.
00
37. lim ~ = lim (2 1/" - 1) = 1 - 1 = 0 < 1, so the series I: ( y/2 - 1)"' converges by the Root Test.
n-too
n-~oo
11.8
n-=1
.
Power Series
1. A power series is a series of the form :L:::'=o enx" = co
+c1x + c2x 2 + c3x + · ··,where x is a variable and the en's are
3
constants called the coefficients of the series.
More generally, a series of the form :L:::'=o c,,(x - a)" =eo+ c1 (x- a)+ c2(x- a) 2 + · · · is called a power series in
(x - a) or a power series centered at a or a power series about a, where a is a constant.
I
I
1
a n+1 = lim ~ (-1)"+
I = lim ( -1)--x
n+1 I = lim [(1 + -1) JxJ] = Jx J. By the Ratio Test, the
( (n+1)x"-H
)
n--+00
Un
n-tOO
- 1 71. nxn
n-H)O
n
n-00
n
'
lim
I
00
.
series I: (-1)"nxn converges when !x i < 1, so the radius of convergence R = 1. Now we' II check !he endpoints, that is,
n= l
x = ± 1. Both series
f: (- 1)"n(±1)" = f: (=F1)"n diverge by the Test for Divergence since lim J(=F1)"nl = oo. Thus,
n=l
·
n=l
TI.--+OO
the interval of convergence is I = (-1 , 1).
x"-, then' lim
. Ia,.+1
. I- x"+l · 2n -11 = !1m
. (2n
5. If an = - - 1 = hm
- - 1 Jx l)
n-+oo
an
n-+oo 2n + 1
x"
n-+oo 2n + 1
2n- 1
.
(2- 1/n
/ lxJ) = JxJ. By
= n-+oo
hm
2+ 1 n
·oo
~
oo
1
·
the Ratio Test, the series I: - - - converges when Jx! < 1, so R = 1. When x = 1, the series I: diverges by
2
1
n=l 2n - 1
n=l n -
.
.
1 - > -1 and -1 ~ -1 d'1verges smce
.
.It .ts a constant mu It .ipIe o fth e harmoruc
. senes.
.
companson
Wit. I1 6~ - 1 srnce
-6
n=1 2n
2n - 1 . 2n
2 n=l
n
When x = - 1, the series
1
) n converges by t~e Alternating Series Test. Thus, the interval of converge~ce is [-1, 1).
n=1 2n -1
f: (-
7. If an= x",, then lim lan+l l
n.
n -+ oo
an
=lim I(x"+\;
x -1 = !xi
+ ·. I=lim 1- +
n-+oo
n
n!
1 .
X"
n -+oo
n
1
1
lim - -
n-+oo n
+1
=lxl·
0 = 0 < 1 for all real x.
So, by the Ratio Test, R = oo and I= ( - oo, oo).
lim
n - oo
Ian+1 I=
. an
~~ =n.-oo
lim Ix(n + 1)2 1 =lim [JxJ (1 + .!.)2] = EJ.(1)2 = ~ JxJ. By the
2n2
n-oo 2
n
2
lim J (n + 1)2 x"+l . 2
2n+l
n x"·
n-CXJ
.
.
© 2012 Cengagc Learning. All Rishts Reserved. May not be scanned. copied. or duplicated. or posted to u publicly occcssiblc website:, in whole or in part.
SECTION 11.8 POWER SERIES
00
n2xn
Ratio Test, the series 2:: (-1)'" .- ,.- converges when
2
n=l
t lx l < 1
¢>
0
75
lx l < 2, so the radius of convergence is R = 2.
00
2:: (=F1)"'n 2 diverge by the Test for Divergence since
n= l
lim I (:fl) n n 2 1 = oo. Thus, the interval of convergence is I = ( -2, 2).
n -oo
11 . If an =
( - 3)"x"
n 312 , then
(- 3) n+l x n+l n I 1' I 3 (. n )3/21 3 1 I 1' . ( 1 )3/2
a,.+t
1'
I-a.,- I= lffi I (n + 1)3/2 · ( - 3)nx n = Im - n + 1 = tm 1 + 1/n .
3/2
.
11m
n-•oo
n~oo
n~oo
X
--
X
n-+oo
I
= 3lx l (1) = 3 lxl
00
By the Ratio Test, the series 2::
(
-3)"
'- x "' converges when 3 lxl < 1 ¢>
,.=1 nyn
(- 1)"
.
1
· I: ~ converges by the Alternating Series Test. When x = - %
, the series 2::
00
n=l
00
n
n
n=l
( p = ~ > 1). Thus, the interval of convergence is [-
S· the series
lx l < %
, so R = %- When x =
312
is a convergent p-series
S, t].
13.Ifan = (-1)" ~ . then lim l a n+t l = lim I
n~oo
4" ln n
xn+l
. _4" ln n i= E11im
1n n
=1.::1 · 1
n-+oo 4n+l ln(n + 1)
x"
4 n~oo ln(n + 1)
4
an
[by !' Hospital 's Rule] = 1: 1. By the Ratio Test, the series converges when 1:1 < 1
xn
00
x = - 4, 2:(-1)" n ln
4
n=2
00
n
2::
n= 2
=
¢:>
lx l < 4, so R = 4. When
00
00
[(-1) (- 4)t
1
.
1
1
1
= 2: -- . Smce lnn < n for n ~2. -- > - and 2:: - is tbe
" ln
4
1n n n n = 2 n
n
n =2 1n n
divergent harmonic series (without then = 1 term),
f -1 ' is divergent
by the Comparison Test. When x = 4,
.
n= 2 1n n
00
xn
00
1
2:: ( - 1)n n ln = 2:: (-1)" l-n , which converges by the Alternating Series Test. Thus, I = ( - 4, 4] .
4
n =2
n
n=2
n
15. If an =
I
I . ·1
+ )1 I= lx- 21 lim
. ( n +) 1 = lx - 21. By the
X- 2 "
n~ oo n + 1 2 + 1
(x - 2)"' , then lim
.
a n+t
(x - 2)"+1
n
)2
- - = lim (
· (
2
n
+1
n ~oo
oo
(x- 2)"
n=O
n2 + 1
Ratio Test, the series 2::
x = 1, the series
an
n-+ oo
n +1
+1
converges when lx- 21 < 1
2
[R = 1]
2
¢:>
-1 < x- 2 < 1
¢:>
1 < x < 3. When
1
f: (- 1)" +
converges by the Alternating Series Test; when x = 3, the series f: - - converges by
1
n +
n +1
n =O
n=O
comparison with the p -series
2
-
f: ~ [p = 2 > 1]. Thus, the interval of convergence is I = [1, 3]. ·
n= l n .
00
1
By the Ratio Test, the series .~
3"'(x + 4)"
Vn
converges when 3 lx + 41 < 1
¢:>
lx + 41 <
1 [R = t ]
¢:>
® 2012 Ccngoge l.e:~ming. All Rights Rcscm :d. M•y no1be scanned, copied, ur duplicated, or posted lo • poblicly accessible wcbsile, in whole or in pnrt.
l
76
CHAPTER 11
0
INFINITE SEQUENCES AND SERIES
-t < x + 4 < t <* - ¥ <; x < -
11
3 • When
~; = - ¥, the series f: (-1)" ~ converges by the Alternating Series
n=1
vn
-¥, the series 'f: ~ diverges [P = ~ ~ 1]. Thus, the interval of convergence is I = [- ¥, - ¥).
Test; when x =
n =l
vn
r, then lim .yljO:j = lim lx -n 2 = 0, so the series converges for all x (by the Root Test).
2
nn
19. If an = (x -
1
n-+oo
n-too
R = ooand I= (- oo, oo).
21. an=; (x - a)", where b > 0.
1
lim lan+ll =lim (n+ 1 ) lx -a.J''+
an
n->oo
.
bn+l
n->oo
b" n =lim (1 +-~) lx .- a l = lx - ai_
n lx -a!
n~oo
71,
b
b
By the Ratio Test, the series converges when lx ~ a l < 1
<*
ix- a l < b [so R = b] <*
- b < x- a < b <*
a- b < x < a+ b. When lx - al = b, H-tOO
lim ian I = lim n = oo, so the series diverges. Thus, I= (a- b, a+ b).
'U-tOO
1
~~~(X2-x -)
)"+ 1 =.lirr~ (n + 1) l2x -11--+ oo as n--+ oo
1n
1
23. lfa, = n! (2x - 1)", then lim jan+! I = lim I (n +
n --+oo
an
u.-oo
n.
for all x :f;
•
t. Since the series diverges for all :c # t, R = 0 and I = { ~} .
(5x- 4)"
25. If a, =
U.-H X)
n3
, then
I a,. I
CLn+l
.
I1m
- = lim
n->oo
n->oo
I(5x4)"+
(n + 1)3
1
3
I = lim l5x - 41 ( _ n _
n
(5 x - 4)"
n-> oo
n+1
)3= lim l5x - 41 (-1 +-11/n-)3
n ->oo
= l5x - 41· 1 = l5x - 41
~
oo
(5x- 4)"
n =l
n3
By the Ratio Test, L
converges when l5x- 41 < 1
<*
lx-
t I < t <* - ~ < x- t < ~ <*
~ < x < 1, so R = t· When x = 1, the series 'f: ~1 is a convergent p-series (p = 3 > 1). When x = ~.the series
n=1 n
f: (-~"'
converges by the Alternating Series Test. Thus, the interval of convergence is I= [~, 1].
n
n =l
x 'n
27. If a, = 1 . 3 . 5 . .. . . ( 2 n _ 1 ) , then ·
lim lan+l l
n->oc
I
lim
x"+l
~ = n~oo- 71--73--:::-5-· - -_-;(72n---,1")"(2::-n---cf--::17)
the Rat1o Test, the senes L
•
•
00
~
n= l 1 · 3 · 5 · · · · ·
(2·
n - 1
_1·3·5· . . . ·(2n - l)l_ r
_ lx_l _ _ 0<1 Tl
b
:c"
- n~oo 2n + 1 . lUS, y
) converges for all rea l x and we haveR= oo and I= ( - oo, oo) .
29. (a) We are given that the power series I:::'=o CnX" is convergent for x = 4. So by Theorem 3, it must converge for at least
-4 < x ~ 4. ln particular, it converges when x = - 2; that is, L~o c,( -2)" is convergent.
© ~0)2 Ccngogc Lcnming. All Ri ghts Rcscrn:J. f.. 1 ~y nol be scanned, copic:d. or Jupliculcd, or posted lit u puhl icly nccC'ssihlc. wc:bsit~. in whole or in fl:trt.
SECTION 11.8 POWER SERIES
0
n
(b) It does not follow that I:::'=o en( - 4)" is necessarily convergent. [See the comments after Theorem 3 about convergence at
t /(n4").]
the endpoint of an interval. An example is Cn = ( - 1
(n!)k n
31. If a,.=-(
) ' x , then
kn.
I
lim a n+ll = lim [(n + 1)!]k (kn)! lxl = lim
(n + 1)k
lxl
a.,
n ..... ex> (n!)k [k(n + 1)]!
,.....,CX> (kn + k)(kn + k- 1) · · · (kn + 2)(kn + 1)
....... oo
[ (n+1) (n+1)
.
(n+1)]
= J~ (kn + 1) (kn + 2) ... (kn + k) lxl
. [n+1 ] lim [ n -+ 1 ] · · · 1liD
n+1 ] IX I
l.t.m [ -
-
n-+ex>
= (
~
kn + 1 ,...... ex> kn + 2
J
lxl < 1
n-oo kn + k
lxl < kk for convergence, and the radius of convergence is R = kk.
¢:}
33. No. If a power series is centered at a, its interval of convergence is symmetric about a. lf a power series has an infinite radius
of convergence, then its interval of convergence must be ( - oo, oo), not [0, oo).
35. (a) If a,. =
r
(-1t X2n+l
) n+l, then
I(
n. n+ 1 1. 22
an+l
I I
li
I
x2n+3
.
,.:.,~ ---a,: = n .....~ (n + 1)!(n + 2)! 22n+3 .
nl(n
l
. + 1)'22n+l
.
x2n+l
(x)2 lim
= 2
1
,. ._..... oo (n + 1)(n + 2) = 0 for all x.
So J 1 ( x) converges for all x and its domain is ( - oo, oo).
(b), (c) The initial terms of J1(x) upton = 5 are ao = ~·
and a(j = -
xu
,
,
. The partial sums seem to
176 947 200
approximate J1 ( x) well near the origin, but as lx l increases,
we need to take a large number of terms to get a good
approximation.
2
3
4
5
2 1
2 2
37. S2n-l = 1 + 2x + x + 2x + x + 2x + · · · + x "- + 2x "= 1(1 + 2x) + x 2 (1 + 2x) + x 4 (1 + 2x) + · · · + x 2" - 2(1 + 2x) = (1 + 2x)(1 + x 2 + x 4 + ... + x 2 ~- 2 )
2
- x "
= (1 + 2:c)-11- [by (11.2.3) with r = x 2]
- X2
1 + 2x
--+ - - 2 as n--+ oo by (11.2.4),
1- X
when lx l < 1.
&:
1+2x
.
A Iso S2n = S2n-1 + x 2n --+ 1 +2x.
_ x 2 smce x 2n --+ 0"•Or Ix I < 1Th
. ere,ore,
Sn --+
_ x 2 smce
S2n .and S2n-1 both
1
1
1 + 2x
.
.
1 + 2x
approach - - as n --+ oo. Thus, the mterval of convergence IS ( - 1, 1) and f (x) = - - 2 .
. 1 -x2
1- x
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78
D
CHAPTER 11.
INFINITE SEQUENCES AND SERIES
y'jCnT = c !x! < 1 for convergence, or
39o We use the Root Test on the series I:; c..x" We need lim \/lc,x"l =I xi lim
0
n.- oo
n--too
lxl < 1/c, so R = 1/c.
41 . For 2 < x < 3, I:; cnx" diverges and I:; dnx" .convergeso By Exercise 11.2.69, I.:;( en+ dn) x" diverges. Since both series
converge for Ixi < 2, the radius of convergence ofi.:;(en + dn) x" is 2.
11.9 Representations of Functions as Power Series
00
00
n=O
n= l
10 If f(x) = I:; c..xn has radius of convergence 10, then!' (x) = I:; nc..x"- 1 also has radius of convergence 10 by
·
Theorem 20
1
3. Our goal is to write the function in the form - · , and then use Equation ( I) to represent the function as a sum of a power
1- r
1
1
00
00
series. f(x) = - - =
( ) = I:; (-xr' = I:; (-1)"x" with 1-xl < 1
1 +X
1 - -X
n =O
n= O
5o f(x) =
3
2 2( 1)
x/ = 32 (x)"
3
_ x =3
00
1
_
n~o
3
oo
or, equivalently, 2 fo
3
1
,.+1 x"
0
0
¢:?
lxl < 1, so R = 1 and I= ( - 1, 1).
The series converges when
lxl
3 < 1,
that is, when Ixi < 3, so R = 3 and I = ( -3, 3).
~o [- (~)
The geometric series ...
conver~es when 1- (~) 1< 1
2
2
2
] "
¢:?
~~ 1 < 1
2
¢:?
lxl < 9
¢:?
lxl < 3, so
R = 3and I= (-3,3).
The series converges when lxl < 1, so R = 1 and I = ( - 1, 1).
·
( )
1 +X
A second approach: f x :::; - - =
1- X
-(1 - x) + 2 = - 1 + 2 ( -1- )
1- X
1- X
~ x n = 1 + 2 LJ
~ x n.
= -1 + 2 nLJ
= O
n=l
A third approach:
f(x)=
1
1
+x =(l +x)(-- ) = (l +x)(l +x+x2 +x3 + ·o·)
1- x
1 -x
= (1 + x + x 2 + x 3 + .. o) + (x + x 2 + x 3 + x 4 + ooo) = I+ 2x + 2x2 + 2x3 + o.. = 1 + 2
f:
Xno
n=l
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SECTION 11.9
11. f (x) =
3
x 2 - x- 2
3
..,....------,-,-,---"7
(x- 2)(x + 1)
A
B
=- + - - =>
x- 2 x+ 1 ·
REPRESENTATIONS OF FUNCTIONS AS POWER SERIES
D
79
3 = A(x + 1) + B(x - 2). Let x = 2 to get A = 1 and
x = - 1 to get B = - 1. Thus
E (=)" _n=O
E(-x)n
3
_ _1_ _ _ 1_=~(
1
)1
=-~
x2- x - 2 - X - 2 X+ 1
- 2 1- (x/2)
1 - (- x)
2 n=O 2
=
f: [-~2 (1:)"
-1(-1)"']xn = E [(- 1t+l- n~l]x"
2
n=O
2
n=O
We represented f as the sum of two geometric series; the first converges for x E (- 2, 2) and the second converges for ( -1, ~) .
Thus, the sum converges for x E ( - 1, 1) =I.
13. (a) f(x ) =
1
(1+x) 2
d [ L.....
~ ( - 1)n x " ]
= ·-d ( --1-) =, - dx
1+x
dx n=O
~
[from Exercise 3]
~
= L (-1)"+1nx"- 1 [from Theorem 2(i)] = L (-1)"(n + l)x" with R = ~ .
n=l
n=O
In the last step, note that we decreased the initial value of the summation variable n by 1, and then increased each
occurrence ofn in the term by 1 [also note that (-1)"+2 = (- 1)" ].
(b)f(x)=
1
1
=-1:j_[
] =-1:j_[E(-1)"(n+ 1)x"]
(1+x) 3
2dx (1+x) 2
2dx n=O
~
[frompart(a)]
~
= -~ L (-1)"(n + 1)nxn- 1 = ~ L (- 1)n(n + 2)(n + 1)x" with R = 1.
n=l
n=O
x
1
2
2 1 ~
(c)f(x) = ( +x) 3 =x · (l+x) 3 = x · 2nJ; {- 1)''(n+2)(n+1)x"
1
0
2
[frompart(b)]
= 1: E (-1)"{n+2)(n+ l)x"+~
2 n=O
To write the power series with x " rather ~an ·x"+ 2 , we will decrease each occurrence of n in the term by 2 and increase
1 .CX>
the initial value of the summation variable by 2. This gives us 2 ..~ ( -1)"(n)(n- 1)x,. with R = 1.
2
15. f(x) = ln(5 - x) = -~~ = _1: ~~ =
5-x
5
1 - x/5
-1:1[f: (::\"]
5
n=O 5}
Putting x = 0, we get C = ln 5. The series converges for Jx/51 < 1
<=>
dx = C- ~ E
xn+l
= ·C..:...
x"
5n=o5"(n+ 1)
n=tn5n
f:
Jx J < 5, so R = 5.
00
1
1
17. We know that =
( ) = L (-4x)". Differentiating, we get
1 + 4X
1- - 4X
n=O
f(x) = {1+x4x)2 = · -4x.
for J-4xJ < 1
<=>
-4 2 = - X f;{ - 4)n+1 (n+1)xn = f;(-1)n4n(n+1)xn+l
(1 + 4x)
4 n=O
n=O
Jxl < ;}, so R = ;} .
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80
0
CHAPTER 11
INFINITE SEQUENCES AND SERIES
1
19. By Example 5, (
00
1- X
)2 = L: (n + 1}xn. Thus,
n =O
1+ X _
1
X
_ ~ (
} ,n
~(
n+l
!(x ) -_ (1x)2 - (1- x)2 + (1- x)2 - ~o n + 1 x + n~o n + 1}x
00
00
n=O
n =l
= 2::(n+1}xn+ E nxn
[make the starting values equal]
= 1 + f; [(n + 1) + n ]xn = 1 + f; (2n + 1}xn = f; (2n + 1}xn with R = 1.
n= l
n= l
<=> lxl < 4, so R = 4. The partial sums are 9 1 = ~.
16
The series converges when l-x2/ 16l < 1 <=> x 2 < 16
5
3
82
n=O
1
9
x
x
x
x
, 83 = 82 +
= s1 - 162
, 84 = 83 - 164 , 8s = S4 +
, .... Note that 8 1 corresponds to the first term of the infinite
165
163
sum, regardless of the value of the summation variable and the value of the exponent.
As n increases, 8n(x) approximates f better on the interval of convergence, which is ( - 4, 4}.
23. f(x) = In ( -1 +
-
x) ln(1 + x) - ln(1 - x) I 1 - + I - - I 1 ( ) + I -1 -
1- X
=
=I L~0(-1)nxn + n~o
=
- dx
dx
4
2
3
xn] dx = 1 [(1- x + x - x +x -
= 1 (2 + 2x2 + 2x4 + ·· ·) dx
=I f;
,
But /(0) = ln t = 0, soC = 0 and we have f(x) =
-
• .. ) + (1
-X
dx
- X
+ x + x 2 + x 3 +x4 + ... )] dx
2
2x2 " dx = C +
n=O
dx
=
1- X
+X
f; 22Xn n++ 1l
n=O
oo
zx2n+l
oo
1
E - - w i t h R = 1. If x = ±1, then f(x) = ± 2 E -2 1 ,
n = O 2n + 1
n =O n +
which both diverge by the Limit Comparison Test with b,. = .!. .
n
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SECTION 11.9
REPRESENTATIONS OF FUNCTIONS AS POWER SERIES
0
81
2x
2x 3
2x 5
The partial sums are s1 = l ' s2 = s1 + 3 ' S3 = s2 + 5 ' · · · ·
As n increases, s,.(x) approximates f better on the interval of
convergence, which is {-1, 1).
1- t
1-
when lt8 l < 1
I
t
n=O
. t
tB·n+2
- converges
j -1 - - dt = C + I: -8n--.
+ 2 The series for 1- t
00
25. _ t_ B = t . _1_ 8 = t f: (tB )" = f: t8n+l
=>
n =O
t8
1
8
n =O
It I < 1, so R = 1 for that series and also the series for t/{1 - t 8 ). By Theorem 2, the series for
¢}
_!_a dt also has R = 1.
1- t
x"
oo
x"+2
2
for lx l < 1, so x ln{1 + x) = I: {- 1t - 1 - -and
n
n= 1
n
oo
27. From Example 6,ln{ 1 + x) = E ( - 1)" - 1 n =1
I
oo
x"+3
.
x 2 ln(1 + x) dx = C + I: (-1)"' (
)' R = 1 for the series for ln{1 + x), so R = 1 for the series representing
n=1
n n+ 3
x 2 ln{1 + x) as well. By Theorem 2, the series for
I
2
x ln{l + x) dx also has R = 1.
1
29 · 1 +1x 5 = 1 - (- x·5 ) = f;(-x 5 )"=f:{-1)"xu"
=>
o
n =O
I
1
- -.. dx =
1 + x~
I=
0.2
£
. 0
n=
;· oo
oo
,&n+l
. Thus,
2:{- l )"'x5 " dx = C + 2:(-1)" -x
5n + 1
,.=o
n=O
1
[
- - 5 dx =
1+x
x6
x- -
6
x lt
.
] 0.2
+ -11 - ·· ·
{0.2)6
{0.2) u
6
11
.
.
= 0.2- - - + - -- -· ··. The senes IS alternating so if we use
0
.'
the fi rst two terms, the error is at most (0.2) 11/11 ~ 1.9 x 10-9 . So I ~ 0.2 - {0.2) 6/6 ~ 0.199. 989 to six decimal places.
31. We substitute 3x for x in Example 7, and find that
.
I
(3x?"+l
I ".32..+1 x2n+2
32n+ l x2n+3
j xarctan(3:c)d:v = x 2:(-1) n -1- dx= 2:: (- 1) n + dx = C+ E(- 1) ( )(2n )
00
n= O
0 .1
So
1
0
"'
00
2
1
n=O
00 .
2
1
n =O
..
2n -1- 1
[ 3 3
] 0 .1
33 G 35 7 37 !)
x arctan (3x) dx = .....:::.__ -~ +~-__E._+ ···
1 ·3
3·5
5·7
7·9
0
1
9
243
2187
= 103 - 5 X 10° + 35 X 107 - 63 X 1Q9 + .. ..
• •IS a 1ternatmg,
.
The senes
so 1'f we use three terms, t11e error •IS at most
2187
8
x
~ 3 .5 x 10- . so
63 109
1
9
x
Jr ·l x arctan (3x) dx ~ 103
x
+ 243
~ 0.000 983 to six decimal places.
5 105 35 107
0
<[) 10 12 Ccngagc Leming. 1\ ll Rip;hts Rcscn·cd. May not be scrumcd, copied, or duplicated, or poslt.'ll to a publicJy nccc.ssible website, in wbolc or in parL
-j- 3
82
0
CHAPTER 11
INFINITE SEQUENCES AND SERIES
5
1
x3
x
(0.2?
(0.2) 5
(0.2) 7
33 . By Example 7' arctan x = x - - 3 + -5
- -x7
+ · ·'
· so arctan 0.
2 =.
0 2 - -3- + -5- - -7- + ... .
The senes
. 1s
. aIternatmg,
. so 1'f we use t11ree terms, t 11e error ·ts at most -(0.2)
7
7
~ 0.000 002.
(0 2?
(0 2?
Thus, to five decimal places, arctan0.2 ~ 0.2- - · - + -·- ~ 0.197 40.
3
5
00
00
(- 1)"2n(2n- 1)x2 "
(-1t2nx2 "'
(- 1)"x 2"+2
X Jo (x) + XJo(X) +X Jo(x) = n~l
+ ~ 22"(n!)2 + n~O 22" (n!)2
22n(n!)2
2
,
2
1
00
1 1
2
(- 1)"2n(2n-1)x "
oo (- 1)"2nx ''
oo
(- 1t- 1 x 2"
2
2
= n~l
2 "(n!)
+ ,~1 22"(n!J2 + ,~1 22n- 2 [(n- 1)!] 2
2
00
_
-
00
00
(- 1)"2n(2n -1)x2"
(- 1t2nx2"
(-1)"(-1 )- 1 22n 2x 2"
r~l
22"(nl)2
+ n~l 22"(n!)2 + ,~1
2 2 "(n!)2
00
= ~ (- 1)" [2n(2n - 1)
+ 2n - 2 n ] x 2"
2
2 2
n~l
2 "(n!)2
,
2
= ~ (- 1)" [4n - 2n + 2n- 4n ]
2
22" (n.1)2
Ln=l
2n
X
=0
.i
Since 16 28 ~ 0.000062, it follows from The Alternating Series Estimation Theorem that, correct to three decimal places,
J
1
0 Jo (x) dx ~ 1 00
1;
Xn
37. (a) f(x) = n~O nf
+ 3~0 ~ 0.920.
I
::}
00
nxn- l
00
Xn-1
oo
x.,."
I: - 1 = f( x)
f (x) = •~1 ~ = •~1 (n- 1)!
n=O n
(b) By Theorem 9.4.2, the only solution to the differential equation df(x)/d.'C = f(:c) is f(x) = Ke" , but f(O) = 1, so
K = 1 and f(x) = e".
Or: We could solve the equation df(x) / dx = f(x.) as a separable differential equation.
I I= I(X+
39. If a,.. = Xn2' then by the Ratio Test, lim an+I
n
n.-oo
an
convergence, so R = 1. When
lim
n-oo
n
n+l )2 .
1
. lxl lim ( ....2::_1 ) 2 = lx l < 1. for
~21 =
xn
n----+oo
n+
x= ± 1, f Ix:n I= f ~which
is a convergentp-series = 2 > 1), so the interval of
n
(p
n= l
" =1
convergence for f is [-1, 1]. By Theorem 2, the radii of convergence off' and f" are both 1, so we need only check the
00
x"
endpoints. f(x) = I; 2
n=l n
00
nxn- 1
00
x"
=> . f'( x ) = I: - -2 - = I; - - , and this series diverges for x = 1 (harmonic series)
n=l
n
n=O n
+1
@ 2012 Cengage Lc:tminiJ. All Rights Reserved. May not be sc:~.Mcd, copied, or duplicated, or posted w n publicly accessible wcbsile, in whole or in part.
SECTION 11.10
TAYLOR AND MACLAURIN SERIES
00
D
n.-1
!" (x) = I: ~ diverges
n=l n+ 1
and conve rges for x = - 1 (Alternating Series Test), so the interval of convergence is [-1, 1).
at both 1 and - 1 (Test for Divergence) since lim ~ = 1 =/= 0, so its interval ofc~nvergence is ( - 1, 1).
n--.oo n+ 1
•
41. By Example 7, tan- 1 x =
7r
have 6 = tan
7r=~f
1
n=O
2n+ 1
y3
r.;• we
2
00
y3
(-1)"
y'3 n = O (2n + 1)3"
11.10
· X2n+ l
n (1/ v'3) n +l
~ ( )" (1)" 1 1
= n~O ( - 1)
2n + 1
= n~O - 1 3 y3 2n + 1' SO
1)
- 1(
00
L: (-1)" - - for lxl < 1. In particular, for x =
=2v'3f
n=O
(-1)" .
(2n + 1)3"
Taylor and Maclaurin Series
J<">(a)
oo
j <B>(5)
- - .
81
1. Using Theorem 5 with n~o b.. (x - 5)", bn = -----n!' so ba = -
3. Since j <">(o) = (n + 1)!, Equation 7 gives the Maclaurin series
oo
(n + 1) 1·
n=O
n.1
oo J (n ) (O)
L: - -1-x" = L
n.
n=O
. I
.
• x" =
oo
L: (n + 1)xn. Applying the Ratio Test with an= (n + 1)xn gives us
n=O
I . I n +1
. n +2 = IxI · 1 = IxI. For convergence, we must have lxl < 1, so the
hm -an+l
- = lim (n( + .2)x"+ll
)
= IxI hm
n-oo
an
n--.oo
xn
+1
n-+oo n
radius of convergence R = 1.
4
5.
n
f(n) (:c)
j<">(o)
0
{1- x)- 2
1
1
2(1- x)- 3
2
2
6(1 - x)- 4
6
3
24(1 - x)- 5
24
4
6
120
120(1- x)-
{1 - x)- 2 = f(O) + f' (O)x + !"(O) x 2 + f"'(O) x 3 + j< >(o) x4 + ...
2!
3!
41
= 1 + 2x + ¥x2 + ¥x3 + l224ox4 + ...
= 1 + 2x + 3x + 4x3 + 5x 4 + · · · =
2
f: (n + 1)xn
n=O
I a,.
lim a,.+l l :;= lim
n-+00
n-oo
I(n(n++2)xn+ll
= lxl lim n + 2 = lxl (1) = lxl < 1
1)xn
n+1
n --.oo
for convergence, so R = 1.
7.
n
J<n>(x )
/ {n ) (0)
0
sin 1rx
0
1
7rCOS 7rX
7r
2
sin 1rx
2
-1r
3
- 7r 3 COS 7rX
4
1r
s in 1rx
0
5
7r° COS7rX
7f
4
0
-7r3
5
sin 7rX = f(O) + !' (O)x + !" (O) x 2 + !"' (O) x 3
2!
3!
/ ( 4 ) (0) ~
!(5) (0) 5
+ -4! -x + - 5! -x +·· ·
7ra
3!
lim
a
n+l
7
5
7r 7
= 1rx - -7r31x3 + -7r/j
x - - x + ·· ·
5!
71
oo
= L (- 1)"
n= O
n--.oo I a,. I
7ro 5
x + ...
5!.
= 0 + 1rx + 0 - - x 3 + 0 + 3
2n+l
7r
x2n+l
(2n+ 1)!
·
7r2n+3 x2n+3
(2n + 1)1 I
7r2 x2
= nlim
·
= .,._,00
lim
=0<1
--. oo
(2n + 3) !
7r2n+l x 2n+l
(2n + 3)(2n + 2)
1
83
for all x, so R = oo.
..
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I
84
0
CHAPTER 11
INFINITE SEQUENCES AND SERIES
9.
n
t<">(o)
f (nl (x)
0
2"'
1
1
2"'(ln 2)
ln2
2
2"'(1n 2) 2
(In 2) 2
3
2"'(1n 2) 3
(In2?
4
2"' (In 2) 4
(In 2) 4
lim
n-oo
I a,. I= I (n +
= (In = <
a ..+!
lim (ln2)n+lxn+l .
n!
n- oo
1)!
(In 2)"x"
2
lim
) !xi
n-oo n -1-1
J<~> (o) = { 1
n
f (n)(x)
J (n) (0)
0
sinh x
0
1
cosh x
1
2
sinhx
0
3
cosh x
1
4
sinhx
0
n
J Cnl(x)
4
x - 3x2 + 1
J(nl (l)
-2
2
4x3 - 6x
12x2 - 6
3
24x
24
4
24
24
5
0
0
6
0
0
n
J(nl(x)
J(n)(2)
0
In x
ln2
1
2
1/x
-1/x2
- 1/2 2
3
2/x3
2/23
1 for all x , so R
=
oo.
x2n+l
00
so sinh x= I: (2 _ ) 1•
n=o n+ 1.
0 if n is even
11.
0
I
if n is odd
x2n+l
Use the Ratio Test to find R. If an = (
)I, then
2n + 1 .
r
I II r ·I
an+
x2n+3
(2n + 1)!
"~ ~ = ,..:.,~ (2n + 3)! · x 2 n+l
=0<1
I x · n~
r
2
=
1
(2n + 3)(2n + 2)
for all x, so R = oo.
f (n) (x) = 0 for n ;:: 5, so f has a finite series expansion about a = 1.
13.
0
1
15.
4
<!
- 6/x
5
24/x5
- 1
6
- 1·
= 0! (x - 1)
t<"l (1)
0
- 2
1
6
.
+ lT (x - 1) + 2f (x - 1)
2
24 ('
+ 3f
x - 1)3 + 24 (x- 1)4
41
4
3
= - 1 - 2(x- 1) + 3(x - t? -1- 4(x- 1) + (x- 1)
A finite series converges for all x , so R = oo.
f(x) = In x = f: t <n>?) (x- 2)"
n=O
n.
ln 2
1
-1
2
2
)3
= 0! (x- 2) 0 + 1! 21 (x- 2) 1 + 2! 22 (x- 2) + 3! 23 (x- 2
1/2
- 6/24
4
f(x) = x4 - 3x 2 + 1 = 2::::: - - (x - 1t
n=O
n!
+ 4J2d
~ (x - 2) 4 + ~
(x - 2)5 + ·· ·
5!25
= ln2 +
f:(-1)"+dn ~:)! (x -2)''
n=l
n.2
24/2 5
c:E> :!012 Cengnge Lc:unin~. All Rights Rcscn1:d. ~ tay not be SCnMCd. copk-d. or duplicated, or postct.l loa publicly ncccs.~iblc wcbsilt:, in whole or in part.
TAYLOR AND MACLAURIN SERIES
SECTION 11.10
.
r
lan+l l
J~ ~ = n~
=
l(-lt+2(x~2)"+
1
(n + 1) 2n+l
lx - 21 < 1
1- ,._n;.,
n2"
. ( - 1)"+ 1 (x- 2)" -
n
t<">(x)
t<"l(3)
0
e2"'
e6
1
2e2 "'
22e2"'
2e6
2ae2x
24e2"'
8e6
2
3
4
n)
l(- l)(x-2)n,_ r . (
(n + 1)2
- ":~ n + 1
lx-21
- 2-
for convergence, so lx - 21 < 2 and R = 2.
2
j(x)= e 2"'· =
17.
li
0
f J'"',n.(a)· (x- 3)"
n= O
e6
= 0! (x - 3)
4e0
0
2e6
4e 6
8e6
16e 6
+ l ! (x- a? +. 2! (x - 3)2
+ 3! (x - 3) 3 + 4! (x- 3) 4 + ...
16e6
=
oo
2n 6
n=O
n.
L -+<x - 3)n
19.
n
J.'"'(x)
j (n ) (7r)
0
cosx
- 1
1
- sin x
0
2
- cosx
1
3
sin x
0
4
COSX
- 1
= -1 + (x - 7r?- (x-7r)4 + (x - 7r)G
2!
oo
4!
{
= L (-1)"+1 X -'lr
(2n)!
n=O
lim an+l _ lim
n -oo
I a,. I-
n-oo
= lim
[I x(2n- +12)!
••-oo (2n
with a= 0 and M = 1r"+1, IRn(x) l ::=; (
.
.
6!
)2n
1r 2n+2 .
lx - 7rl2
(2n ) 1. ]
lx- 7rl2 "
+ 2)(2n + 1)
= 0 < 1 for all x, so R = oo.
7rn+l
17rx l" +l
1
) • Thus, IRn(x) l -+ Oas n -+ oo by Equation 10.
l )l lxl" + = (
.
n+ 1 1.
n+
So n.lim
R,.(x) = 0 and, by Theorem 8, the series in Exercise 7 represents sin 1TX for all x .
--+c:x:>
.
23. If f(x) = sinhx, then for all n, j <n+ll(x) = cosh x or sinhx. Since Isinh xl < lcoshxl ·= coshx for all x, we have
jJ<n+ ll (x)l ::=; coshxforall n. Ifdis any positive numberand lxl ~ d, then l!(n+ll(x)l ~ cosh x ~ coshd, so by
Formula9 with a= 0 and M = coshd; we have I.Rn.(x) ! ~ (~ :~! lxln+ . 1tfollowsthat IR.,(x) l -+ 0 asn-+ oofor
0
1
Ixi ::=; d (by Equation 10). But d was an arbitrary positive number. So by T heorem 8, the series represents sinh x for all x.
© 2012 Ccngage uaming. All Righi:< Resc" ·cd. Moy not be sconned, copied, or duplicated. or poSlcd lo • publid y ocecs.ible website, in whole or in pan.
85
86
0
CHAPTER 11
INFINITE SEQUENCES AND SERIES
25. {/1 -x = [1 +(-x)FI4 =
f (1n/ 4) ·(-x)"= 1 +t(-x)+ H-,2 ~)(-x) 2 + H-~),(-D(-x?
+ · ..
3.
n=O
= 1 _ !x + ~ (- 1)"- (-1)" · [3 · 7 · · · · · (4n - 5)]x"
4
n=2
4n · n!
1
00
1
3 · 7 .. · .. (4n - 5)
= 1- - x- I:
xn
4
n=2
4n. n !
and 1-xl < 1
27.
lxl < 1, so R = 1.
¢}
1
1
=
= -81 ( 1 +
(2 + x) 3
[2(1 + x/2)] 3
~2 ) - 3 = -18 n=O
f: (-n3) (~2 ) ". The binomial coefficient is
I
- 3) = ( -3)( -4)( - 5) ..... (-3 - n + 1) = ·c-3)( - 4)( -5) ..... [- (n + 2)]
( n
n!
n!
_ ( - 1)" · 2 · 3 · 4 · 5 · · .. · (n + l )(n + 2) _ ( - l)"(n + l )(n + 2)
2 ·n!
2
Thus
1
= ! ~ ( -l)"(n + 1)(n + 2) x" = ~ ( -1)"(n + 1)(n + 2)x" for ~ ~~ < 1
' (2 + x) 3
8 n=O
2
2n n=O
2"+4
2 .
X2n+ l
00
) ~
2n + 1 1·
29. sin x= L::(-1)" (
n=O
31. ex =
oo
x"
oo
( 1fX) 2n+ 1
00
¢}
7r2n+ 1
lxl < 2, so R = 2.
) = L::(-1)" (
) x ~+ ,R = oo.
2n + 1 1· n = O
2n + 1 1·
f(x) =sin(1rx) = L::(- 1)" (
n=O
2
1
~ (2x)"
~ 2" x"
1 n
~ 2" n
~ 2" + 1 n
- = L.J => e2x = L.J
- ,so J( x ) =ex + e2o: = ~
L.J
- -x,
1 x + nL.J
1 x = nL.J
n=O
n.1
n=O
n1
n=O n.
=O n.
=O
n.1
EI
n = O n.
R =oo.
x2n
00
33. cosx =
I: (-1)" (2 ·),
n ·
n=O
35. We must write the binomial in the form (1 +expression), so we' ll factor out a 4.
X
:i;
X
X (
J4 + x 2 = y'4(1 + x2 /4) = 2 y'1 + x2 /4 = 2
X
X ~(
2
2 n= l
X
00
= - + - ~
_
-
- 2 + ..~ {-1)
. 2
37. sm x
1
1 •
X2)-l/2
X
oo
= 2 ,~0
+4
(-1)(X2)"
4
2
n
l )"' 1 · 3 · 5 · · · · · (2n- 1) X 2n
2" . 4" . n !
n 1 · 3 · 5 · · · · · (2n- 1) 2n+l
X
n! 23,.+ 1
x
and
2
!xj
4 < 1 ¢> 2 < 1 ¢>
lxl <2, soR = 2.
= !
- ! [ - oo (- 1)" (2x?" ] - ! [ - - oo (- 1)"(2x?"] = oo (-1)n+l22n- l x2n
(2n)!
- 2 1 1 ,~1
( n)!
n~l
(2n)!
'
2 (1 cos2x)- 2 1 ..~o
2
R = oo
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TAYLOR AND MACLAURIN SERIES
SECTION 11.10
D
87
1.5
- 1.5 t--r.'-----+---~
.. :--11.5
\ . T 8 = T9 = T 10 = T11 •
= 1- ~x'' + fixs- 7~oxl2 + ...
\ 't
.
The series for cos x converges for all x , so the same is true of the series for
\
f(x), that is, R = c;o· Notice that, as n increases, Tn(x) becomes a better
- 1.5
approximation to f(x).
oo
n
oo
(
)"
n
oo
X , so e - x = ""
.:._,SO
41 . e:r; (11)
= ""
L- I
L- ....=::___1 ·= ""{-1)"
Ln =O n.
n =O
n.
n =O
n1
f: (-1)" ~n. x"+l ·
f (x) = xe- "' =
n=O
The series for e"' converges for all x, so the same is true of the series
for f(x); that is, R = oo. From the graphs off and the fi rst few Taylor
polynomials, we see that T,. (x) provides a closer fit to f (x) near 0 a!; n increases.
7T
43. 5° = 5° ( 1800
7r
)
=
36
.
oo
x2"
radians and cos x = ..~o ( -1t (2n) ! = 1 -
x4
x2
x6
2f + 41 - 6f + ·· ·,so
.
7T
(7r/ 36) 2
(7r/ 36)4
(7r/ 36)6
(7r/ 36) 2
(7r/36)~
cos - = 1 - - -- +
+ ... Now 1 ~ 0 99619 and addmg
~ 2 4 X w-G
36
2!
4!
6!
.
21
.
4!
~ .
does not affect the fifth decimal place, so cos 5° ~ 0.99619 by the Alternating Series Estimation Theorem.
1
(b)sin- lx= l
dx=C + x +
V1- x 2
f: 1·3 · 5 ·· .. · (2n-1)x n+l
2
(2n + 1)2n · n !
n= l
= x + f; 1 · 3 · 5 · · · · · {2n - 1) x 2 n+l
(2n + 1)2". nl
n= l
(16)
since 0 = s in- 1 0 = C.
x 2n
00
47. COSX = "~ (- 1 )" {2n) !
0
I
(16)
49. cosx
I
oo
n
x2"
= .~o ( -1) ( 2n)!
COS X -
X
1
oo
n =l
3
oo
x2"
=> cosx - 1 = .~ ( - 1)" {2n)!
dx = C + 2:::{-1)
n
X 2n
2n
·(
X Gn+2
oo
x cos(x ) dx = C + .~o (- 1)" (6n + 2 )(2n)! , with R = oo.
1
.
cos x X
1
=
oo
2n - 1
n=l
(2n) !
2: {-1)" .::_____ =>
•
) , wtth R = oo.
2n 1.
@ 20 12 Ccngagc Lcunlin g. All Rights Ri:st:rvc:J. May not be scwmct.l. cupicd, or duplicated, or pO!'Itr.:d to a publicly IU:cc~ siblc website, in whole or in pan.
88
0
CHAPTER 11
•
INFINITE SEQUENCES AND SERIES
oo
x2n+l
00
x2n+ 4
51 . arctanx = 2: (- 1)" for lxl < 1, so x 3 arctanx = 2: (-1)" - for Jxl < 1 and
2n + 1
n=O
n =O
2n+ 1
I
oo
x2n+5
3
x arctanx dx = C + ,..;;:o ( -1)"' (2n + 1)( 2n + 5)' Since ~ < 1, we have
5
9
11
(1 /2)
(1/ 2r (1/ 2)
·
· {1/ 2)
1.5
-~
+ ~ ~ 0.0059
and subtracting T i l ~ 6.3 x 10- 6 does not affect the fourth decimal place,
so J0112 x 3 arctan x dx ~ 0.0059 by the Alternating Series Estimation Theorem.'
53. v'1 + x4 = (1 + .x 4 ) 112 =
00
2: (1/ 2) (x 4)", so
n
n =O
I
oo ( 1/ 2) x4n+1
}1 + x 4 dx = C + L:
- and hence, since 0.4 < 1,
,
n =O
n 4n + .1
we have
I=
1
0.4
o
00
} 1+x4 dx = E
n=O
(1/ 2) (0.4) 4n+l
n
4n + 1
(o.4) 1
~ (o.4) 5
H-4) (o.4) 9 H-t)(-~) (o.4) 13 H-4)(-~)(-~) (o.4) 17
=( 1) o r + l ! - 5 - +-2-,---9-+
3!
~+
4!
---rr- +· ··
(0.4) 5
= 0.4 + ----w- Now
5(0.4) 17
----;:;2 + 208 - 2176 + ...
(0.4) 0
(0.4) 1 3
5
(O~~)o ~ 3.6 x 10- < 5 x 10- so by the Alternating Series Esti~ation Theorem, I ~ 0.4 + (0~~) ~ 0.40102
6
6
,
(correct to five decimal places).
x -ln(1 + x)
x- (x - l x2 + l x 3 - lx4 + lx 5 - • • ·)
lx2 - lx3 + l x 4 - l x 5 + · · ·
= lim
2
3
., 4
fi
= lim 2
:l
4
5
x-0
x2
x -.0
x·
.,_,o
X2
55. l im
2
- ix3 + : .. ) = !
= m-+0
lim (!2 - l3 x + lx
4
5
2
since power series are continuous functions.
57. lim
x-o
sinx- x + l6 x 3
x5
since power series are continuous functions.
.
_59. From Equation 1 I, we have e-x
2
~
~
~
~
= 1 - I~
+ - - I + · · · and we know that cos x .= 1 - - + -41 - · · · from
21.
21.
1.
3.
.
2
Equation 16. Therefore, e-"' cosx = (1 - x 2 + ~x4 - ·- ·) (1 - ~x2 + i4 x 4 -
- - • ).
Writing only the tenns with -
2
2
4
2
2
4
4
degree -< 4 ' we get e-"' cos x = 1 - lx
+ .l.x
- x + lx" + lx + . · . = 1 - !!.x + ~x +- ·.
2
24
2
2
2
24
.
© 2012 Cc.ngace Learning. All Rights Rcscn·cd. May not be SCOJ:mcd, oopicd~ or duplicated, or po.stcd too publiCly accessible website, in whole or in part.
SECTION 11 .10
61 _ x_ <~>
· sin x -
x
x- tx + 1~0 x 5
3
TAYLOR AND MACLAURIN SERIES
0
· ···
1
I
2
7
4
+ 6x +aGo X + · · ·
x- * x3 + l~o xs - .. . j x
x - ~ x3 + rlo xs - .. .
l x3 _ ....Lxs
6
1
3
oX -
2
120
1 5
36X
+ .. .
+ .. .
4
From the long division above, ~ = 1 + ~x + 3 ~x + · · ·.
·
oo
4n
oo
I
SlllX
·
I
( - x~~) n
a
63. L; (- 1) " ~ = L; - -1 - = e- x· , by (I I ).
n=O
n.
n =O
n.
65. ~(-1)"-J
n=l
3 5
3
nn = ~(- 1)"_ 1 ( / )" = ln(1 + ~) [fromTable l ] = ln~
n5
n= 1
n
5
5
71. lfp is an n th-degree polynomial, then p<'l (x) = 0 for i> n, so its Taylor series at a is p(x)
Put x- a.= 1, so that x = a+ 1. Then p(a + 1) =
This is true for any a, so replace a by x : p(x
n.
p(i) (a)
i=O
t.
n
p(i)(x)
i =O
t.
+ 1) = iL:
-.-1 =O
t.
1
=>
(i ) ( )
2: - .-1 - .
73. Assume that lf" (x)l ~ M , so /" (x) ~ M for a ~ x ~a + d. Now fa"' !'11 (t) dt ~
f"(x) - J"(a) ~ M (x - a)
n.
= L; ~(x - a)i.
f"(x) ~ J"(a) + M(x - a). T hus,
J'(x) - f '(a) ~ J"(a)(x - a)+ ~M(x- a )2
J: M dt =>
I: J"(t) dt ~ I: [!"(a)+ M(t- a)] dt =>
=> J'(x) :::; /'(a) + J"(a)(x - a)+ ~M(x- a) 2 =>
.r: f(t )dt ~ .r: [!' (a) + J"(a) (t - a)+ tM(t - a?] dt =>
f(x) - f(a) :::; · f' (a)(x- a) + tf"(a)(x - a? + iM(x- a) 3 . So
f(x) - f(a) - J'(a)(x- a)- ~f"(a)(x - a) 2 ~ ~ M(x - a)3 . But
R 2(x) = f (x)- T2(x) = f (x) - f(a) - J'(a)(x- a)- ~f"(a) (x- a) 2 , so R2(x) :::; !M(x - a) 3 •
A sim ilar argument using f"'(x ) ~ -M shows that R2(x) ~ - t M(x- a) 3 . So IR2 (x2)l ::; tM lx - a j3 .
Although we have assumed that x > a, a similar calculation shows that.this inequality is a lso true if x < a.
© 2012 Ccngngc Lc:uning. All Righls Rcscr'\'cd. May nol be scanned, copied, or duplicated, or posted lo • publicly accessible wcbsilc. in whole or in pan.
89
90
0
CHAPTER 11
00
75. (a) g(x) = :L;
n~
INFINITE SEQUENCES AND SERIES
(k)n x"' => g'(x) = :L; (k)
nx"n
00
1
n~
00
(1 + x) g'(x) = (1 + x) :L;
(k)n nxn- = :L; (k)n nx"'- + :L; (k)n nxn
1
n=1
, so
1
00
00
n=1
n= 1
. .
I
=
f: (n+k1) (n + 1)xn + f: (k)nxn
n
n=O
00
_
-
n~o (n + 1)
00
= n~O
[
n=O
Replace n with n + 1 ]
in the first series
00
k(k-1)(k-2) .. ·(k-n+1)(k-n) n
[
k(k- 1)(k - 2)···(k-n+ 1) ] ,,
(n + 1)!
x + n~O (n)
nl
x
(n + l )k(k - '1)(k - 2) ... (k- n + 1)
(n+ 1)!
.
n
((k -n) _+n]X
= k n~o k(k - 1)(k- 2l!... (k- n + 1) x" = k
fo (~) =
x"
kg(x)
Thus, g'(x) = k g(x).
1 +x
(b) h(x) = (1 + x)-k g(x)
=>
h'(x) = - k(l + x) - k- 1g{x) + (1 + x) -k g'(x)
[Product Rule]
= -k(l + x)-i.:- 1 g(x) + (1 + x) - k kg(x)
[from part (a)]
1 +x
= -k(l + x)-k- 1 g(x) + k{1 + x)-"'- 1 g(x) = 0
(c) From part (b) we see that h(x) must be constant for x E (-1 , 1), so h(x)
Thus, h{x) = 1 = (1 + x)-k g{x)
11.11
<::>
= h(O) = 1 for x E ( -1, 1).
g(x) = {1 + x)k for x E ( -1, 1).
Applications of Taylor Polynomials
. Tn(x)
1. (a)
n
J <n>(x)
t <"'>(o)
0
cosx
1
1
-sinx
0
2
-cosx
- 1
3
sinx
0
4
cosx
1
5
-sin x
0
6
-cosx
-1
T,= r,
2
T6
-2
1
1
1- ~x2
1 - ~x2
1- ~ x2 + -f4x 4
1- ~x 2 + -f4x4
1 - ~x2 + -f4x4 - 7~oxs
@ 2012 COIJ8li£O Looming. All RigbiS Reserved. May not be scannod. copied. or duplioatcd, or posted too publicly dcccssible website, in wllolo or in pan.
SECTION 11.11
APPLICATIONS OF TAYLOR POLYNOMIALS
(b)
To =T1
T2 = Ts
T4 =Ta
Te
0.7071
1
0.6916
0.7074
0.7071
2
0
1
- 0.2337
0.0200
- 0.0009
1T
- 1
1
-3.9348
0.1239
-1.2114
f
X
1r
4
7f
(c) As n increases, Tn (x ) is a good approximation to f (x) on a larger and larger interval.
3.
n
f(nl(x)
j(n)(2)
0
1/x
- 1/ x 2
2
- 4
2/x3
4
3 · - 6/x
4
3
- 8
1
1
1
1
2
3
J (n)(2)
Ts(x) = 2::::: - - (x- 2)"
·n =O
n.1
1
1
= 2 - l
0!
=
1!
1
3
2!
3!
(x - 2) + 4 (x- 2? - j_ (x- 2?
t- H x - 2) + ~(x - 2) 2 - fa(x- 2) 3
5.
n
J<"l(x)
f(n)( 1T/ 2)
0
cosx
0
1
-sinx
- 1
2
-cosx
0
3
sin x
1
0 1-----~'1.;;--------j '•'fr
_!!__
2
-1.1
= - (X -
¥) + i (X - ¥)3
2
7.
f (n) (x)
J<"l(1)
0
ln x
0
1
1/x
n
2
- 1/x
3
2fx3
2
1
- 1
2
1
- 1
)2
2
3
= 0 + li(x - 1) + T!(x - 1 + 3!(x- 1)
= (x -1) -
tCx - 1) 2 + ~(x -1?
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D
91
I
92
0
INFINITE SEQUENCES AND SERIES
CHAPTER 11
9.
J<n>(o)
J<">(x)
n
2
xe- "'
0
0
1
2
(1- 2x)e- "'
1
2
2
4(x - 1)e- "'
-4
3
2
12
4(3 - 2x)e- "'
-4
11. You may be able to simply find the Taylor polynomials for
J(x) =cot x using your CAS. We will list the values of J<"J (rr / 4)
for n = 0 to n = 5.
-2
64 (X - 4,. )5
= 1 - 2 (X - 4,. ) + 2 ( X - 4,. )2 ·- 38(X - 4")3 + 310 (X - 4,.. )4 - 15
For n = 2 to n = 5, Tn (x) is the polynomial consisting of all the terms up to and inc luding the (X - .-i
13.
n
J<n>(x)
f {n)(4)
0
Vx
2
1
4
2
lx- 1/2
2
- l x - 3/2
3
1x-5/2
4'
1
(a) f(x) = Vx ~ T2(x) = 2 + !(x- 4) 4
= 2 + i(x - 4) -
M
2
/~
(x- 4?
2.
i4(x- 4) 2
(b) IR2(x)l ~ 3! Jx- 4J , where lf'"(x) l ~ M. Now 4 ~ x ~ 4.2
3
1
-32
8
lx- 41 ~ 0.2
=> Jx - 41 3 ~ 0.008. Since f'"(x) is decreasing
on [4,4.2], we can take M
IR2(x)l ~
(c)
1
r term.
3
= 1/"'(4) 1 = i4- 5 / 2 = 2~6 , so
/~56 (0.008) = 0~~~8 = 0.000015625.
0.00002
r - -- - - -- ,
From the graph of IR2 (x)l = lvx- T2(x)l, it seems that the
error is less than 1.52 x 10- 6 on (4, 4.2].
0 2012 Cengnse U:.vnins. All Rights Rescn-.:d. Moy not be: scanned. copied, oc duplicnted. or posted 10 o publicly accessible website, in whole or in·p:ut,
=>
SECTION 11.11
f (n) (x)
J (n){1)
0
x2/ 3
1
1x- l /3
3
1
2
3
2
_1x- 4/ 3
2
- g-
3
.§..x- 7/3
8
4
- ~ x- 10 / 3
9
27
93
= 1 + ~(x- 1) -·~(x-I?+ 841(x - 1)3
4
(b) IR3(x)l5 ~ lx - 11 , where Jj< 4 >(~)~5 M. Now 0.8 5 x 5 1.2
=?
lx- l l 5 0.2 ::} lx - 114 5 0.0016. Since jJ <4>(x) l is decreasing
27
on [0.8, 1.2], we can take M
81
~(0 8)-10/3
IR3(x) l 5 81 '
0.00006
(c)
0
. 2
2/9
8/27
(a) f(x) = x 13 ~ Ta(x) = 1 + ~(x- 1) - 2!(x - 1) 2 + 3!(x - 1) 3
15.
n
APPLICATIONS OF TAYLOR POLYNOMIALS
24
=I
j <4 >(o.s)l = ~(0.8) - 10 1 3 , so
(0.0016) ~ 0.000096 97.
From the graph of IRa(x) l = Jx 2/ 3 - Ta(x) I• it seems that the
error is less than 0.000 053 3 on [0.8, 1.2).
17.
n
f(n) (x)
J (n){O)
0
secx
1
1
secx tanx
0
2
secx (2sec2 x- 1)
1
(a) f(x) = secx ~ T2(x) = 1 + ~x 2
2
sec x tan x (6 sec x - 1)
3
(b) IR2(x) l 5
~ lx l3, where It <3>(x) l 5 M . Now - 0.2 5 x 5 ~.2 =:>
lx l 5 0.2
=?
lxl 3 5 (0.2) 3.
f(3 ) (x) is an odd function and it is increasing on (0, 0.2) since secx and tanx are i"ncreasing on [0, 0. 2) ,
I
so J C3>(x) l 5 J <3>(o.2)
3
~ 1.085 158892. Thus, IR2(x) l 5 J< >i~· 2) (0.2) 3 ~ 0.001447.
0.0004
(c)
· From the graph ofiR2(x)l = isec x - T2(x) l, it seems that the
error is less than 0.000 339 on [-0.2, 0.2).
0
19.
J<nl (x)
n·
0
e"'
1
2
1
e"' (2x)
2
e"' (2 + 4x )
3
ex (12x + 8x 3)
4
f (nl(o)
2
2
0
2
2
2
2
e"' (12 + 48x
2
0
(b) IR3(x)l $ ~ lxl , where Jt < >(x) J 5 M. Now 0 5 x $ 0.1
4
4
4
x 4 5 (0.1) , and letting x = 0.1 gives
IR3(x)l $ eo.Dl (12 + ~!8 + 0.0016) (0.1)4 ~ 0.00006.
+ 16x )
4
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=?
94
D . CHAPTER11
(c)
INFINITE SEQUENCES AND SERIES
0.00008
~------------~
From the graph of jR3(x) l = le~
2
- T3(x) l, it appears that the
y = IR,(x)l
error is less than 0.000 051 on [0, 0.1].
21.
n
J<nl(x)
t<" l(o)
0
xsinx
0
1
sinx+xcosx
0
2
2cosx- xsinx
2
3
- 3sin x- xcosx
0
4
- 4cosx + xsin x
- 4
5
5sinx + x cosx
(b) jR4(x) j ~ ~~ jxj , where I f(")(x)l ~ M. Now - 1 ~ x ~ 1
5
=>
jxj ~ 1, and a graph of j1 l(x) show; that I j <5 l(x) j ~ 5 for - 1 ~ x ~ 1.
5
5
r;
Thus, we can take M = 5 and get IJ4(x) l ~ 5T · 1 =
1
24
= 0.0416.
0 .009
(c)
From the graph of jJ4(x) j = jxsinx- T.1(x) j, it seems that the
I
error is less than 0.0082 on [- 1, 1].
23. From Exercise 5, cosx = - (x- ~) + Hx IJ< 4 l(x) l
~)3 + Rs(x), where jRs(x)l ~ ~ !x - ~ l4 with
= jcosxj ~ M = 1. Now x = 80° = (90° - 10°) = (~-fa-) = ~11' radians, so the error i.s
IR 3 ( 4; ' ) I ~ f.;_ (fa-) ~ 0.000 039, which means our estimate would not be accurate to five decimal places. However,
4
Ts = T4, so we can use !Rot'(-;) I ~ 1 ~ 0 (fa-t ~ 0.000 001. Therefore, to five decimal places,
25. All derivatives of e"' are e"' , so IRn (x)l ~ (n ~ I)! lxl"+ , where 0 < x < 0.1. Letting x = 0.1,
1
.R,,(O.l)
~ (neO
.l ) ' (0.1)"+1 < 0.00001, and by trial and error we find that n = 3 satisfies this inequality since
+I .
Rs(O. l ) < 0.0000046. Thus, by adding the four terrns of the Maclaurin series for e"' corresponding ton = 0, 1, 2, and 3,
we can estimate e0 · 1 to within 0.00001. (In fact, this sum is 1.10516 and e0 · 1 ~ 1.10517.)
@) 2012 Ccn~agc Lcamins;. All Rights Reserved. May not be scanned, cop ic~ or dupliC4ted, or posted to a publicly accessible website. in whole or in p:u1.
SECTION 11.11
1 3 +Six
1 s - · · ·. Bth
· senes
·
27. sinx = x- 3!x
y e A Iternatmg
APPLICATIONS OF TAYLOR POLYNOMIALS
~x3 is less than I ;, x 5 1 < 0.01
lx 5 l < 120(0.01)
¢:}
95 ·
0.9
r-------------~~~~--~
y=sinx+ o.oV
/
/
Estimation Theorem, the error in the approximation
sin x = x -
D
lxl < (1.2) 115 ~ 1.037. The curves
0.9
3
y=x-kx
/~
'fooox-
¢:}
y = x - ix3 andy= sin x- 0.01 intersect at x ~ 1.043, so
/
O.QJ
"-L,__L,.:...__
_ . __ _ _ _ _ __ __ . /
1.2
0.8
the graph confirms our estimate. Since both the sine function
and the given approximation are odd functions, we need to check the esti~ate only for x > 0. Thus, the desired range of
. values for xis - 1.037 < x < 1.037.
29. arctan x = x -
x3
3
+
x5
7
x
5 - 7 + · · · . By the Alternating Series
Estimation Theorem, the error is less than l -~ x 7 1 < 0.05
jx 7 j < 0.35
¢:}
¢:}
y =arctan x + 0.05
lxl < (0.35?1 7 ~ 0.8607. The curves
y = x- tx 3 + kx 5 andy= arctan x + 0.05 intersect at
x ~ 0.9245, so the graph confirms our estimate. Since both the
arctangent function and the given approximation are odd functions,
we need to check the estimate only for x > 0. Thus, the desired
-1
y = x - .!. x 3 + .!. xs
3
5
range of values for xis - 0.86 < x < 0.86.
31 . Let s(t) be the position function of the car, and for convenience set s(O) = 0. The velocity of the car is v(t) = s' (t) and th~
acceleration is a(t ) = s" (t), so the second degree Taylor Polynomial is T2(t) = s(O) + v(O)t + a~O) t 2 = 20t + t 2. We
estimate the distance traveled during the next second to be s(1) ~ T2(1) = 20 + 1 =21m. The function T 2 (t) would not be
accurate over a full minute, since the car could not possibly maintain an acceleration of 2 m/s 2 for that long (if it did, its final
speed would be 140 m/ s ~ 313 mi/ hl).
33. E =
~2 - (D!d)2 = ~2- D2(1:d/ D)2 = ; 2 [1 - (1+ ~r2l
We use the Binomial Series to expand (1 + d/ D ) - 2 : .
when D is much larger than d; that is, when P is far away fro m the dipole.
35. (a) If the water is deep, then 27rd/ L is large, and we know that tanh x -+ 1 as x -+ oo. So we can approximate
tanh(27rd/ L) ~ 1, andsov2 ~ f!L/ (27r)
¢:}
v ~ ..jgL j (21r).
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96
D
CHAPTER 11
INFINITE SEQUENCES AND SERIES
(b) From the table, the first term in the Maclaurin series of
j (n) (x)
f (nl(o)
0
tanhx
0
1
sech2 x
1
2
-2sech xtanhx
2
0
3
2sech2 x (3tanh 2 x- 1)
- 2
n
tanhx is x, so ifthe water is shallow, we can approximate
(c) Since tanh x is an odd function, its Maclaurin series is alternating, so the error in the approximation
3
3
2trd
2trd.
. . IJ"'(O)I (2trd)
1 (2trd)
tanh L ~ L IS less than the first neglected term, wh1ch IS - -, - L
=
.
3
3 L
~ ( 2~d) < ~ ( 2tr · 110 ) = ; 5 , so the error in the approximation v = gd is less
3
If L > lOd, then
3
3
2
.
than
.
gL tr3
7r •
~ 0.0132gL.
2
375
37. (a) Lis the length of the arc subtended by the angle B, soL = RB
B=L/R.NowsecB=(R+C)/R. =>
=>
RsecB=R+C =>
C = RsecB- R = Rsec(L/R)- R.
(b) First we'll find a Taylor polynomial n(x) for f(x) = secx at X = 0.
n
/(nl(x)
/(n) (0)
0
secx
1
1
secx tanx
0
2
secx(2tan x + 1)
1
3
secxtanx(6tan 2 x + 5)
0
4
secx(24tan x + 28tan x + 5)
5
2
Thus, f(x) = secx ~ T4(x) = 1 + ~(x-
4
2
o? + -fi(x- W = 1 + ~x + f4x 4 • By part (a),
2
(c) Taking L = 100 km and R = 6370 km, the formula in part (a) says that
C = Rsec(L/ R)- R = 6370 sec(100/6370)- 6370 ~ 0.785 009 96544 km.
£2
5£4
1002
5. 1004
.
. .
The formula m part (b) says that C ~ R + R 3 = .
+
~ 0.785 009 957 36 km.
.
2
24
2 6370
24 63703
The difference between these two results is only 0.000 000 008 08 km, or 0.000 008 08 m!
39. Using f(x) = Tn(x) + Rn(x) with n = 1 and x = r, we have J(r) = T1(r) + R 1(r) , where T1 is the first-degree Taylor ,
polynomial off at a. Because a= Xn, J(r) = f(xn) + f'(x n)(r- x,.. ) + R1(r). But r is a root off, so f(r) = 0
and we have 0 = f (Xn) + J' (Xn) (r - Xn) + R1 (r). Taking the first two terms to the left side gives us
© 2012 Ccngagc Learning. AU RighlS Rcscr.·cd. May not be scanned, copied. or duplicated, or poslci.lto u publicly accessible website, in whole or in pan.
CHAPTER 11 REVIEW
J'(xn)(xn - r) - f (xn) = R 1 (r). Dividing by f'(xn), we get Xn - r -
0
;,~:~) = :(~::).By the formula for Newton's
method, the left side of the preceding equation is Xn+I - r, so lxn+l - r l =
I·
I ;(~:}) Tay lor's Inequality gives us
IR 1 (r)l ::=; I J'~~r)llr - xnl2 • Combinin~ this inequality with the facts lf"(x) l ::=; M and IJ'(x)l ~/(gives us
lxn+l - rl ::=; 2M/( \xn - r \2 ·
11
Review
CONCEPT CHECK
1. (a) See Definition 11.1. 1.
(b) See Definition 11.2.2.
(c) The terms ofthe sequence {an} approach 3 as n becomes large.
(d) By adding s ufficiently many terms of the series, we can make the partia l sums as close to 3 as we like.
2. (a) Sec the definition on page 721 [ET page 697].
(b) A sequence is monotonic if it is either increasing or decreasing.
(c) By Theorem 11.1.1 2, every bounded, monotonic sequence is convergent.
3. (a) See (4 ) in Section 11.2.
(b) The
p-se~ies f: J.. is converge~! ifp > 1.
n=l 77,1'
4. If
2: a ,. = 3, then nlim
an = 0 and lim S n = 3.
- oo
n ---t ex>
5. (a) Test for Divergence: If lim a,, does not exist or if lim an =I 0, then the series 2:"':.. 1 an is divergent.
n---+oo
n --+ oo
n-
(b) Imegral Test: Suppose f is a continuous, positive, decreasing fun ction on [1, oo) and let an= f(n). Then the series
:2:;::'=1 an is convergent if and only if the improper integral f1 f(x) dx is convergent. In other words:
00
(i) If
f 1 f(x) dx is convergent, then 2:;::'=1 a, is convergent.
00
(ii) If f 1
00
f( x) dx is divergent, then I::;"=1 a,. is divergent.
(c) Comparison Test: Suppose that E an and
2: b,. are series with positive terms.
(i) If E b,. is convergent and an ::=; bn for all n, then E a,. is also convergent.
(ii) If E br. is divergent and an ~ bn for a ll n, then
97
2: an is a lso divergent.
(d) Limit Comparison Test: Suppose that E a,. and E bn are series with positive tenns. If lim (a ,/bn) = c, where c is a
n.- oo
finite number and c > 0, then either both series converge o r both diverge.
(e) Alternating Series Test: Ifthe alternating series :2:;::'= 1 ( - 1)n- 1 bn = b1 - b2 + b:3 - b4 + bo - ba + · · · [b,. > OJ
satisfies (i) bn+l :::; bn for all n and (ii) lim b,. = 0, then the series is convergent.
n-<oo
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98
0
CHAPTER 11
INFINITE SEQUENCES AND SERIES
(f) Ratio Test:
(i) If lim
n-oo
Ian+l )
I
(.ll") If 1"un -an+1
n -oo
= L < 1, then the series
an
an
f an is absolutely convergent (and therefore convergent).
no=l
I= L > 1 or 1" IO.n+l I= oo, then the senes
. ~ a n .IS d"Ivergent.
1m
--
n--+oo
an
I an I
(iii) If lim an+l = 1, the Ratio Test is inconclusive; that is, no
n-+co
Ln=l
c~nclusion can be drawn about the convergence or
divergence of 2:: an.
(g) Root Test:
(i) If lim
\li(i:j = L < 1, then the series 2::;:'= 1 a,. is absolutely convergent (and therefore ~nvergent).
(ii) If lim
\li(i:j = L > 1 ~r n--+oo
lim ~ = oo, then the series 2::;:'= 1 an is divergent.
(iii) If lim
~ = 1, the Root Test is inconclusive.
n -co
n-.oo
11.--+ 00
6. (a) A series L an is calledab~olutely convergent if the series of absolute values 2:: Iani is convergent.
(b) If a series L an is absolutely convergent, then it is convergent.
(c) A series L:al' is called conditionally convergent if it is convergent but not absolutely convergent.
7. (a) Use (3) in Section 11.3.
(b) See Example 5 in Section 11.4.
(c) By adding terms until you reach the desired accuracy given by the Alternating Series Estimation Theorem.
00
8. (a) 2:: en(x - a)"
n=O
00
(b) Given the power series L en(x- a)", the radius of convergence is:
n=O
(i) 0 if the series converges only when x = a
(ii) oo if the series converges (or all x, or
(iii) a positive number R such that the series converges if lx - al < Rand diverges if lx- al > R.
(c) The interval of convergence of a power series is the interval that consists of all values of x for which the series converges.
Corresponding to the cases in part (b), the interval of convergence is: (i) the single point {a}, (ii) all real numbers, that is,
the reaL number line ( -oo, oo ), or (iii) an interval with endpoints a - R and a + R which can contain neither, either, or
both of the endpoints. In this case, we must test the series for ·convergence at each endpoint to determine the interval of
convergence.
9. (a), (b) See Theorem 11 .9.2.
n f(i)(a)
10. (a) Tn(x) = L - .-1 - (x - a)'
i =O
~.
oo j (n){ )
(b) I: __a_ (x - a)"
n=O
nl
® 2012 Cengnge Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted ton publicly ncccssiblc website, in wbole or in part.
CHAPTER 11 REVIEW
oo f (n) (O)
(c) L
n=O
- -1 - xn
n.
0
99
[a= 0 in part (b)]
(d) See Theorem 11.10.8.
(e) See Tay.lor's Inequality ( 11.10.9).
11 . (a)-(f) See Table I on page 786 [ ET 762].
12. See the binomial series ( 11 .10.17) for the expansion. The radius of convergence for the binomial series is 1.
TRUE-FALSE QUIZ
1. False.
See Note 2 after Theorem I 1.2.6.
3. True.
If lim an
5. False.
For example, take en = ( - 1)" j (n6n) .
n - oo
= L, then as n -7+ 00, 2n + 1 -+ oo, so a2n + l -+ L.
. . I_+_
an I= lim
. I(n +1 1) · n3~1 1 = lun
.· I
.
1
. · -1/n31
- = hm
= 1.
a,.
(n + 1) 1j n
(1 + 1/ n)
1
7. False, smce hm
n-oo
n3
3
n - oo
3
n--+oo
9. False.
See the note after Example 2 in Section 11.4.
11. True.
See (9) in Section II . I .
13. True.
By Theorem 11.10.5 the coefficient of x 3 is ~ =
111
/
(0)
3
n -oo
3
1
3 => !'" (0) = 2.
Or: Use Theorem 11 .9.2 to differentiate f three times.
For example, let a.. = bn = (-1
15. False.
r. Then {an} and {bn} are divergent, but a,.b,. = 1, so {a,.b.. } is convergent.
[2:: (-1)" a,. is absolutely convergent and hence convergent.]
17. True by Theorem 11.6.3.
0.99999 . . . = 0.9 + 0.9(0.1) 1 + 0.9(0.1? + 0.9(0.1) 3 + ·· · =
19. True.
f: (0.9)(0.1)n-l = __2.:Q_
= 1 by the formula
1 - 0.1
n=l
for the sum of a geometric series [S = a 1 / (1 -;- r)] with ratio r satisfying lrl < 1.
21 . True.
A finite number of terms doesn't affect convergence or divergence of a series.
EXERCISES
1. {
2 + n3 }
1+
2n 3
.
.
converges smce lim
2 + n3 ·
n --+00 1
+
2n 3
.
= lim
n--+00
2/ n 3 + 1
11n 3 + 2
1
= - .
2
3
3. lim a,. = lim ~ = lim / ~
= oo, so.the sequence diverges.
n--+OC
n-+ oo 1 + n
n-+ oo 1 n + 1
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100
0
CHAPTER 11 INFINITE SEQUENCES AND SERIES
nsinn l::; ~
n
1
·
.
< - ,so Jan !-> 0 as n-> oo. Thus, lim a,.= 0. The sequence .{an} is convergent.
1
1
+
n
+
n
l
5. !an i = - 2-
-
n
7.
n --+oo
•
{(1 + 3)4"}is convergent. Let y (1 +;3)4:t. Then
~
~
lim lny = lim 4xln(1 + 3/x) = lim ln( 1 + 3/x)
:r:-oo
:z:--+oo
x --+ oo
1/ (4x}
(1+ -n3)""
limy= lim
.:c-oo
n---.oo
~ lim ~
( -~)
- 1/ (4x2)
.
IliD
x--+oo
x - oo
12
1 + 3/x
= 12,so
=e12 .
9. We use induction, hypothesizing that an- 1 < an < 2. Note first that 1 < a2 =
t (1 + 4) = 1< 2, so the hypothesis holds
fo r n = 2. Now assume that ak-1 < a k < 2. Then ak = ~(ak- 1 + 4) < ~(a~.:+ 4) < ~(2 + 4) = 2. Soak < ak+l < 2,
and the induction is complete. To find the limit of the sequence, we note that L =
lim a n = lim an+l
'ft.-oo
L = HL + 4)
n
n
11. --g--- < 3
n +1
n
n-oo
=>
=> L = 2.
n
··
1
= 21 , so L; --g--converges by the Comparison Test with the convergent p-series L; 2 [ p = 2 > 1].
n
n=l n + 1
n=l n
00
00
.
1 )3 · -1 =-1 < 1, so LJ
~ -na converges .by the Rat1.o .,.,,est. ·
13. )1m
- - = 1'1m [ (n+1)3
+ · 35n ] = l'rm (1 +Tl.___.OO
an
n - 00
n
n-oo
n
5
5
n=l 511.
5n 1
lan+ll
~. Then f is continuous, positive, and decreasing on [2, oo), so the Integral Test appl ies.
15. Let f (x) =
xv ln x
00
1
2
f (x)dx= lim
t --+oo
1t
2
1
~
dx
XV In X
[u = ln x. du=_: <L7:]
x
!.In
!n t
= t-oo
lim ln 2 u- 1/ 2 du = t-+oo
lim (2Fu]
In 2
L
= lim (2 vlfrit - 2v1ii2) = oo,
t--+oo
. LJ
~
so t I1e senes
1
~
n=2 nv lnn
17. Jan! =
eliverges.
11 : (1~;)n I::; 1 + (~. 2 )" < (l.~)n = (~J, so n~1 ianl converges by comparison with the convergent geometric
0
00
00
n=l
n =l
series L; ( ~ )" [r = ~ < 1] . lt fo llows that L; an converges (by Theorem 3 in Section I 1.6).
19. lim
Tt-+ 00
Ian+l I= n--+oo
lim 1 . 3. 5 . . . . . (2n - 1)(2n + 1)
5n n !
5n+J (n + 1)!
1 • 3 • 5 • • • • • (2n- 1)
Qn
·lim 2n + 1
2
th
.
n--+oo 5(n + 1 ) = S < 1, so e scnes
converges by the Ratio Test.
21. bn =
Fn > 0, {bn} is decreasing, and lim bn = 0, so the series E (- 1)n- 1 Vn converges by the Alternating
n--+oo
n= l
n +1
n +1
Series Test.
@ 2012 O:ng•gc U:;uning. All RighlS Reserved. Moy not be seaMed, copied, or duplicoted, or posted to Dpublicly occcssiblc website, in whole or in p;u1.
CHAPTER 11
En
23. Consider the series of absolute values:
3
-I/ is a p-series with p
REVIEW
0
101
= t ~ 1 and is therefore divergent. But if we apply the
1'1=1
Alternating Series Test, we see that bn =
converges. Thus,
E(- 1)"- 1 n - 1 / 3
3~ > 0, {bn} is decreasing, and lim bn = 0, so the series n=l
yn
n-Hxl
E(-1)"- n- 13 is conditionally convergent.
1
1
tt = l
2
an+ll
~ ( - 1t+ 1 (n+2) 3n+l .
2 "+1
l - n+2 3 _ 1+(2/n) 3
3
.
25. ~ ·~
· (- 1)"(n + 1) 3" .- n+ 1 · 4 - 1 + ( 1/ n) · 4 --+ 4 < 1. asn--+ oo, so by the Ratro
22n+3
l
oo
Test, I:
n=l
oo
27· I:
n=I
+
( -1)" (n
1)3"
2 "+ 1
2
is absolutely convergent.
(- 3)n- 1
oo (- 3)n-l
oo (- 3)"-1
1 oo (- 3)n- l
1 oo ( 3 ) n- l
1( ·
1
)
23n
= n=l
I: (23)n = n=l
L
8"
= -8 n=
I:l 8"- 1 = -8 n=l
L - -8
= -8 1 - ( - 3/8)
1
8
1
=s· u = u
00
29. :L[tan- 1 (n+ 1)-tan- 1 n ) = lim Bn
n= l
n-oo
= lim [(tan- 1 2- tan- 1 1) + (tan- 1 3- tan- 1 2) + · · · + (tan- 1 (n + 1)- tan- 1 n)J
n-oo
= lim [tan- 1 (n + 1)- tan- 1 1) = 1!. n-oo
2
1!. = 1!.
4
4
e2
e3
e4
oo
e"
oo ( -e)n
oo x"
31. 1 - e + -21. - 31. + -41. - .. · = L (-1)"1 = E --1- = e- e since e"' = I: I for all x .
n=O
n.
n=O n.
.
n=O n .
1
·
1 ( oo x"
oo (-x)" )
33. cosh x = -(e"' +e-"') = - I: 1 + L --12
2 n=O n.
n=O n.
oo
' 35· n~l
( -1) n+l
n5
· 1
1
1
1
1
1
1
= 1 - 3 2 + 243 - 1024 + 3125 - 7776 + 16,807- 32,768 + ... ·
.
1
Srnce ba = 85 =
1
oo (- 1)"+1
7 (- 1)"+1
< 0.000031, I:
~ E
~ 0.9721.
5
32' 768
n=l
n
n=l
n5
00
1
8
1
1
1
" ~ E " ~ 0.18976224. To estimate the error, note that - - ·- < -,so the remainder term is
37. I: 2 + 5"
n=l 2 + 5
n=l 2 + 5
5n
.
00
00
1
1
1/5 9
Ra = n~o 2 + 5" < n~o S" = 1 _
= 6.4 ~ 10,- 7 [geometric series with a=
115
-b and r = i ] .
© 20 I2 Ceng~ge lc::tming. All Rights Resc:r•cd. May not be sc:umcd, copied, or duplicated. or posted io n publicly ncccs.siblc website, in ,whole or in part.
l
102
~HAPTER 11
0
INFINITE SEQUENCES AND SERIES
Limi~ Comparison Test. lim I(~an)a,, I = lim n +n 1 = Jim (1 + .!)
= 1 > 0.
n
39. Use the
n--+00
n - oo
Since E Ia,. I is convergent, so is I: \ ( n :
1
n-+oo
I,
) a,. by the Limit Comparison Test.
+
I
+
41. lim a,.+ll = lim [ lx 21"+1 . n4" ,.] = lim [__2!:___ lx 2[] = [x + 21 < 1
n - oo an
!'-oo (n + 1) 4" + 1 lx + 21
n-oo n + 1 4
4
lx + 21 < 4
¢:?
-4 < x + 2 < 4
¢:?
¢:}
lx +21 < 4, so R = 4.
oo (x + 2)n
becomes
n 4"
- 6 < x < 2. lf x = - 6, then the series E
.
•
n =l
f: (-44r = f: (- nl )", the alternating harmonic series, which converges by the Alternating Series Test. When x = 2, the
n=l
71.
n=l
series becomes the harmonic series
1
2
f: .!, which diverges. Thus, I= [- 6, 2).
n= l n
3
43. lim I Un+l l = lim 1 n+ (x- )"+1 ·
a,.
n-oo
../n + 4
n-oo
soR = ~- lx- 31< ~
¢:?
y'n+3 ,., = 2 jx- 31 lim Jn + 3 = 2lx- 31< 1
2"(x - 3)
n-oo n + 4
-~ < x- 3 < ~
·
¢:}
¢:}
jx- 31 < ~.
·
·
oo 2"{x - 3}"
~ < x < ~ · For x =~. the series I: .;n+3 becomes
n =l
n +3
I: ~
= I: ;/ which diverges [P = ~ $ 1), but for X = ~' we get I: ~ .which is a convergent
+3
n
n +3
n=O y n
n=3
2
,
n=O v
alternating series, so I = [~ , ~).
45.
n
/("l(x)
/ (n)(:g)
0
sinx
2
1
cosx
.il
2
- sin x
- 2
3
- cosx
4
sin x
1
2
1
_ _il
2
1
2
•.
"C)
{3) (7r)
{4}C)
4
2
3
sinx = f(~)+f'(~)(x-~)+!.____i_(x-~)
+
/
6
(x-~)
+/
6
(x
~)
+ ...
6
6
. 6
2!
6
3!
6
4!
6
1
47. -1 +--X = 1 _ ~-X) = n~O (- x)" = n~O {-1)" x" for lxl < 1
2
00
1+x
n=O
=> ~ =I: {-1f'x"+2 withR= 1.
© 2012 Ccngnge Learning. All RighiS R=n-.:d. May not be S<31lncd, copied, or duplicotcd, or posted too publicly accessible website, in wbolc or in port .
CHAPTER 11
49./I
REVIEW
D
1
- dx = -ln{4- x) + C and
4-x
I
-1- dx = -1
4- X .
4
- 1dx = -1
1 - x/4
4
I L: (X)"
=- E 00
dx
n=O 4
1 / oo x" dx = -1 E
oo
xn+1 + C So
4 n=O 4"
.4 n=O 4"(n + 1}
·
1 00
xn+1
00
xn+1
00
x"
.
ln{4 - x )=-- E ,.(
n +C.Puttmg x=O,wegetC = ln4.
) +C= - E ,.+1( + ) +C = - E
1
4
4 n=O 4 n+ 1
4
n=O
n
n=1 n
oo x"
" . The series converges for Jx/41 < 1
. Thus, f(x) = ln{4- x) = ln4- E
rt=1 n 4
<:::}
JxJ < 4, so R = 4.
Another solution:
ln{4 - x) = ln[4(1 - x/4)] = In 4 + ln{1 - x/4) = ln 4 + ln[1 + ( -x/4))
f
4
= ln4 + 'f(-1t+ 1 (-x/ )n (fromTablel ) =ln 4+ f{ - 1) 2"+1 x "n = ln4 x" .
n=1
n
n=1
n4
n=1 n4"
51. sin x =
oo
(- 1)"x2n+1
•
E ,__,.~-,..,.­
n =O
(2n + 1)!
{-1)"(x4?n+l
oo ( - 1 )"x8n+4
.
)
(
+
=
E
(
for
all
x
,
so
the
rad1us
of
)
1
1
2n 1 ·
n=O
.n=O 2n + 1 ·
,
oo
4
=* sm(x ) = E
convergence is oo.
1
53. f(x)= ~
1
1
- 1(1-...!..x)-1/ 4
1e
V'i6 (1 - ie x) l / 4 - 2
{/16(1 - x/16)
= ~ [1 + (-~) (-~) + (-t)(-~) (-~)
2
4
16
2!
16
2
+ (- t) (-~) (-~) (-~)3 + .. ·]
3!
16
.
.
1
oo 1 . 5 . 9 ..... (4n - 3)
1
oo 1 · 5 · 9 · .. · · ( 4n - 3) x"
= - + 'E
xn = - + 2:: ----::-::--:-:-'-:
- ----!..
2 n=l
2 . 4" . n ! . 16"
2 n=1
26n+1 n !
1
for J- ~ J < 1
x"
00
55. e"' =
I
#
Jxl < 16, so R = 16.
e"'
1 00 xn
00
xn-1
00
xn-1
1
00
xn-1
E I• so -X = -X n=O
E 1n. = ><=0
E - n 1.- = x- 1 + n=1
E - n.1- = -X + n=1
2:: - n.1-· and
n=O n.
x"
ex
oo
X
n=l n·~
-dx = C+ lnJxJ+ E -
57. (a)
-,.
n
f(n)(x)
j (n)( 1)
0
x1/2
1
1
~x - 1 12
2
- ±x-3/2
3
l!x- 5/2·
4
-wx - 7/2
8
15
Vx :=::::: T3(x) = 1 + 1/ 2 (x - 1) - 1/ 4 (x - 1? + 3 / 8 (x - ·1)3
1!
2!
31
= 1 + Hx - 1)- Hx - 1? + -fo(x - 1) 3
1
2
..,.. 41
3
8
15
- Iii
© 2012 Cengoge Le=ing. All rughiS Reserved. Mlly not be sconned. copied, or duplicated, or posted to a poblidy accessible website, in "1101e or in prut.
103
104
0
CHAPTER 11
INFINITE SEQUENCES AND SERIES
(c) IR3 (x) l $ ~~ ix - 11 , where J! ( ) (x) J $ M with
4
1.5
(b)
4
j (4>(x) = - ~~x- 7 1 2 . Now 0.9 ~ x $ 1.1
- 0.1 ~ X - 1 ~ 0.1
=>
(x- 1) $ ·(0.1) 4 ,
. x = 0.9 gtves
. M =
.
and Ietting
IRa(x) l $
2
0
=>
16 (0.~~712 41(0.1)
4
15
( _ )7 , so
16 0 9 12
4
:::::: 0.000005648
:::::: 0.000 006 = 6 X lO- G
5 X 10- 6
(d)
From the graph of IR3(x)l = IJX - T3(x)l. it appears that
the error is less than 5 x
w- 6 on [0.9, 1.1].
oo
• x2n +l
x3 x6 x7
.
x3 xs x7
59.sin x = 2:(- 1t (
) 1 = x- - + - - - +· ··,sosmx - x=- - + - - - +··· and
2
1
n=O
n+ .
3!
5!
7!
3!
5!
7!
61. f(x ) =
00
2: Cn x"
00
00
=> f( -x)= l: cn(-xt = 2:(- 1)"c.,x"
n =O
n=O
n=O
(a) Iff is an odd function, then f( - x) = - f(x)
=>
00
00
u=O
n=O
2: (- 1)"cnx" = 2: - c,,x". The coefficients of any power series
are uniquely determined (by Theorem 11.10.5), so ( -1)" Cn = - en.
If n is even, then ( - 1)" = 1, so en = -c,.
=> 2en = 0 => c,. = 0. Thus, all even coefficients are 0, that is,
Co = C2 = C4 = · · · = 0.
00
00
n=O
n=O
If n is odd, then ( - 1 )" = -1, so -en = Cn
=> 2c,.. = 0 =>
(b) If f is even, then f (-x) = f(x)
that is, Cl = C3 = C5 = · · ·
=>
'
2: {-lt c,. xn = 2: Cn x"
Cn =
=> {-1)" Cn = Cn.
0. Thus, all odd coefficients are 0,
= 0.
@) 2011 Ccngoge Lcorning. All Rigbls Reserved. Moy not be scanned. copied, or<luplicotcd, or J'OSted to u publicly occessiblc website, in whole or in port.
D
PROBLEMS PLUS
1. It would be far too much work to compute 15 derivatives of f. The key idea is to remember that j <••l (0) occu~s in the
.
x3
coefficient ofx" in the Maclaurin series of f. We start with the Maclaurin series for sin: smx = x- -
3!
·
Then sin(x3 ) = x 3 -
/ (15) (O)
xl5
xg
3f + 5f - ···,and so the coefficient of xlu is
151
x5
+- ... .
5!
1
= 5f· Therefore,
151 = 6 . 7 . 8 . 9 . 10 . 11 . 12 . 13 . 14 . 15 = 10,897,286,400.
f ( 1"") (0) = 5f
. Append.IX D , Wit
" h x = y = u,
n
3. (a) From Formu Ia 14a tn
we get tan 28 =
2 cot 20 =
1 - tan2 8
tan
e
=cot 8.- tan8. Replacing 8 by
2tanB., , so cot 28 = 1 - tan28
1-tan- 8
2tan8
tx. ~e get 2 cot
X= cot ~X- tan
=?
!x. or
tan ~x =cot ~x - 2cotx.
X
'
X
X
00
X
1
X
(b) From part (a) with - 1 in place of x, tan - = cot - - 2 cot - 1 , so the n th partial sum of L -tan - is
ra.=l 2n
2n
2n.
2n
2tt 2nSn
=
tan(x/ 2)
tan(x/ 4)
tan(x/ 8)
tan(x/2")
2
+ 4
+ 8 + .. •+ 2n
= [ cot~/2) _
cotx]+ [cot(:/4) _ cot(;/2) ] + [ cot~/8) _ cot~/4)] + ...
_ cot(x/2n- l) ] _ _
.
cot(x/ 2")
+ [ cot(x/2")
.. _
cot x +
,.
n
2
2 1
2
Now cot(x/2n) = cos(x/2")
2"
2" sin(x/ 2n)
cos(x/2n)
X
[telescoping sum]
x/2"
1
, 1
. (
) -> - · 1 = - as n -> oo since xf2n -> 0
SIDX 12 n
X
X
for x "1- 0. Therefore, if x "1- 0 and x "1- k1r where k is any integer, then
~ - 1 t anX = }"1m s,. = Iim ( -cotx+ - 1 cot -X ) = -cot x+ -1
n
n-too
n-+oo
n
2n
X
0
n=l 2n
2
2
If x = 0, then all terms in the series are 0, so the sum is 0.
5. (a) At each stage, each side is replaced by four shorter sides, eacb of length
k of the side length at the preceding stage. Writing s 0 and Eo for the
number of sides and the length of the side of the initial t1:iangle, we
generate the table at right. In general, we have sn = 3 · <! n and
so= 3
eo = 1
81
= 3·1
= 3 · 42
3
S3 = 3 · 4
e1 = 1 / 3
S2
e2 =
1/ 3 2
e3 = 1/ 3
3
e,. = ( ~) n' so the length of the perimeter at the nth stage of construction
is Pn = Snfn = 3 ·. 4" · (~)" = 3 · (1)".
(b) P n =
4"
n-l = 4
3
(4)n-l
3
.
Since
1 > 1, Pn -> oo as n-> oo.
(c) The area of each of the small triangles added at a given stage is one-n inth of the area of the triangle added at the preceding
stage. Let a. be the area of the original triangle. Then the area an of each of the small triangles added at stage n is
e 2012 CcngUl_;C Lcamin&- All Rig)tts RL"SCr\'cd. Ma)' nol be scanned. copied. or duplic31Cd. or pos1l-d to a publicly accessible websile, in whole or in part.
105
106
CHAPTER 11 PROBLEMS PLUS
0
an = a · _.!:_ = .!::.. . Since a small triangle is added to each side at every stage, it follows that the total area An added to the
9n
9n
4n-1
9
figure at the ntl1 stage is An = Sn-1 ·an = 3 · 4n- 1 · ~, = a· 32 n_ 1 . Then the total area enclosed by the snowflake
.
2
3
curve is A = a + A1 + A2 + As + · · · = a + a · ~ + a · ; + a · : 5 + a · : 7 + · · · . After the first term, this is a
3
. senes
. w1t
. h common ratlo
. !)
4 , so.A = a'+ a/3
8a But t I1e area of .the ongma
. . I eqUI'IateraI
_ .1 = a + a · 9 = S.
geometric
1 9
3 5
.
2 J3
. Ie w1t
. h s1'd e 1 .IS a = '12 · 1 · sm
. 7r = J3 . So t he area enc Iosed by th e snowfla ke curve IS
. 8 · J3 = T
tnang
.
4
3
5 4
7. (a) Let a= arctanx and b =arctan y. Then, from Formula 14b in Appendix D,
(b)
tana-tanb
tan(arctan x)-tan(arctan y)
x-y
tan a = 1 +tan a tan b = 1 + tan(arctanx) tan(arctan y) = 1 + xy
.
x-y
Now arctan x - arctan y = a - b = arctan( tan( a - b)) = arctan - - - since
1 +xy
--t < a - b < ¥.
(b) From part (a) we have
120
t
1 t
arc t an 120
119 - arc an 2311 -.arc .an
1
28,561
Wi - 239 = arc t an 28,441
120
1
28 _561
1 + Wi . 239
28,441
= arct an 1 = 4
7r
(c) Replacing y by-yin the formula of part (a), we get arctanx +arctan y = arctan x + y . So
1 -xy
4 arctan 51 = 2(arctan 51 + arctan 51) = 2 arctan
= arctan
..!!...
+..!!...
12
12
5
5
1 -12'12
Thus, from part (b), we have 4 arctan
1
+1
5 5 -- 2 arc t an 5 -- arc t an 12
5 + arc t an 12
5
1
12
1-5.
51
= arctan g~
t - arctan 2i 9 = arctan ~ i~ - arctan 2; 9 = "i.
x3 · x5
x7
x9
x 11
(d) From Example 7 in Section 11.9 we have arctanx = x- 3 + 5 - 7+ 9 -~ +· ··,so
11
1
1
1
1
1
arctan 5 = 5 - 3 · 53 + 5 · 55 - 7 · 57 + 9 · 59 - 11 · 511 + .. .
This is an alternating series and the size of the terms decreases to 0, so by the Alternating Series Estimation Theorem,
the sum lies between ss and s6, that is, 0.197395560 <arctan
(e) From the series in part (d) we get arctan
t < 0.197395562.
2 ~9 = 2~9 - 3 . ~393 + 5 . ; 395 - -- ·. The third term is less than
2.6 x 10- 13 , so by the Alternating Series Estimation Theorem, we have, to nine decimal places,
arctan 2 ; 9 ~ s2 ~ 0.00~184076. Thus, 0.004184075 < arctan 2 ~ 9 < 0.004184077.
(f) From part (c) we have 1r = 16 arctan
k- 4arctan 2~9 , so from parts (d) and (e) we have
16(0.197395560) - 4(0.004184077) < 1f < 16(0.197395562)- 4(0.004184075)
=?
3.141592652 < 1r < 3.141592692. So, to 7 decimal places, 1r ~ 3.1415927.
© 2012 Ccngagc Lt;.uning. All Rights Rcscn·cd. Mny not be scanned. copied, or duplicntcd, or posted to a publicly accessible website, in whole or in pan.
CHAPTER 11 PROBLEMS PLUS
9. We start with the geometric series
D
107
l
f; x" = -1-1 -X , lx l < 1, and differentiate:
n. =O
d( }: xn = -dd (-11
_ ) = ( - 1 ) for lx l < 1 => L nxn = X L nxn- l =
1
(1 - x)2
oo
oo
n =1
n =l
00
oo nxn- l = -d
}:
n=l
X
..,.---~
)
X
X
n= O
X
X
2
for lx l < 1. Differentiate again:
00
(1 - x? - x · 2(1 - x)( - 1)
x+ 1
= (1 - x)3
dx (1 - x)2 (1 - x) 4
-
-
n= 1
x
d
_
}: n 2 x" 1 -
=>
-
~
L- n
n =l
2
,..
x =
x2 + x
(1 - x)S
=>
f; n3xn - 1 = !!:.._ x 2 + x = (1 - x? (2x + 1) - (x2 + x)3(1 - x)2 ( - 1) = x 2 + 4x + 1 ==>
dx (1 - x) 3
n=l
t
(1 - x)~
(1 - x )6
f; n 3 x" = x 1-4 xX )~ x , lx l < 1. The radius of convergence is 1 because that is the radius of convergence for the
3
n=l
2
geometric series we started with. If x = ± 1, the series is }:n3 (±1)n, which diverges by the Test For Divergence, so the
interval of convergence is (-1, 1).
11 . ln(1- : 2 )
= ln(n~~ 1 ) ,;, In (~+ ll~n - 1) = ·in[(n +1)(n- 1)]- ln n 2
= ln(n + 1) + ln (n - 1) - 2ln n = ln(n - 1) - In n- ln n + In(n + 1)
n- 1
n- 1
n
= In - n - - [ln n - ln(n+ 1)] = In - n - -In n+ .
1
Ek In ( 1 - 2n1) = nE=k 2 ( In -n-1
- - In - n - ) fork ~ 2. Then
n
n+ 1
Let Sk =
n= 2
3) + · · · + (In - -1 - In -k+1- = 21 In -k+-1 ' so
. (In-12 -In -+k-1 ) = In -12 1 = 1 -In 2 - In 1 = - In 2.
}: In 1 - - 1) = .
= hm
Sk =
00
1
2)
( 2
( ln -2 - In -3 + In -3 - In -4
(
n2
n= 2
lim Sk
k- oo
k k
)
In- -
-ln
k
k-oo
13. (a)
k
k
·I
ln
The x-intercepts of the curve occur where sin x = 0
<=> x = mr,
nan integer. So using the formula for disks (and either .a CAS or
l-+-l-~~+-''r7"""'-'...,.._---t 40
sin2 x = ~ (1 - cos 2x) and Formula 99 to evaluate the integral),
the volume of the nth bead is
( - :r:/10 sm
. x )2 dx = rr Jrmr
-:r:/ 5 sm
. 2 x dx
v.n = rr Jf"''
(,.-l ) .. e
(n- 1 ) .. e
- I
= 2;g,rr (e- (n-l)lf/5- e- mr/5)
(b) The total volume is
rr .{000 e- :r:/S sin 2 x dx =
f V,, = 2~it f; [e- (n-1)"/
n =l
5 -
;g;
e-n"/ 5 ] = 2
[telescoping sum].
n=l
Another method: If the volume in part (a) has been written as Vn =
~g; e-n"/ 5 (e"/5 - 1), then we recognize f; Vn
2
·
as a geometric series with a= 2~g; (1 - e-"15 ) and r = e-"/ 5 .
© 2012 Ccngo.ge Learning. All Rights Rcsr.n·ed. l\.Jay nol be settnr~-d, copiW, or dupl icuiL-d, or posted to a publicly accessible website. in whole or in part.
n=l
108
D
CHAPTER 11 PROBLEMS PLUS
15. If Lis the length of a side of the equilateral triangle, then the area is A= ~L · 4L = 4£2 and so £ 2 =fa A.
Let r be the radius of one of the circles. When there are n rows of circles, the figure shows that
L = v'3r+r+(n-2)(2r)+r+v'3r=r(2n-2+2../3),sor =
L
(
y'3
)'
2 n+ 3-1
1
The number of circles is 1 + 2 + · · · + n = n(n + ) , and so the total area of the circles is
2
An =
=
n(n + 1)
1fT
2
2
=
n(n + 1)
· 7r
2
£2
2
4 (n+v'3- 1)
n(n+ 1)
4A/J3
n(n+1)
1rA
2
2
2
1f 4(n+v'3 - 1)
= {n+../3-1) 2../3
n(n + 1)
1r
A = (n+../3-1) 2 2.;3
1 + 1/n
1r
=
=>
An
1r
-> - - asn ->oo
2
[1+(J3-1)/n) 2v'3
r
2../3
2r
2r
~-----L------~
17. As in Section 11.9 we have to integrate the function x"' by integrating series. Writing x"' = (e1" "')"' = e"' 1""' and using the
Maclaurin series for e"', we have x"' = (eln "')"' = e"' 1" "' =
integrate this series term-by-term:
·1
x"' dx = 2:
11
0
n=O
0
1
oo
withu = (lnx)",dv =x" dx,sodu=
n(ln x)"- 1
·
x
E (x Inn.x) = E x" (~n. x) As with power series, we can
n
1
n=O
n .
n=O
x"(ln xt
oo 1 1 1
dx = 2: I
x"(lnxf dx. We integrate by parts
1
n.
n=O n. 0
x"+1
dxand v = - - : .
n+ 1
1
1
f \"(lnx)"dx = lim /. x"(ln xfdx= lim [x"+l (lnx)"]
t-o+ t
}0
=
t-o+ n + 1
.
1
t
lim ( ___!!:__ x"(1nx f- 1 dx
t-o+
it n + 1
o- _ n _ t x"(Inxf- 1 dx
n+ 1 } 0
(where !'Hospital's Ru le was used to help evaluate the first limit). Further integration by parts gives
1
(
}0
1
1
x"(ln x)k dx = -~ 1 x"(ln x)k- 1 dx and, combining these steps, we get
n +1 0
1
1x"(ln x)"dx = (-1)"n!11 x"dx= (-1)"n!
(n+1)"
0
"' d
1
, 0
X
X
0
=~
..!_ 1 1 "(In ·)"' d = ~ ..!_ (- 1t n! = ~ (-1)" = ~ (- 1)"-1
L...
I
X
X
X
L...
I (
L... (
L...
·
) +1
) +1
n=O n.
0
.
19. By Table 1 in Section 11. 10, tan- 1 x =
n=O n.
00
n +1 n
have 6 = tan-1
(
1 )
y'3
00
= n~O ( - .1t
n=O
n +1 "
n"
x 2n+l
1
2
n+ 1
v3
1
00
(1/../3) "+
(1)" i
1
2n + 1
= n~O ( -1)" 3 y'3 2n + 1' SO
6 00
(-1)"
00
(-1)"
(
00
'
(-1)" )
- 2 3
- 2 3 1
- ../3 n~O (2n + 1)3" - y'3 n~O (2n + 1)3" - y'3
+ f1 (2n + 1)3n
·rr - -
n=l
E (- l t -2' - - f or JxJ < 1. In particular, for x = r.;• we
n=O
7r
=?
(n+ 1)n+l
·
.
=>
E
00
(-1)"
7r
=--1.
+ 1)3" 2 y'3
n=1 (2n
<[) 2012 Ccngogc Learning. All Rights Reserved. Moy not be sconncd. copio;cJ, or dUJ>licutcd, or posted to a publicly oeccssiblc website, in whole or in p:tr1.
CHAPTER 11
2
x < 0 are given by x =
x2
x4
xs
xs
2! + 4! - 6! + 8f - · · · = cosx. The solutions of cosx = 0 for
i - 1rk, where k is a positive integer. Thus, tl1e solutions off(x) 0 are x = - (i - 1rk) where
2
=
,
23. Call the series S. We group the terms according to the number of digits in their denominators:
Now in the group gn , since we have 9 choices for each of then digits in the denominator, there are 9n terms.
Furthermore, each term in 9n is less than 10 ~_ 1 [except for the first term in gl]. So 9n < gn ·
00
Now L: 9 ( 190 ) n - l is a geometric series witl1 a
10~_ 1 = 9( {0 )"'- •
= 9 and r = fa < 1. Therefore, by the Comparison Test,
n=l
9
oo
n= l
n=l
S = L: 9n < L: 9(w)
n- 1
9
= 1 _ 9110 = 90.
Use the Ratio Test to show that the series for u, v, and w have positive radii of convergence (oo in each case), so
Theorem 11'.9.2 applies, and hence, we may differentiate each of these series:
du
3x 2 6x5 9x8
x2
x5
x8
- = - + - +-+··· = - +-+ - +· ·· = w
dx
. .
x3
dv
x6
3!
6!
x9
9!
2!
5!
8!
dw
x4
x7
x 10
S imilarly, dx = 1 + 3! + 6f + 9! + · · · = u, and dx = x + 4 ! + 7T + lO! + · ·· = v.
So u' = w, v' = u, and w' = v. Now differentiate the le ft-hand side of the desired equation:
d
dx (u 3 + v 3 + w 3 - 3uvw) = 3u2 u' + 3v2 v' + 3w 2 w' - 3(u'vw + uv'w + uvw')
= 3u w + 3v u + 3w v - 3(vw
2
2
2
2
+ u 2 w + u~ 2 ) = 0 =>
u + v + wa - 3uvw = C. To find the value of the constant C, we put x = 0 in the last equation and get
3
109
x 2 · x3
x4
! + 6f + 8f + .. · = 0. lfx;:::: ~· then f(x ) ;:::: 1 and t~ere are
4
k is a positive integer.
oo
0
x
21. Let f (x ) denote the left-hand side of the equation l + 2T +
no solutions oftheequation. Note that / (- x ) = 1 -
PROBLEMS PLUS
3
13 + 03 + 03 - 3(1 · 0 · 0) = C
=> C = 1, so u 3 + •v 3 + w3 - 3uvw = 1.
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1
12
D VECTORS AND THE GEOMETRY OF SPACE
12.1
Three-Dimensional Coordinate Systems
1. We start at the origin, which has coordinates (0, 0,. 0). First we move 4 units along the positive x-axis, affecting only the
x-coordinate, bringing us to the point ( 4, 0, 0). We then move 3 units straight downward, in the negative z-direction. Thus
only the z-coordinate is affected, and we arrive at {4, 0, - 3).
3. The distance from a point to the yz-plane is the absolute value of the x-coordinate of the point. G(2, 4, 6) has the x -coordinate
with the smallest absolute value, soC is the point closest to the ·y z-plane. A( - 4, 0, -1) must lie in the xz-plane since the
distance from A to the xz-plane, given by the y-coordinate of A, is 0.
5. The equation x + y = 2 represents the set of all points in
y=2 - x
JR3 whose x- andy-coordinates have a sum of2, or
equivalently where y = 2 - x. This is the set
y=2 -x,z= O
{(x, 2 - x , z) I x E JR, z E R } which is a vertical plane
that intersects the xy-plane in the line y = 2 - x, z = 0.
X
7. We can find the lengths of the sides of the triangle by using the distance formu la between pairs of vertices:
IPQI = )(7 - 3)2 + [0 - ( -2)]2 + [1 - ( -3)] ~ = yf16 + 4 + 16 = 6
IQRI = j(l- 7)2 + {2 - 0) 2 + {1 - 1)2 = vf36 + 4 + o = .J40 = 2 v'15
IRPI = )(3 - 1)2 + (- 2 - 2)2 + (- 3 - 1) 2 = vf4+ 16+ 16 = 6
The longest side is QR, but the Pythagorean Theorem is not satisfied: IPQI
2
+ IRPI 2 =f:. IQRI 2 . Thus PQR is not a right
triangle. PQR is isosceles, as two sides have the same length.
9. (a) First we find the distances between points:
IABI = ){3- 2)2 + {7- 4)2 + (-2 - 2)2 = v'26
IBGI = ){1- 3) 2 + (3 - 7) 2 + [3- ( - 2)]2 = v'45 = 3 v'5
IACI = )(1- 2)2 + (3 - 4) 2 + (3- 2) 2 = J3
In order for the points to lie on a straight line, the sum of the two shortest distances,must be equal to the longest distance.
Sin<;:e v'26 + J3 =f:. 3 v'5, the three points do not lie on a straight line.
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111
112
0
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
(b) First we find the distances between points:
IDEl = J(l- 0)2 + [- 2- (- 5)]2 + (4 - 5) 2 ::::: VIT
IEFI = )(3 - 1) 2 + [4 - ( - 2)]2 + (2 - 4) 2 = v'44 = 2 VIT
IDFI = )(3 - 0) 2 + [4- (-5)]2 + (2 - 5) 2 = v'99 = 3Vll
Since IDEl + IEFI = ID F I, the three points lie on a straight line.
11. An equation of the sphere with center ( -3, 2, 5) and radius 4 is [x - ( - 3)]
2
+ (y - 2) 2 + (z - 5) 2 = 4 2 or
(x + 3) 2 + {y - 2) 2 + (z- 5) 2 = 16. The intersection of this sphere with the yz-plane is the set of points on the spher~
whose x -coordinate is 0. Putting x = 0 into the equation, we have 9 + (y - 2)
2
+ (z- 5) 2 = 16, x = 0 or
(y- 2)2 + (z- 5) 2 = 7, x = 0, which represents a circle in the yz -plane with center (0, 2, 5) and radius -/7.
13. The radiusofthesphere is the distance between (4,3, - 1) and (3,8, 1): r = )(3 - 4)2 + (8 - 3)2 + [1 - (-1)]2 = V3o.
Thus, an equation of the sphere is (x - 3) 2 + (y - 8)
2
+ (z- 1)2 = 30.
15. Completing squares in the equation x 2 + y 2 + z 2 - 2x - 4y + 8z = 15 gives
recognize as an equation of a sphere with center (1, 2, - 4) and radius 6.
17. Completing squares in the equation 2x2 - 8x + 2y
2 + 2z 2 + 24z = 1 gives
2(x2 - 4x + 4) + 2y2 + 2~z 2 + 12z + 36) = 1 + 8 + 72
:::}
2(x - 2) 2 + 2y2 + 2(z + 6? = 81
:::}
(x - 2)2 + y 2 + (z + 6) 2 = 821 , which we recognize as an equation of a sphere with center (2, 0, -6) and
radius
j¥ = 9/ v'2.
19. (a) lfthe midpointofthe line segment from P 1(x 1 ,y1,z1) to P2 (x 2
,y ,z
2
2 ) is
Q = (x 1 ;x2 , Y1
;v
2
, z
1
2
; z ),
then the distances IPl Q l and IQ P21are equal, and each is half of IPt P21· We verify that this is the case:
IP1QI =
v[~ +
(x1
X2) - X1]
2
+ a<v1 + Y2 ) - Y1t + [t(z1 + Z2 )- Z1]
2
1
1
)2
1
1
)2
= y / ( 2X2
- 2X1
+ ( 2Y22Y1
+ ( 21 Z2 - 21 Z1) 2
= /(~) 2 ((x2 - x l) 2 + (y2 - vd + (z2 - z1)2 ] = ~J(x2 - xd + (y2- y1) 2 + (z2 - z1?
= ~ IPtP2 1
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SECTION 12.1 THREE-DIMENSIONAL COORDINATE SYSTEMS
J[x2 - ~(X 1 + X2)] 2 + [y2- ~(Yl + Y2)] 2 + [z2 - ~ (z1 + z2)) 2
(~x2- ~ x1) + (h2- ~·yt) + (tz2 - ~z1) = JG)
xd +
=
0
113
1
!QP2\ =
2
2
.= ~J(x2 -
2
2
[(x2-
(y2 - yl)
2
+ (z2 - z1) 2)
xd + (y2- y1) 2 + (z2- zd = ~ \P1P2\
So Q is indeed the midpoint of P 1P2.
(b) By part (a), the midpoints of sides AB, B C and C A are P1 ( -~, 1, 4), P2 (1, ~. 5) and P3 (~ , ~, 4). (Re~all that a median
of a triangle is a line segment from a vertex to the midpoint of the opposite side.) Then the lengths of the media ns are:
IAP2\ =
\BP3\ =
\CP1\ =
Jo + (~ - 2) + (5 - 3) = jf+4 = -fj = ~
J(~ + + (~) +
J¥ + *+ = J2j = ~JW
J(-~- +
+
J¥ + ~.J85
2
2
2)
2
2
4)
2
2
2
(4- 5) =
1
2
1=
(1 :__ 1}
(4- 5? =
21 . (a) Since the sphere touches the xy-plan.e , its radius is the distance from its center, (2, -3, 6), to the xy-plane, namely 6.
Therefore r = 6 and an equation of the sphere is (x-
2? + (y + 3? + (z - 6? = 6 = 36.
2
(b) The radius of this sphere is the distance from its center (2, - 3, 6) to the yz-plane, which is 2. Therefore, an equation is
(~- 2) 2 + (y + 3) 2 + (z - 6) 2 = 4.
(c) Here the radius is the distance from the center (2, - 3, G) to the xz-plane, which is 3. Therefore, an equation is
(x- 2) 2 + (y + 3) 2 + (z- 6} 2 = 9.
23. The equation x = 5 represents a plane parallel to the yz-plane and 5 units in front of it.
25. The inequality y < 8 represents a half-space consisting of all points to the left of the plane y = 8.
27. The inequality 0 ~ z ~ 6 represents all points on or between the horizontal planes z = 0 (the xy-plane) and z = 6.
29. Because z = - 1, all points in the region must lie in the horizontal plane ~ = -1. In addition, x 2 + y 2 = 4, so the region
consists of all points that lie on a circle with radius 2 and center on the z-axis that is contained in the plane z = -1.
31. The inequality x 2 + y 2 + z 2 ~ 3 is equivalent to jx 2 + y 2 + z 2 ~ J3, so the region consists of those points whose distance
from the origin is at most V3. This is the set of all points on or inside the sphere with radius ../3 and center (0, 0 , 0).
33. Here x 2 + z 2 ~ 9 or equiva lently Jx2 + z 2 ~ 3 which describes the set of all points in JR3 whose distance from they-axis is
at most 3. Thus, the inequality represents the region consisting of all points on or inside a circular cylinder of radius 3 with
axis the y-axis.
35. Thi s describes all points whose x-coordinate is between 0 and 5, that is, 0 < x < 5.
37. This describes a region all of whose points have a distance to the origin which is greater than r, but smaller than R. So
inequalities describing the region are r < jx 2 + y 2 + z 2 < R, or r 2 < x 2 -1- y 2 + z 2 < R 2 .
@ 2012 Ccngugc l earnlnu. All R i~lus RescrvcU. May nut be scanned, copit."t.l. or duplicated, or poSicd ton public\)' :~cccssibtc website. in whoJc or in part.
114
0
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
39. (a) To find the x- andy-coordinates of the point P , we project it onto £2
I
and project the resulting point Q onto the x- and y-axes. To find the
z-coordinate, we project P onto either the xz-plane or the yz-plane
(using our knowledge of its x - or y-coordinate) and then project the
resulting point onto the z-axis. (Or, we could draw a line parallel to
QO from P to the z-axis.) The coordinates of Pare (2, 1, 4).
(b) A is the intersection of £ 1 and £ 2, B is directly below the
y-intercept of L2, and 0 is directly above the x-intercept of £2.
41. We need to find a set of points { P(x, y , z) II API = IBPI}.
J(x + 1)2 + (y - 5)2 + (z- 3)2 = J(x - 6)2 + (y - 2)2 + (z + 2)2 ~
(x + 1) 2 + (y- 5) + (z - 3) 2 = (x - 6) 2 + (y - 2) 2 + (z + 2) 2 =>
x
2
+ 2x + 1 + y 2 - lOy + 25 + z 2 - 6z + 9 = x 2 - 12x + 36 + y2 - 4y + 4 + z2 + 4z + 4 '=>
14x - 6y - lOz = 9.
Thus the set of points is a plane perpendicular to the line segment joining A and B (since this plane must contain the
perpendicular bisector of the line segment AB).
43. The sphere x 2 + y 2 + z 2 = 4 has center (0, 0 , 0) and radius 2. Completing squares in x 2 - 4x + y 2 - 4y + z 2 - 4z = -11
gives (x
2
-
4x + 4) + (y2 - 4y + 4) + (z 2 - 4z + 4) = - 11 + 4 + 4 + 4 =>
(x - 2) 2 + (y - 2? + (z- 2) 2 = 1,
so this is the sphere with center (2, 2, 2) and radius 1. The (shortest) distance between the spheres is measured along
the line segment connecti ng their centers. The distance between (0, 0, 0) and (2, 2, 2) is
.J(2 - 0)2 + (2- 0)2 + (2 - 0)2 = ffi = 2 J3, and subtracting the radius of each circle, the distance between the
spheres is 2 v'3 - 2 - 1 = 2 .J3 - 3.
12.2 Vectors
1. (a) The cost of a theater ticket is a scalar, because it has only magnitude.
(b) The current in a river is a vector, because it has both magnitude (the speed of the current) arid direction at any given
location.
(c) If we assume that the initial path is linear, the initial fli ght path from Houston to Dallas is a vector, because it has both
magnitude (distance) and direction.
(d) T he population of the world is a scalar, because it has only magnitude.
3. Vectors are equal when they share the same length and direction (but not necessarily location). Using the symmetry,of the
.
-----+
--+ ---+
parallelog~am as a guide, we see that AB = D C, DA
------+ --+
--+
__.
-----+
= CB, DE = EB, and EA = CE.
® 2012 Ccngogc l.cnming. All Rights Reserved. May not be sc.mncd, copied, orlluplicatcd. or posted to a publicly accessible website. in whole or in part.
SECTION 12.2 VECTORS
5. (a)
un
(b)
(c)
w~
uNw
u
(d)
u
u
v
~
(e)
u
0
v+w
(f)
Dw
v+u+w
u
-:Gv
u-w-v
--->
7. Because the taif of dis the midpoint of QR we have QR = 2d, and by .the Triangle Law,
a + 2d = b
=?
2d=b-a
c = b - d = b - (~ b -
d=!{b-a) = th - ~a. AgainbytheTriangleLawwehavec + d = bso
=?
~a) = ~ ~ + ~b.
9. a= (3 - (- 1), 2 - 1}=(4, 1}
11.a =(2 - (-1), 2 - 3} = (3,-1}
y
y
B(3, 2)
0
A(- 1, 3)
X
13. a = ·(2 - 0, 3 - .3,-1 - 1} = (2, 0,-2}
X
15. (-1 , 4} + (6, - 2} = (-1 + 6, 4 + (- 2)} = (5, 2)
y
A(0, 3, 1)
X
X
17. (:~, 0, 1) + (0, 8, 0) = (3 + 0, 0 + 8, 1 + 0)
= (3, 8, 1)
(3, 0, 1)
X
© 2012 Ceng3ge Learni ng. All Rights Reserved. May not be scallll(((. copied. or duplicoltcd. or posted too publicly accessible wcbsilc. in who le or in part
115
116
0
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
19. a + b = (5 + (-3) ,-12 + (-6)) = (2, - 18)
2a + 3b = (10, -24) + (-9, -18) = (1, -42)
lal = y'5 2 + ( - 12) 2 = .jl69 = 13
Ia- b l = 1(5- (-3), - 12- (-6))1 = 1(8, -6) 1= y'82 + ( - 6)2 = v'100 = 10
21. a+ b = (i'+ 2j - 3 k ) + ( -2 i - j + 5 k ) = - i + j + 2k
2a + 3b = 2 (i + 2j - 3 k ) + 3 (-2 i - j + 5 k ) = 2 i + 4j - 6 k - 6 i - 3j + 15 k = - 4 i + j + 9k
lal = y'12 + 22 + ( -3)2 = v'i4
Ia - b l = l(i+ 2j - 3 k) - (-2 i -j +5 k )l = l3 i + 3 j - 8 k l = y'32 +3 2 + (-8)2 = v'82
23. The vector - 3 i + 7 j has length l-3 i + ~ j I = y'( -3)2 + 72 = v'58, so by Equation 4 the unit vector with the same
.
. .IS
d1rect1on
1
r,:n
v58
( - 3.1 + 7 J' ) = - r.;o
3 1. + r.;oJ
7 .·
v58
v58
25. The vector 8 i - j + 4 k has length 18 i - j + 4 k l = y'82 + (- 1) 2 + 4 2 = J8I = 9, so by Equation 4 the unit vector with
the same direction is ~ (8 i - j + 4 k ) = ~ i - ~ j + ~ k.
.
y
27.
.j3
From the figure, we sec that t a n 0 = - 1
= J3 =>
B = 60°.
X
29. From the figure, we see that the x-component ofv is
v1
= Ivi cos(1r/ 3) = 4 ·. ~ = 2 and they-component is
V2
= Ivi sin(1r / 3) = 4 · .,;; = 2 v'3. Thus
V
= (th, V2) = (2, 2 .j3 ).
y
v
Vz
v,
X
31. The velocity vector v makes an angle of 40° with the horizontal and
has magnitude equal to the speed at which the footba ll was thrown.
From the figure, we see that the horizontal component of v is
Ivi cos 40° = 60 cos 40° ~ 45.96 ft/s and the vertical component
is Ivi sin 40° = 60 sin 40° ~ 38.57 fils.
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SECTION 12.2 VECTORS
0
117
33. The given force vectors can be expressed in terms of their horizontal and vertical components as -300 i and
~00 cos 6~0 i + 200 sin 60°j = 200 ( ~) i + 200 ( ~) j = 100 i + 100 v'3 j . The resultant force F is the sum of
these two vectors: F = (-300 + 100) i + (0 + 100 v'3) j = - 200 i
+ 100J3j. Then we have
IFI ~ V(-200) 2 + (100v'3) 2 = .j70,000 = 100 V7 ~ 264.6 N. Let() be the angle F makes with the
· · x-ax1·s. Then tan () = lOO
. I pomt
. of F 11es
· m
. t he second qua drant, so
positive
_ J3 = -. J3 an d the termma
2
200
() = tan- 1 ( -
~) + 180° ~ -40.9° f. 180° = 139.1° .
35. With respect to the water's surface, the woman's velocity is the vector sum of the velocity of the ship with respect
to the water, and the woman's velocity with respect to the ship. If we let north be the positive y-direction, then
v = (0, 22) + (-3, 0) = (-3, 22). The woman 's speed is lvl = .jg + 484 ~ 22.2 mi/ h. The vector v makes an angle()
with the east, where()= tan- 1 ( ~;) ~ 98°. Therefore, the woman's direction is about N(98- 90) 0 W = N8°W.
37. Let T 1 and T 2 represent the tension vectors in each side of the
clothesline as shown in the figure. T 1 and T 2 have equal vertical
components and opposite horizontal components, so we can write
T 1 = - a i + b j and T 2 = a i + bj [a, b > 0]. By si milar triangles, !!_ =
a
0 08
·
=> a = 50b. The force due to gravity
4
acting on .Ule shirt has magnitude 0.8g ~ (0.8)(9.8) = 7.84 N, hence we have w = - 7.84j. The resultant T 1 + T 2
=> (-a i + bj ) + (a i + bj ) = 7.84 j
of the tensile forces counterbalances w , so T 1 + T 2 = -w
( -50b i + bj ) + (50b i + bj ) = 2bj = 7.84j
=> b =
7
·:
4
=>
= 3.92 and a= 50b = 196. Thus the tensions are
T 1 = - a i + bj = - 196i + 3.92j and T 2 = ai + bj = 196i + 3.92j .
Alternatively, we can find the value of() and proceed as in Example 7.
y
39 .. (a) Set up coordinate axes so that the boatman is ·at the origin,- the canal is
bordered by the y-axis and the line x = 3, and the current flows in the
negative y-direction. The boatman wants to reach the point (3, 2). Let() be
the angle, measured from the positive y-axis, in the direction he should
steer. (See the figure.)
X
In still water, the boat has velocity v b = (13 sjn B, 13 cos()) and the velocity of the current is V c (0, -3.5), so the true path
of the boat is determined by the velocity vector v = v b + V c = (13 sin B, 13 cos() - 3.5). Let t be the time (in hours)
after the boat departs; then the position of the boat at time t is given by tv and the boat crosses the canal when
tv = (13sin0, 13cos0- 3.5) t = (3, 2). Thus 13(sinO)t = 3
=>
3
. ()and (13 cosB- 3.5) t = 2.
13 sm
·
t = -
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118
0
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
Substituting gives {13 cos (} - 3.5)
13
:in(} = 2 => 39 cos 8- 10.5 = 26 sin(} (1). Squaring both sides, we have
+ 110.25 = 676 sin2 () = 676 ( 1 - cos2 9)
2
1521 cos 8 - 819 cos 8
2197 cos 2 (} - 819 cos(} - 565.75 = 0
The quadratic formula gives
_ 819 ± v'C--819)2- 4{2197)(- 565.75)
cos 8. -
·2{2197)
·= 819 ± ~5g~42 ' 572 :::::: 0. 72699 or - 0.35421
The acute value for(} is approximately cos-- 1 (0.72699) :::::: 43.4°. Thus the boatman should steer in the direction that is
43.4° from the bank, toward upstream.
Alternate solution: We could solve ( 1) graphically by plotting y = 39 cos 8- 10.5 andy = 26 sin 8 on a graphing device
and finding the appoximate intersection point (0.757, 17.85). Thus 0:::::: 0.757 radians or equivalently 43.4°.
(b) From part (a) we know the trip is completed when t
= ·-13 sm
~ (} .· But 8 :::::: 43.4°, so the time required is approximately
3
.
:::::: 0.336 hours or 20.2 minutes.
13sm 43 .40
41 . The slope of the tangent line to the graph ofy = x 2 at ,the point {2, 4) is
·
I
dy
=
dx x=2
2xl
= 4
x=2
and a parallel vector is i + 4 j which has length Ji + 4j J = .)1 2 + 42 = ..JPi, so unit vectors parallel to the tangent line
are ±~ (i + 4 j ).
~
_..
--t
,_..
-+
------+-
------+-
----+-
._.
---+
---+
43. By the Triangle Law, AB + BC = AC. Then AB + BC + CA = AC + CA, but AC + CA = AC +
--+ --+
(
--+) = 0 .
-AC
--+
SoAB+BC+CA = 0.
(c) From the sketch, we estimate that s :::::: 1.3 and t:::::: 1.6.
45. (a), (b)
(d) c = sa + t b
#
7 = 3s + 2t and 1 = 2s - t.
Solving these equations gives s =
t and t = V.
47. Jr- ro J is the distance between the points (x, y , z) and (xo, y 0 , z 0 ), so the set of points is a sphere with radius 1 and
center (xo, Yo, zo).
Alternate method: Jr - ro J = 1
#
J(x- xo) 2 + (y- vo)2 + (z- zo)2 = 1
#
(x - xo? + (y - yo) 2 + (z - zo? = 1, which is the equation of a sphere with radius 1 and center (xo, yo, zo).
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SECTION 12.3 THE DOT PRODUCT
0
49. a+ (b + c) = (at, a2} + ((bt , b2} + (c1, c2}) = (at , a2} + (b1 + c1, ~ + c2)
= (a 1 + b1 + c1, a2 + ~ + c2} = ((a1 + bt) + c1, (a2 + ~) + c2)
= (a1 + b1 ,a2 + b2} + (ct, c2} = ((a1,a2} + (b1 , b2}) + (c1,c2}
.= (a+ b } + c
_____.
-+
-+
51 . Consider triangle ABC, where D and E are the midpoints of AB and BC. We know that AB. + BC = AC
---+
---+
_.
I
.-;.
---+
---+
-+
(1)
----+
..
and
---t
DB +BE= DE (2). However, DB = ~AB, and BE = ~BC. Substituting these expressions for DB and BE into
---+
---+
-+
---t
----+ .
.-.
--+
· (2) gives ~AB + ~BC = DE. Comparing this with (1) gives DE = ~AC. Therefore·AC and DE are parallel and
12.3 The Dot Product
1. (a) a · b is a scalar, and the dot product is defined only for vectors, so (a · b) · c has no meaning.
(b) (a . b) c is a scalar multiple of a vector, so it does have meaning.
(c) Both [a[ and b · care scalars, so [al (b ·c) is an ordinary product of real numbers, and has meaning.
(d) Both a and b +care vectors, so th~ dot product a · (b +c) has meaning.
(e) a · b is a scalar, but cis a vector, and so the two quantities cannot be added and a · b -1:' c has no meaning.
(f) Ia ! is a scalar, and the dot product is defined only for vectors, so lal · (b + c ) has no meaning.
3. a· b = ( - 2, ~) · (- 5,1 2} = (-2)(-5} + (~)(12) = 10 + 4 = 14
5. a · b = (4, 1, i ) · (6, -3, - 8) = (4)(6) + (1)(- 3) + (i) (- 8) = 19
7. a· b = (2 i + j ) · (i - j + k ) = (2)(1) + (1) ( - 1) + (0)(1) = 1
' - 1\
' '•
9. By Theorem 3, a · b = lal lb1 cos 0 = (6)(5) cos 2; = 30 ( - ~) = -15.
11. u , v , aJ)d ware all unit vectors, so the triangle is an equilateral triangle. Thus the angle between u and v is 60° and
u · v = lullvl cos60° = (1)(1) (~) = ~. lfw is moved so it has the same initial point as u , we can see that the angle
between them is 120° and we have u · w = lul lw l cos 120° = (1)(1) ( -~) = - ~.
13. (a) i · j = (1, 0, 0) · (0, 1, 0) = (1)(0) + (0)(1) + (0)(0) = 0. Similarly, j · k = (0)(0) + (1)(0) + (0) (1) ,;, 0 and
k . i = {0){1) + {0){0) + (1)(0) = 0.
Another method: Because i, j , and k are mutually perpendicular, the cosine factor in each dot product (see Theorem 3)
is cos~= 0.
(b) By Pr~perty I of the dot product, i · i = lil = 1 2 = 1 since i is a unit vector. Similarly, j · j = lj l = 1 and
2
2
k . k = lkl 2 = 1.
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1-19
120
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
D
15. ja j = vW + 32 = 5, jbj = }22 + (- 1)2 = v'5, and a· b = {4){2) + {3)(- 1) = 5. ~rom Corollary 6, we have
a· b
cos(}= jajjb/ =
5
5
. v'5 =
1
· .
1 ( 1 ) ·
vg·
So the angle between a and b IS(}= cosv'5 ~ 63°.
17. jaj = .,j32 + (-1)2 + 52 = .;35, jbj = .,j(- 2) 2 + 42 + 32 = v'29, and a · b ~ {3){ -2) + (-1)(4) + {5)(3) = 5. Then
cos B = l:l·l~l = .;35 ~ v'29 =
v'1~ 15 and the angle between a and b is.B = cos- 1 ( v'1~15 ) ~ 81° .
19. jaj = .J42 + ( -3)2 + 12 = .;26, jbj = .,j22 + 02 + ( -1)2 = v'5, and a. b = {4){2) + ( - 3)(0) + (1)( -1) = 7.
a· b ·
7
7
d
. _1 ( 7 )
o
T hen cos (} = jajjbj = v'26 . v'5 = v'130 an 8 = cos
v'130 ~ 52 .
21. Let p, q, and r be the angles at vertices P, Q, and R respectively.
-->
-'-+
Then p is the angle between vectors PQ and P R , q is the angle
-->
-->
between vectors QP and QR, and r is the angle between vectors
---+
--> .
RPaildRQ.
-->
-->
.
2
4
2
(- • 3 ) · (1 • )
= - + 12 =___!Q_and =cos- 1 (___!Q_) ~48° . Similarl,
.,j(- 2)2+32 .j12+ 42
y'l3ffi v'ffi p
y'ffi
y
!?O!!nl
PQ·PR =
Thuscos =
p
---+
---+
QP . QR .
cosq =
· (3, 1)
6- 3
3
IQ?IIQRI= v'4(2, -3)
+ 9 v'9 + 1 = v'13 v'f5 = v'13Q so =cos
- 1(
q
3 )
o
v'l3Q ~ 75 an
d
r ~ 180° - {48°+ 75°) = 57°.
.
.
Alternate solution: Apply the Law of Cosines three times as follows: cos p =
,___.,2
,___.,2 1--,-+12
~-~-n
,__. , ,__.
2 PQ
2
2
_i?R( -I?QI
.- IQJil. , and
,___.,, __. ,
cos q ·
2 PQ
2
_
cosr -
j?Q( - lnl -IQ'RI
---.
QR
1
2
.-.
2 PR QR
1
PR
11
1
23. (a) a . b = (-5){6) + (3)( - 8) + {7){2) = -40 -:/= 0, so a and bare not orthogonal. Also, since a is not a scalar multiple
ofb, a and bare not parallel.
(b) a · b = (4) ( -3) + (6)(2) = 0, so a and bare orthogonal (and not parallel).
(c) a· b = (-1){3) + (2){4) + {5)(-1) = 0, soaand bare orthogonal (and not parallel).
(d) Because a = - ~ b , a and b ~e parallel.
--+
---+
----+
--+
----+
----+
25. QP = (-1, - 3, 2), QR = (4, - 2, - 1), and QP · QR = -4 + 6-2 = 0. Thus QP and QRare orthogonal, so the angle of
the triangle at vertex Q is a right angle.
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SECTION 12.3 THE DOT PRODUCT
21. Let a = a 1 i + a 2 j + a 3 k be a vector orthogonal to both i + j and i + k , Then a · (i + j ) = 0
a . (i + k ) = 0
<=>
<=>
a1
0
121
+ a2 = 0 and
a 1 + a3 = 0, so a 1 = - a2 ~ -a3. Furthermore a is to be a unit vector, so 1 = ai + a~+ a§ = 3ai
.
.
1 . Thus a= l 1
.
1.
l k
d
1 '+ 1 ' + 1 k
tw . h .
1mphes
a 1 = ± 73
73 - 7a J- 73 .an a = - 73 1 7a J 73 are o sue umt vectors.
29. The line 2x - y = 3 <=>
y ·= 2x - 3 has slope 2, so a vector parallel to the line is a = (1, 2}. The line 3x + y = 7
<=>
y = - 3x + 7 has slope - 3, so a vector parallel to the line is b = (1, - 3). The angle between the lines is the same as the
angle 8 between the vectors. Here we have a · b = (1)(1) + (2)( - 3) = - 5, lal = v'P + 22 = v'5, and
-5
a ·b
[hi = J1 2 + (-3)2 = v'lQ, so cos 8 = Ialibi = y'5. v'1Q =
- 5
5
1
v'2
J2 =- J2 or ""'2· Thus 8 = 135°, and the
acute angle between the lines is 180° - 135° = 45°.
<=> x3 - x2 = 0 <=> x 2 (x - 1) = 0 <=> x = 0, x = 1. We have
31. The curves y = x 2 andy= x3 meet when x 2 = x 3
i:-x2 = 2x and .!£x3 = 3x2 , so the tangent lines of both curv~s have slope 0 at x = 0. Thus the angle between the curves is
dx
dx
0° at
th~ point (0, 0). For x = 1,
! x 1:c=l
2
= 2 and
d~ xt:=l = 3 so the tangent lines at the point (1, 1) have slopes 2 and
3. Vectors parallel to the tangent lines are (1, 2} and (1, 3}, and the angle (} between them is given by
cos(}
= (1, 2} . (1, 3} = 1 + 6 = _ 7_
1(1, 2} 1 1(1,3}1
v'5v'W 5v'2
~3. Since 1(2, 1, 2) 1 = v'4 + 1 + 4 = V9 = 3, using Equations 8 and 9 we have cos a=~ . cos/3 =~. and cos-y = ~· The
direction angles are given by a = cos- 1 ( i) ~ 48°, /3 = cos- 1 (~) ~ 71°, and 1 = cos- 1 (~) = 48°.
35. Since I i - 2 j - 3kl = v'1 + 4 + 9 = v'l4, Equations 8 and 9 give cos a =
7IT• cos/3 = * ' and cos-y = 7&, while
a = cos- 1 (~ ) ~74° , {3 = cos- 1 (-~) ~ 122°, and /=cos- 1 (- vih) ~ 143°.
37. l(c, c, c) I = v'c2 + c2 + c 2 = v'3c [since c > 0], so .c os a = cos/3 = cos:-y =
~ = ~ and
v3c
v3
r;an
a ·b
-5 · 4 + 12 · 6
39. IaI = J( -5)2 + 122 = v w:.r = 13. The scalar projection of b onto a is comp 11 b = - - =
= 4 and the
13
1a 1
vector projection ofb onto a is, proj 11 b = ( j~~)
: = 4·
1 1
f3 (-5, 12} = (- ~, - ~ ).
41 . la l = v'9 + 36 + 4 = 7 so the scalar projection of b onto a is compab = j~~ = ~ (3 + 12..:. 6) =
l The vector
. . of b onto a .IS,. prOJa
. b = 7j;f
9 a = 79 · 7l (3 , 6 , - 2} = 4o
o (3 , 6 , - 2} = ( 49
27 54
18 )
proJeCtiOn
• 49• -49
·
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122
0
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
43. lal = J4 + 1 + 16 = J2I so the scalar projection of b onto a is compa b = a · bl =
a
1
0
-; ;
21
2
1
= - - while the vector
J2I
. .
fb
.
. b
1 a
1
2i- j + 4 k
prOJeCtion 0
Onto a IS prOJa = J2T jaj = J2T ·
J2T = 2l1 ( 2 1• - J• + 4 k ) = 212 i - 2T1 j + 2T4 k.
'
a ·b
a·b
2
45. ( ortha b ) · a = (b - p roja b ) · a = b · a - (proja b } · a = b · a - - 2 a· a = b · a - - -2 Ia I = b · a - a· b = 0.
Ja l
lal
So they are orthogonal by (7).
a·b
47. compa b = j;f = 2
~
a · b = 2lal = 2 v'IO. lfb = (b1, b:l, b3) , then we need 3bl + Ob2- 1b3 = 2 v'IO.
One possible solution is obtained by taking b1 = 0, b2 = 0, bs = - 2 v'IO. In general, b = ( s, t, 3s - 2 v'10 ), s, t E llt
49. The displacement vector is D = (6 - 0) i + (12 - 10} j + (20- 8) k = 6 i + 2j + 12 k so, by Equation 12, the work done is
W = F · D = (8 i - 6j + 9 k ) · (6i + 2j + 12k} = 48 - 12 + 108 = 144joules.
51. Here ID I = 80 ft, IF I = 30 lb, and B = 40°. Thus
w =F . D = IF IIDI cos B = (30)(80) cos 40° = 2400 cos 40° ~ 1839 ft-lb.
53. First note that n = (a, b) is perpendicular to the line, because if Q 1 = (a 1, bt) and Q2 = (a2 , b2) lie on the line, then
-----+
n · Q1 Q2 = aa2 - aa1 + ~ - bb1 = 0, since aa2 + bb:l = - c = aa1 + bb1 from the equation of the line.
I
Let P2 = (x2, y2) lie on the line. Then the distance from P1 to the line is tJ:te absolute value of the scalar projection
----+p.
(P
-----t
P. )
In · (x2- x 1, Y2 - Yl ) l
lax2- ax1 + by2 -lnJ1 I
lax1 + by1 + cl
ofp1 2 onto n . compn
1 2
=
I I
=
=
n
vr:?'7l:?
a2 + b2
vr::?7'l:?
a2 + b2
since ax2 + by2 =-c. The required distance is 1( 3)( - 2) + (- 4 )(3 ) + 5 1 = 13 .
5
)32 + (- 4)2
.
55. For convenience, consider the unit cube positioned so that its back left corner is at the origin, and its edges lie along the
coordinate axes. The diagonal of the cube that begins at the origin and ends at (1, 1, 1} has vector representation (1, 1, 1).
The angle B between this vector and the vector of the edge which also begins at the ori~in and runs along the x-liXis [that is,
. .
b
B
(1, 1, 1) · (1, 0, 0)
1
(1,0,0)] 1sg1ven ycos = I( , , }I I( , 0, 0) I = J3
1 1 1
1
57. Consider the H- C-H combination consisting of the sole carbon atom and the two hydrogen atoms that are at (1, 0, 0) and
(0, 1, 0) (or any H - C -H combination, for that matter). Vector representations of the line segments emanating from the
carbon atom and extending to these two hydrogen atoms are ( 1 -
t,0 - t, 0 - t ) = ( t ,- t, -t) and
(0 - t.1- t.0- t) ~ (- t, t, - t ). The bond angle, B, is therefore given by
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,I
SECTION 12.4 THE CROSS PRODUCT , 0
123
59. Let a = (a1, a2, a3) and= (b1, b2, b3).
Property 2:
a· b = (a1, a2 , a3 ) · (b1, b2, bs) = a1b1 + a2b2 + asbs
= b1a1 + ~a2 + b3a3 = (b1, ~. bs ) · (a1 , a2, as ) = b ·a
Property 4:
(ca) · b = (ca1, ca2, ca3) · (b1, b2, bs) = (ca1)b1 + (ca2)b2 + (ca3)b3
= c(a1b1 + a2b2 + asbs) = c (a· b) = a1(cb1) + a2(~) + a 3(cbs)
= (a1 , a2, as)· (cbt, c~, cbs) = a· (c b )
Property 5:
0 ·a = (0, 0, 0) · (a1, a2, a3) = {O)(al) + (O)(a2) + (O)(a3) = 0
61 . Ia · b l = l lal lbl cos8l = lallbl lcos 81. Since lcos81 ~ 1, Ia · bl = lal lbl lcos 81 ~ lal lbl.
Note: We have equality in the case ofcos8 = · ± 1, so 8 = 0 or 8 = 1r, thus equality when a and bare parallel.
t he Parallelogram Law states that the sum of the squares of the
63. (a)
lengths of the diagonals of a parallelogram equals the sum of the
squares of its (four) sides.
2
2
2
2
(b) Ia + b l =(a + b) · (a + b )= lal + 2(a ·b)+ lbl and Ia - b l = (a- b )· (a- b)= lal
2
-
2(a · b )+ lb l2 •
2
Adding these two equations gives Ia + b l2 + Ia- b l2 = 2 lal + 2 lbl 2 .
12.4
The Cross Product
k
j
= I~ -20
1. a x b = 6 0 -2
0 8 0
.
1 1
6 - 2 .
6 oI
J +
k
0
0
0 8
1
1
1
= [0 - (- 16)]i - (0 - O)j + (48 - O) k = 16 i + 48k
Now (a x b)· a = (16, 0, 48) · (6, 0, -2) = 96 + 0 - 96 = 0 and (a x b)· b = (16, 0, 48) · (0, 8, 0) = 0 + 0 + 0 = 0, so
a x b is orthogonal to both a and b .
j
3. a x b =
k
1 3 - 2
-1 0
5
= I ~ ~~I i - ~-~ -~
lj l~~ ~I
+
k
·= (15 - O)i - (5 - 2) j + [0 - (- 3))k = 15 i -3j +3 k
Since (a x b) · a= {15i - 3j + 3k) · (i + 3j - 2k) = 15 - 9-6 = O, a x bisorthogonal to a .
Since (ax b)· b = (15 i - 3j + 3 k) · ( - i + 5 k) = - 15 + 0 + 15 = 0, a x b is orthogonal to b .
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124
0
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
k
j
5. ax b = 1 - 1 -1
1
1
1
2
= ,-~ -~
li 11
1
2
2
- 11, J. + 111
2
2
-~ lk
Now {ax b )· a= {~ i - j + ~ k ) · (i - j - k) = ~ + 1- ~ ~ 0 and
(a x b ) · b = {~ i - j + ~ k) · Gi + j + ~ k) = ~ - 1 + ~ = 0, so a x b is orthogonal to both a and b .
k
j
7. a x b =
t
1 1/t
e t
2
1
= 1 1/t Ii -I t
1t21
I· I t t2 I
1/t
+
t21 J
t2
1 k
= (1 - t) i - (t- t) j + (t3 - t 2 ) k = (1 - t) i + (t3 - t 2 ) k
Since {a X b ). a = (1 -t,O,t3 -
t2). (t, 1, 1/t) = t - e +0 +t2 - t = 0, a X b is orthogonal to a.
Since (a x b ) · b = ( 1 - t, 0, t 3 - t 2 ) • ( t 2 , t 2 , 1) = t 2 - t 3 + 0 + t 3 - t 2 = 0, a x b is orthogonal to b .
9. According to the discussion precedin.g Theorem II , i x "j = k, so (i x j ) x k = k x k = 0 [by Example 2).
11. (j - k) X (k- i) = (j - k) X k + (j - k) .X ( - i)
by Property 3 of Theorem I I
= j X k + ( - k) X k + j X ( -i) + ( -k) X ( - i)
by Property 4 of Theorem 11
2
= (j X k ) + (-1){k X k) + (- 1)(j X i)+ (-1) (k X i)
= i + (- 1) 0 + (-1)(- k) + J = i + j + k
by Property 2 of Theorem I 1
by Example 2 and
the discussion preceeding Theorem II
13. (a) Since b x cis a vector, the dot product a· (b x c) is meaningful and is a scalar.
(b) b ·cis a scalar, so a x (b ·c) is meaningless, as the cross product is defined only for two vectors.
(c) Since b x cis a vector, the cross 'product a x (b x c) is meaningful and results in another vector.
(d) b · cis a scalar, so the dot product a · (b ·c) is meaningless, as the dot product is defined only for two vectors.
(e) Since (a . b ) and (c . d) are both scalars, the cross product (a . b ) X ( c . d) is meaningless.
(f) a x band c x d arc both vectors, so the dot product (ax b )· (c x d ) is meaningful and is a scalar.
15. If we sketch u an~ v starting from the same initial point, we see.that the
angle between them is 60°. Using Theorem 9, we have
lu x vi = lullvl sin 0 = {12)(16) sin 60° = 192 ·
v; = v'3.
v
96
By the right-hand rule, u x v is directed into the page. .
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SECTION 12.4 THE CROSS PRODUCT
j k
17. axb =
2 - 1 3 =
4
2 1
.0
i - (2- 12)j + [4- (- 4)] k = - 7i+10j+8 k
~-~ :Ii - I~ : lj + I~ -1,2 k = ( -1-6)
.
j k
b xa=
4
21 k = [6- ( - 1)]
. i- {12 - 2)j + ( -4 - 4) k = 7 i - lOj- 8 k
=1
-~
~
li
-I:
~
lj
+ 1: -1
3
2 1
2 -1
Notice a x b = - b x a here, as we know is always true by Property I of Theorem II .
19. By Theorem 8, the cross product of two vectors is orthogonal to both vectors. So we calculate
3i
2j
k1 = 12
- 1
1
0
(3,2,1)x(- 1, 1, 0) =
1
11 i -
0
I-13 011· j + I - 31 21
k= - i-j +5 k.
1
.
I b h
± (-1, -1,5)
±(-1, -1,5) h . (
1
1
5 )
So two umt vectors orthogona to ot are v' + +
=
../3 , t at 1s, - M,
-M ~ 37:3
1 1 25
3
21 . Let a= (a1, a2, as). Then
O x a=
0
j
k
0
0
= Ia: :31i
a1 a2 as
j
125
k
axO = a1 a2 as =
Ia~ I
0
- ~~ :sl j + 1:1
as i _
0
I a1 as, . +
0 0 J
0 lk= O,
a2
I
a1 a21
0 O k = O.
0 0 0
25. ax (b+c) =a x (b1 +c1,b2 +c2,bs +c3)
= (a2(bs + cs)- a3(b2 + c2), as(b1 + c1)- a1(b3 + cs) , a1(b2 + c2)- a2(b1 + q))
= ((a2bs - asb2) + (a2cs - asc2), (asb1- a1bs) + (asc1 - a1c3), (a1b2 - a2b1) + (a1c2 - a2c1))
= (a2bs- asb2 , aab1 - a1bs, a1b2- a2b1 ) + (a2cs - asc2, asc1- a1c3, a1c2 - a2c1 )
= (a x b) + (a X c)
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126
0
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
y
27. By plotting the vertices, we can see that the parallelogram is determined by the
---+
---+
vectors AB = (2, 3) and AD= (4, -2}. We know that the area of the parallelogram
determ ined by two vectors is equal to the length of the cross product of these vectors.
.
---+
In order to compute the cross product, we consider the vector AB as the three---+
dimensional vector (2, 3 , 0) (and similarly for AD), and then the area of
parallelogram ABCD is
j
IA13 x ml=
2 3
4 - 2
k
o =l(o) i- (o) j+ (-4 - 12)kl=l-16kl = 16
0
---+
---+
29. (a) Because the plane through P, Q, and R contains the vectors PQ and P R, a .vector orthogonal to both of these vectors
---+
---+
(such as their cross product) is also orthogonal to the plane. Here PQ = (- 3, 1, 2} and P R = (3, 2, 4}, so
---+
---+
PQ x PR = ((1)(4)- (2)(2), (2)(3)- (-3)(4) , (-3)(2)- (1)(3)) = (0, 18, - 9)
Therefore, (0, 18, - 9) (or any nonzero scalar multiple thereof, such as (0, 2, - 1)) is orthogonal to the plane through P , Q,
andR.
(b) Note that the area of the tr~angle determined by P, Q, and R ls equal to half of the area of the
parallelogram determined by the three points. From part (a), the area of the parallelogram is
I.PQ x
PR I = I(0, 18, -9) I = y'O + 324 + 81 = v'4o5 = 9v'5, so the area of the triangle is ~ · 9v'5 = ~ .J5.
---+
---t
31 . (a) PQ = (4, 3, - 2) and P R = (5, 5, 1}, so a vector orthogonal to the plane through P, Q, and R is
---+
---+
PQ x P R = ((3)(1) - ( -2)(5), ( -2)(5) - (4)(1), (4)(5) - (3) (5)) = (13, - 14, 5) [or any scalar mutiple thereof].
---+
---+
(b) The area of the parallelogram determined by PQ and PR is
IPQ x P'RI = 1(13,-14, 5)1 = yl132 + (- 14)2 +52 = v'395,sotheareaoftriangle PQR is ~v'39Q.
33. By Equation 14, the volume of the parallelepiped detennined by a , b , and cis the magnitude of their scalar triple product,
'
1 2 3
which is a · (b x c) = -1 1 2
.
2 1 4
1- 1
= 1 111 21
- 2
24 1 + 3
4
2
1-12 111 = 1(4-2)-2(- 4-4) + 3(-1-2) = 9.
Thus the volume of the parallelepiped is 9 cubic units.
---t
--+
--+
35. a= PQ = (4, 2, 2}, b = PR = (3, 3, -1), and c = P S = (5,5, 1).
so the volume of the parallelepiped is 16 cubic units.
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SECTION 12.4 THE CROSS PRODUCT
1
D
127
5 -2
1 0
3 0
3
0 =1 1- ·
1- 5 1
1+(- 2) 1 -ll = 4 + 60 - 64 = 0,whichsaysthatthevolume
9 -4
5 -4
5 9
.
.
9 - 4
37. u · (v x w)= 3-1
5
of the parallelepiped determined by u , v and w is 0, and jhus these three vectors are coplanar.
39. The magnitude of the torque is Jr J = Jr x FJ = JrJJF JsinO = (0.18 m)(60 N) sin(70 +lOt = 10.8 sin80° r::::: 10.6 N·m.
41 . Using the notation of the text, r = {0, 0.3, 0) and F has direction (0, 3, -4). The angle(} between them can be determined by
cos 9 =
(0, 0.3, 0) . {0, 3, -4) .
J(o, o.3, o)JI{o,3, -4)1
100 = 0.3IFisin53.1°
9
=> cos =
0.9
(0.3)(5)
=> cosO = 0.6 =>
(J r::::: 53.1°. Then Jr l =
lri iFI sin(} =>
=> 1F i r:::::417N.
43. From Theorem 9 we have Ia x bl = la l lb l sin 9, where 0 is the angle between a and b , and from Th,eorem 12.3.3 we have
=>
a· b = lal lb l cosO
Ja l lbl = a · be . Substituting the second equation into the first gives Ia x b l = a · b sin B, so
cos
cos (J
Ia X b l = tan9. Here Ia X b l = 1(1, 2, 2)1 = V1 + 4 + 4 = 3, so tan() = Ia x bbl = ~ = J3
. a· b
a·
v3
=>
() = 60° .
The distance between a point and a line is the length of the perpendicular
45. (a)
from the point to th~ line, here
d=
a
In I=d. But referring to triangle PQS,
In I= IQP' sin = lbl sin 9. But 8 is the angle between QP = b
(J
--+
Ia x bl
and QR = ~·Thus ~y Theorem 9, sinO = lai Jb l
= lb1 . 9 = Jbi Ja x bl = Ia x b J
d
an so d
'I sm
Jai Jbl
laJ
--+
---+
(b) a= QR = (- 1, - 2, - 1) and b = QP = (1, - 5, - 7}. Then
a x b = ((-2)( -7) - ( -1){7 5), (- 1)(1)- ( - 1)( - 7) , ( -1)( -5)- ( - 2)(1)} = (9, - 8, 7).
Thus the distance is d = Ia [:Ibl = ~ JB l + 64 + 49 =
ji¥ = j¥.
47. From Theorem 9 we have Ia x b l = [al[b[ sin 9 so
Ja x b] 2 = [a[ 2 [b[ 2 sin2 () = [a[ 2 lb[2 (1 - cos2 B)
2
= fa i Jb[
2
-
2
2
2
([ai Jb [cos 9) = [a[ lb[ - (a · b)
2
by Theorem 12.3.3.
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128
D
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
49. (a - b ) x (a + b) = (a - b) x a + (a - b ) x b
.
by Property 3 of Theorem 11
=ax a+ (-b)x a +ax b+(- b)x b
by Property 4 ofTheorem I I
=(a X a)- (b x a)+ (a X b )- (b x b)
by Property 2 ofTheorem II (with c =: -1)
= 0 - (b x a ) + (a x b )- 0
by Example2
=(ax b)+ (a x b)
by Property I of Theorem ll
= 2(a x b )
51. a X (b X c) + b X (c X a) + c X (a X b )
= [(a · c)b- (a · b)c] + [(b · a)c- (b · c)a] + [(c · b)a - (c: a )b]
by Exercise 50
= (a · c)b - (a· b )c+ (a · b)c- (b · c)a+ (b · c)a- (a · c)b = 0
53. (a) No. If a · b = a · c, then ·a · (b - c) = 0, so a is perpendicular to b - c, which can happen if b -:f. c. For example,
let a = (1, 1, 1), b = (1, 0, 0) and c = (0, 1, 0).
(b) No. If a x b =ax c then a x (b- c) = 0, which implies that a is para llel to b- c; which of course can happen
ifb -:f. c.
(c) Yes. Since a· c = a· b , a is perpendicular to b - c, by part (a). From part (b), a is also parallel to b - c. Thus since
a -:f. 0 but is both parallel and perpendicular to b - c, we have b - c = 0, so b = c.
12.5 Equations of Lines and Planes
1. (a) True; each of the first two lines has a direction vector parallel to the direction vector of the third line, so these vectors are
each scalar multipl~s of the third direction vector. Then the first two direction vectors are also scalar multiples of each
other, so these vectors, and hence the two lines, are parallel.
(b) False; for example, the x- and y -axes are both perpendicular to the z-axis, yet the x- andy-axes are not parallel.
(c) True; each of the first two planes has a normal vector parallel to ~he normal vector of the third plane, so these two normal
vectors are parallel to each other and the planes are parallel.
(d) False; for example, the xy- and yz-planes are not parallel, yet they are both perpendicular to the xz-plane.
(e) False; the x- and y-axes are not paralle l, yet they are both parallel to the plane z = 1.
(f) True; if each line is perpendicular to a plane, then the lines' direction vectors are both parallel to a normal vector for the
plane. Thus, the direction vectors arc parallel to each other and the lines are paralle l.
(g) False; the planes y = 1 and z = 1 are not parallel, yet they are both parallel to the x-axis.
(h) True; if each plane is perpendicular to a line; then any normal vector for each plane is parallel to a direction vector for the
line. Thus, the normal vectors are parallel to each other and the planes are parallel.
(i) True; see Figure 9 and the accompanying discussion.
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SECTION 12.5 EQUATIONS OF LINES AND PlANES
0
129
(j) False; they can be skew, as in Example 3.
(k) True. Consider any normal vector for the plane and any direction vector for the line. If the normal vector is perpendicular
to the direction vector, the line and plane are parallel. Otherwise, the vectors meet at an angle B, 0° :S B < goo., and the
line will intersect the plane at an angle goo -B.
3. For this line, we have r 0 = 2 i + 2.4j + 3.5 k and v = 3 i + 2 j - k, so a vector equation is
r = ro + t v = (2 i + 2.4 j + 3.5 k ) + t(3 i + 2j - k) = (2 + 3t) i + {2.4 + 2t) j + (3.5 - t) k and parametric equations are
X
= 2 + 3t, y = 2.4 + 2t, Z = 3.5 - t.
5. A Line perpendicular to the given plane has the same direction as a normal vector to the plane, such as
n = (1, 3, 1). So r 0 = i + 6 k , and we can take v = i + 3j + k. Then a vector equation is
r = {i
.
'
+ 6 k) + t(i + 3j + k ) = (1 + t) i + 3tj + (6 + t ) k , and parametric equations are x = 1 + t, y = 3t, z = 6 + t.
7. The vector v = ( 2 - 0, 1 - ~ . - 3- 1) = ( 2, ~ . -4) is parallel to the line. Letting Po = (2, 1, -3), parametric equations
· equat1ons
.
Z +3
are x = 2 + 2t, y = 1 + 21 t , z = -3 - 4.t, wh'1le symmetnc
are -X -- 2 = y - 1 = ---=4
or
2
112
X -
2
2 = 2y _ 2 = Z + 3.
- 4
9. v = (3- ( -8) , - 2- 1,4- 4) = (11, - 3,·0), and letting Po= (- 8, 1, 4), parametric equations are x = -8 + llt,
. number
----u-8 :;: y _- 31 , z = 4. Nottcc. here that the d'trectton
. equations
.
y = 1 - 3t, z = 4 + 0t = 4, wI11·1e symmetnc
are x +
c = 0, so rather than writing z ~
4
in the symmetric equation we must write the equation z = 4 separately.
11. The line has direction v = (1, 2, 1). Letting Po = (1, - 1, 1), parametric equations are x = 1 + t, y = -1 + 2t, z = 1 + t
1
. equatiOns
.
an d symmetriC
are x - 1 = -y +- = z - 1.
2
13. Direction vectors of the lines are v 1 = (- 2 - ( - 4), 0 - ( - 6), - 3 - 1) = (2, 6, - 4) and
v 2 = (5 - 10,3 - 18, 14 - 4) = (-5, - 15, 10), and since v 2 = -:-~ v1 , the direction vectors and thus the lines are parallel.
15. (a) The li ne passes through the point (1, - 5, 6) and a direction vector for the line is ( - 1, 2, - 3), so symmetric equations for
.
x- 1
y+5
the line are ---=1 = - 2
z -6
= --=3.
x- 1
y+5
(b) The line intersects the xy-plane when z = 0, so we need -=1' = -
Y~
5
= 2 =>
we need x = 0
2
-
0-6
x- 1
= --=3
or -=1' = 2
=>
x = - 1,
y = -1. Thus the point of intersection with the xy-plane is ( - 1, - 1, 0). S imilarly for the y z-plane,
=>
1= y~
the x z-plane, we need y = 0
5
=>
= z ~36
=> y = - 3, z = 3. Thus the line intersects the y z-plane at {0, - 3, 3). For
=>
x = - ~, z = - ~ . So the line intersects the xz-plane
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130
D
\~~From
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
Equation 4, the line segment. from ro
~
2 i - j + .4 k to r 1 = 4 i + 6 j + k is
r (t) = (1 - t)ro +tr1 = (1 - t)(2 i - j +4 k ) +t(4 i + 6j + k ) = (2 i - j +4k) +t(2 i + 7j- 3k), 0 ::=; t ::=; 1.
l 19. Since the direction vectors (2, -1, 3} and (4, - 2, 5} are not scalar multiples of each other, the Lfnes aren't parallel. For the
'
\
'
\ -
-
lines to intersect, we must be able to find one value oft and one value of s that produce the same point from the respective
..------______:
----·- ----
--~·-----------"'--
parametric equations. Thus we need to satisfy tlie following three equations: 3 + 2t = 1 + 48, 4 - t = 3 - 28,
...-----
1 + 3t = 4 + 5s. Solving the last two equations we get t = 1, 8 = 0 and checking, we see that these values don't satisfy the
first equation. Thus the lines aren't parallel and don't intersect, so they must be skew lines.
r
·• 21. Since the direction vectors (1, - 2, - 3} and (1, 3, -7) aren't scalar multiples of each other, the lines aren't parallel. Parametric
equations of the lines are L1: x = 2 + t, y = 3- 2t, z = 1 - 3t and L2: x ·= 3 + 8, y = -4 + 38, z = 2 -78. Thus, for the
lines to intersect, the three equations 2 + t = 3 + 8, 3 - 2t = -4 + 3s, and 1- 3t = 2- 78 must be satisfied simultaneously.
Solving the first two ~quations gives t = 2, 8 = 1 and checking,_we see that these values do satisfy the third equation, so the
lines intersect when t = 2 and 8 = 1, that is, at the point (4, -1, -5).
23. Since the plane is perpendicular to the vector (1, -2, 5), we can take (1, - 2, 5} as a normal vector to the plane.
(0, 0, 0) is a point on the plane, so setting a= 1, b = -2, c = 5 and xo = 0, yo = 0, z 0 = 0 in Equation 7 gives
1(x - 0) + (-2)(y- 0) + 5(z - 0) = 0 or x- 2y + 5z = 0 as an equation of the plane.
25. i + 4j + k = (1, 4, 1} is a normal vector to the plane and ( - 1, ~, 3) is a point on the plane, so setting a= 1, b = 4, c = 1,
xo = - 1, yo = ~, zo = 3 in Equation 7 gives 1 (x - ( -1 )] + 4 (11 - ~) + 1(z - 3) = 0 or x + 4y + z = 4 as an equation of
the plane.
27. Since the two planes are parallel, they will have the same normal vectors. So we can taken = (5, -1, - 1), and an equation of
the plane is5(x-1) -1 (y- (- 1)]-1[z- (-1)] = Oor5x -11- z = 7.
29. Since the two planes are parallel, they will have the same normal vectors. So we can taken = (1, 1, 1), and an equation of the
plane is 1(x- 1) + 1 (y - ~) + 1 (z - ~) = 0 or x + y + z =
Jt or 6x + 6y + 6z = 11.
31. Here the vectors a= (1- 0, 0 - 1, 1 1 1) = (1, -1, 0) and b = (1 - 0, 1- 1, 0- 1) = (1, 0, - 1) lie in the plane, so
ax b is a normal vector to the plane. Thus, we can taken= ax b = (1 - 0, 0 + 1, 0 + 1) = (1, 1, 1}. If P o is the point
(0, 1, 1), an equation of the plane is 1(x- 0) + 1(y - 1) + 1(z - 1) = 0 or x + y + z = 2.
33. Here the vectors a = (8 - 3, 2 - ( - 1), 4- 2} = (5, 3, 2) and b = (- 1 - 3; - 2 - ( -1) , -3- 2} = (-4, - i , - 5) lie in
the plane, so a normal vector to the plane is n = a x b = (- 15 + 2, -8 + 25, - 5 + 12) = (-13, 17, 7) and an equation of
the plane is - 13(x - 3) + 17[y- (-1)) + 7(z - 2) = 0 or -13x + 17y + 7z = -42.
35. If we first find two nonparallel vectors in the plane, their cross product wi ll be a normal vector to the plane. Since the given
line lies in the plane, its direction vector a= (-2, 5, 4) is one vector in the plane. We can verify that the given point (6, 0, -2)
© 2012 <:engage Learning. All Rights Rcscm:d. May not be scanned, copied, or duplicated, or posted on • publicly accessible website, in whole or in p.'lrt.
SECTION 12.5 EQUATIONS OF LINES AND PLANES
0
131
does not lie on this line, so to find another nonparallel vector b which lies in the plane, we can pick any point on the line and
find a vector connecting the points. If we putt = 0, we see that ( 4 1 3 1 7) is ·on the line, so
b = (6 - 4, 0- 3, -2 -7) = (2, -3, -9) and n = a x b = (-45 + 12, 8-18,6 -10) = (- 33, - 10, - 4). Thus, an
equation of the plane is -33{x - 6)- 10(y- 0)- 4[z - ( -2)] = 0 or 33x +lOy + 4z = 190.
37. A direction vector for the line of intersection is a= n1 x n 2 = {1, 1, - 1) x (2, -1, 3) = (2 1 - 51 - 3), and a is parallel to the
desired plane. Another vector parallel to the plane is the vector .connecting any point on the line of intersection to the given
point ( - 1 1 2 1 1) in the plane. Setting x = 0, the equations of the planes reduce toy- z = 2 and -y+ 3z = 1 with
simultaneous solution y = ~ and z = ~. So a point on the line is (0 1 ~ 1 ~) and another vector parallel to the plane is
( - 1 1 -~ , -~) .Then a normal vector to the plane is n = (2, - 5 1 -3) x (- 11 -~, -~ ) = (- 2,4, - 8) and an equation of
the plane is -2{x + 1) + 4(y- 2) - B(z - 1) = 0 or x - 2y + 4z = --:1.
39. If a plane is perpendicular to two other planes, its nonnal vector is perpendicular to the normal vectors of the other two planes.
Thus (2, 1, -2) x (1, 0, 3) = (3- 0, -2- 6, 0 - 1) = (3, -8, - 1) is a normal vector to the desired plane. The point
(1, 5, 1) lies on the plane, so an equation is 3(x - 1) - B(y - 5)- (z- 1) = 0 or 3x - By- z = - 38.
41. To find the x-intercept we set y
and obtain 2x = 10 =>
---
= z = 0 in the equation 2x + 5y + z = 10
x = 5 so the x-intercept is (5, 0, 0). When
x = z = 0 we get 5y = 10 => y = 2, so they-intercept is (0, 2, 0).
Setting x = y = 0 gives z = 10, so the z-intercept is (0, 0, 10) and we
graph the portion of the plane that lies in the first octant.
X
43. Setting y = z = 0 in the equation 6x - 3y + 4z = 6 gives 6x = 6
x = 1, when x = z = 0 we have -3y = 6
implies 4z = 6
=>
=> y = - 2, and x = y = 0
=> z = ~.so the intercepts are (1, 0, 0), {0, -2, 0), and
(0 1 '0, ~). The figure shows the portion of the plane cut off by the coordinate
planes.
X
I
@
ubstitute the parametric equations ofthe line into the equation of the plane: (3- t) - (2 + t) + 2{5t) = 9 =>
Bt = 8 => t = 1. Therefore, the point of intersection of the line and the plane is given by x = 3 - 1 = 2, y = 2 + 1 = 3,
and z = 5(1) =5 1 that is, the point (2, 3, 5).
47. Parametric equationS for the line are x = t, y = 1 + t, z = ~t and substituting into the equation of the plane gives
4{t)-{1+t)+3(~t)=8
=> ~t=9 => t=2.Thusx = 2,y=1+2 .= 3,z=H2)=1andthepointof
intersection is (2, 3 1 1).
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132
D
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
49. Setting x = 0, we see that (0, 1, 0) satisfies the equatio~s of both planes, so that they do in fact have a line of intersection.
v
= o1 x 02
= (1, 1, 1} x (1, 0 , 1) = (1, 0, -1} is the direction of this line. Therefore, direction numbers of the intersecting
line are 1, 0, - 1.
51. Normal vectors for the planes are 01 = (1, 4, -3} and 0 2 = (-3, 6, 7), so the normals (and thus the planes) aren't parallel.
.
.
But 01 · 0 2 ,;, -3 + 24 - 21 = 0, so the normals (and thus the planes) are perpendicular.
53. Normal vectors for the planes are 01 = (1, 1, 1) and 02
= (1, -1, 1). The normals are not parallel, so neither are tl1e planes.
Furthe'rmore, o1 · o2 = 1 - 1 + 1 = 1 =F 0, so the planes aren't perpendicular. The angle between them is given by·
1_ = ~
lolllo21 = __
.j3.j3 3 =>
cos8 = O t . 0 2
O=cos-
1(1)
3 ::::::70.5o.
55. The normals are 0 1 = (1, -4, 2) and 0 2 = (2, -8, 4). Since 02 = 2o 1 , the normals (and tlms the planes) are parallel.
57. (a) To find a point on the line of intersection, set one of the variables equal to a constant, say z = 0. (This will fail if the line .o f
intersection does not cross the xy-plane; in that case, try setting x or y equal to 0.) The equations of the two planes reduce
to x + y = 1 and x
+ 2y = 1. Solving these two equations gives x = 1, y = 0. Thus a point on ilie line is (1, 0,·0).
A vector v in the direction of this intersecting line is perpendicular to the normal vectors of both planes, so we can take
v = o 1 x o 2 = (1, 1, 1) x (1, 2, 2) = (2- 2, 1 - 2, 2- 1) = (0, - 1, 1}. By Equations 2, parametric equations for tlle
line are .'1: = 1, y = - t, z = t.
.
. 0 2
(b) The angle between the planes satisfies
cos 0 = I0] II
I = 1 +r;;2 +If\2 = 5r;;. T here fiore 0 = cos -1 ( 5r;; ) :::::: 15.80.
0] 02
v 3 v 9.
3v3
3v3
59. Setting z = 0, tlle equations of the two planes become 5x- 2y = 1 and 4x + y = 6. Solving tllese two equations gives
x = 1, y = 2 so a point on the line of intersection is (1, 2, 0). A vector v in the direction of this intersecting line is
perpendicular to the normal vectors ofbotll planes. So we can use v = o 1 x 0 2 = (5, - 2, -2) x (4, 1, 1} = (0, - 13, 13} or
2
equivalently we can take v = (0, -1, 1}, and symmetric equations for the line are x = 1, y _=- =
1
I x = y- 2= -z.
or
1,
61. The distance from a point (x , y , z) to (1, 0, - 2) is d 1 = ..j(x- 1)2 + y 2 + (z + 2)2 and the distance from (x, y, z) to
(3 ,4 , 0) is d2 = ..j(x- 3) 2 + (y- 4) 2 + z 2 . The plane consists of all points (x, y, z) where d1 = d2
=> d{ = di
<*
(x- 1)2 + y 2 + (z + 2) 2 = (x - 3) 2 + (y - 4) 2 + z 2 <*
x 2 - 2x + y 2 + z 2 + 4z + 5 = x 2 -: 6x + y 2 - By + z 2 + 25
<*
4:1;
+ By + 4z = 20 so an equation for the plane is
4x + By + 4z = 20 or equivalently x + 2y + z = 5.
Alternatively, you can argue that the segment joining points (1, 0, -2) and (3, 4, 0) is perpendicular to tlle plane and the plane
includes the midpoint of the segment.
63. The plane contains tlle points (a, 0, 0), (0, b, 0) and (0, 0, c) . Thus the vectors a = (-a, b, 0} and b = (- a, 0, c) lie in tlle
plane, and o = a x b = (be- 0, 0 + ac, 0 + ab) = (be, ae, ab) is a normal vector to the plane. The equation of the plane is
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SECTION 12.5 EQUATIONS OF LINES AND PLANES
0
133
there fore bcx + aetJ + abz = abc + 0 + 0 or bcx + acy + abz = abc. Notice that if a ol 0, b ol 0 and c ol 0 then we can
'
.
. to remem ber.I
rewnte
the equat1on
as -X + -'1j + -Z = 1. Th'ts 1.s a goo d equat1on
a
b
c
65. Two vectors which are perpendicular to the·required line are the normal of the given plane, (1, 1, 1), and a direction vector for
the given line, (1, - 1, 2). So a direction vector for the required line is (1, 1, 1} x (1, -1, 2} = (3, - 1, -2}. Thus L is given
by (x, y, z} = (0, 1, 2} + t(3, -1 , - 2}, or in parametric form, x = 3t, y = 1- t , z = 2- 2t.
67. Let Pi have normal vector D i. Then n1 = (3, 6, -3}, n 2 = (4, -12, 8), n a = (3, -9, 6), ll4 = (1, 2, - 1}. Now n 1 = 3D4,
so n 1 and n4 are parallel, and hence P1 and P4 are parallel; similarly P 2 and P3 are parallel because n 2 = ! n 3. However, n 1
and n 2 are not parallel (so not all four planes are parallel). Notice that the point (2, 0, 0) lies on both P 1 and P 4 , so these two
planes are identical. The point ( ~ , 0, 0) lies on P2 but not on P3, so these are different planes.
69. Let Q = (1, 3, 4) and R = (2, 1, 1), points on the line corresponding to t = 0 and t = 1. Let
.
.
--+
--->
P = (4, 1, - 2). Then a = QR = (1, - 2, -3), b = QP = (3, - 2, - 6) . The distance is
1(6, - 3,4)1 =
J62 + (-3)2 + 42 = J6I = {61
1(1, - 2, - 3}1
JP+(- 2)2 + (-3)2
v'I4 V14'
d = Ia x bl = 1(1, -2,-3} x (3,-2,-6}/ =
/al
.
1(1, -2, -3}/
tl
71. BE
y quat1on 9, 1e
d'ta
. D
iax1+by1 +cz1 +d/
18
IS nee IS
=
= /3(1) +2( - 2)+6(4)-5/
= /181
r.n = - .
Ja2+b2+ c2
.
J 32 + 22 + 62
v49
7
73. Put y = z = 0 in the equation of the first plane to get the point (2, 0, 0) on the plane. Because the planes are parallel, the
distance D between them is the distance from (2, 0, 0) to the second plane. By Equation 9,
D = /4(2) - 6(0) + 2(0) - 31 = _ 5_ = _5_ or 5 VT4.
J42 + ( - 6)2 + (2)2
V56 2 VI4
28
75. The distance between two parallel planes is the same as the distance between a point on on~ ofilie planes and the oilier plane.
Let Po = (xo, yo, zo) be a point on the plane given by ax+ by + cz + dt = 0. Then axo + byo + czo + d 1 = 0 and the
distance between Po and ilie plane given by ax + by + cz + d2 = 0 is, from Equation 9,
D = /ax o + byo + czo + d2/ =
Ja2 +b2+c2
77. £ 1: x = y = z
=?
/-dl + d2 /
Ja2 + b2+ c2
, /d1 - d2 /
Ja2+ b2+c2
x = y (1). £2: x + 1 = y/2 = z/3 =?
x = y = - 2. However, when x = -2, x = z
=?
x + 1 = y/2 (2).
z = - 2, but x + 1 = z/3
=?
The solution of (l ) and (2) is
z = - 3, a contradiction. Hence the
lines do not intersect. For L1. v1 = (1, 1, 1), and for L2, v2 = (1, 2, 3}, so the lines are not parallel. Thus ilie lines are skew
lines. If two lines are skew, they can be viewed as ly ing in two parallel planes and so ilie distance between the skew lines
would be the same as the distance between these parallel planes. The common normal vector to the planes must be
perpendicular to boili (1, 1, 1) and (1, 2, 3), the direction vectors of the two lines. So set
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134
D
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
n = {1, 1, 1) x {1, 2, 3) = (3- 2, - 3 + 1, 2- 1) = {1, - 2, 1). From above, we know that ( - 2, -2, - 2) and ( -2, - 2, -3)
are points of £1 and~ respectively. So in the notation of Equation 8, 1{- 2)- 2( - 2) + 1(- 2) + d 1 = 0
1( -2) - 2( -2) + 1(-3) + d2 = 0
~
~
d 1 = 0 and
d2 = 1.
By Exerci~e 75, the distance between these two skew lines is D =
.
IO - II
v 1 +4 + 1
Alternate solution (without reference to planes): A vector which is perpendicular to both of the lines is
n = (1, 1,1) x (1, 2, 3) = (1, - 2, 1). Pick any point on each of the lines, say ( - 2, -2, -2) and ( -2, - 2, - 3), and form the
vector b = (0, 0, 1) connecting the two points. The distance between the two skew lines is the absolute value of the 'sca,lar
. .
..
ln·bl = 11 · 0-2·0+1·11 = 1u;·
proJection of b along n , that IS, D = - - .
1n 1
v 1 +4+1
v6
79. A direction vector for £1 is v1 = (2, 0, -1) and a direction vector for £ 2 is v 2 = (3, 2, 2). These vectors are not parallel so
neither are the lines. Parametric equations for the lines are £1 : x = 2t, y = 0, z = - t, and £2: x = 1 + 3s, y = - 1 + 2s,
z = 1 + 2s. No values oft and s satisfy ·these equations si multaneously, so the lines don't intersect and hence are skew. .we
can view the lines as lying in two parallel planes; a common normal vector to the planes is n = v 1 X v2 = (2, - 7, 4). Line
£ 1 passes through the origin, so (0, 0, 0) lies on one of the planes, and.(1, - 1, 1) is a point on £2 and therefore on the other
plane. Equations of the planes then are 2x- 7y
between the two skew lines is D =
+ 4z = 0 and 2x- 7y + 4z- 13 = 0, and by Exercise 75, the distance
I0-(-13)1
= 13
rcn.
v4+49 +16
v69
Alternate solution (without reference to planes): Direction vectors of the two lines are v1 = (2, 0 , -1) and v 2
Then n
= (3, 2, 2).
= v 1 x v 2 = (2, - 7, 4} is perpendicular to both lines. Pick any point on each of the lines, say (0, 0, 0) and (1, -1, 1),
and form the vector b = (1, -1, 1) connecting the two points. Then the distance between the two skew lines is the absolute
value of the scalar projection of b a long n , that is, D = ln · bl =
1D 1
81 . If a =F 0, then ax + by+ cz + d = 0 =?
3
12 + 7 + 4 1 = ~
.
V4 + 49+16
69
YO::!
a(x + dj a) + b(y - 0) + c(z - 0) = 0 which by (7) is the scalar equation of the
plane through the point ( -d/ a, 0, 0) with normal vector (a , b, c). Similarly, if b =F 0 (or if c =F 0) the equation of the plane can
be rewritten as a(x - 0) + b(y + djb) + c(z - 0) = 0 [or as a(x- 0) + b(y- 0) + c(z + djc) = 0] which by (7) is the
scalar equation of a plane through the point (0, - d/b, 0) [or the point (0, 0, -d/c)] with normal vector (a, b, c).
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SECTION 12.6
12.6
CYLINDERS AND QUADRIC SURFACES
0
Cylinders and Quadric Surfaces
1. (a) In JR 2 , the equation y = x 2 represents a parabola.
X
z
(b) In R 3 , the equation y = x 2 doesn't involve z, so any
horizontal plane with equation z = k intersects the graph
in a curve with equation y = x 2 • Thus, th~ surface is a
parabolic cylinder, made up of infinitely many shifted
X
copies of the same parabola. The rulings are parallel to
the z-axis.
(c) In R 3 , the equation z = y 2 also represents a parabolic
z
cylinder. Since x doesn't appear, the graph is formed by
moving the parabola z = y 2 in the direction of the x-axis.
Thus, the rulings of the cylinder are parallel to the x-ax.is.
3. Since y is missing from the equation, the vertical traces
5. Since x is missing, each vertical trace z = 1 - y 2 ,
x 2 + z 2 = 1, y = k, are copies of the same circle in
x = k, is a copy of the same parabola in the plane
the plane y = k. Thus the surface x 2 + z 2 = 1 is a
x = k. Thus the surface z = 1 - y 2 is a parabolic
circular cylinder with rulings parallel to they-axis.
cylinder with rulings parallel to the x-axis.
.1'
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135
136
D
CHA~TER 12
VECTORS AND THE GEOMETRY OF SPACE
7. Since z is missing, each horizontal trace xy = 1,
z = k, is a copy of the same hyperbola in the plane
z = k. Thus the surface xy = 1 is a hype~bolic
cylinder with rulings parallel to the z-axis.
9. (a) The traces ofx2 + y 2 - z 2 = 1 in x = k are y 2 - z 2 = 1 - k 2 , a family of hyperbolas. (Note that the hyp!!rbolas are
oriented differently for -1 < k < 1 than for k < - 1 or k > 1.) The traces in y = k are x 2 - z 2 = 1 - k 2 , a similar
family of hyperbolas. The traces in z = k are x 2 + y 2 = 1 + k 2 , a family of circles. Fork = 0, the trace in the
xy-plane, the circle is of radius 1. As Iki increases, so does the radius of the circle. This behavior, combined with the
hyperbolic vertical traces, gives the graph of the hyperboloid of one sheet in Table 1.
(b) The shape of the surface is unchanged, but the hyperboloid is
rotated so that its axis is the y-axis. Traces in y
= k are circles,
while traces in x =. k and z = k are hyperbolas.
(c) Completing the square in y gives x 2
+ (y + 1)2 - z 2 = 1. The
surface is a hyperboloid identical to the one in part (a) but shifted
one unit in the negative y-direction.
11. For x = y 2 + 4z2 , the traces in x = k are y 2 + 4z 2 = k. When k >. 0 we
have a fam ily of ellipses. When k = 0 we have just a point at the origin, and
the trace is empty for k < 0. The traces in y = k are x = 4z 2 + k 2 , a
family of parabolas opening in the positive x -direction. Similarly, the traces
in z = k are x = y 2 + 4k2 , a family of parabolas opening in the positive
x-direction. We recognize the graph as an elliptic parabofoid with axis the
x-axis and vertex the origin.
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SECTION 12.6
CYLINDERS AND QUADRIC SURFACES
0
137
13. x 2 = y 2 + 4z2 . The traces in x = k are the ellipses y 2 + 4z2 = k 2 . The
traces in y = k are x 2 - 4z2 = k 2 , hyperbolas for k =/; 0 and two
intersecting lines if k = 0. Similarly, the traces in 'z = k are
x 2 - y 2 = 4k2 , hyperbolas for k =/; 0 and two intersecting lines if k = 0.
We recognize the graph as an elliptic cone with axis the x-axis and vertex
the origin.
15. - x 2 + 4y 2 - z 2 = 4. The traces in x = k are the hyperbolas
4y 2 - z 2 = 4 + k2 . The traces in y = k are x 2 + z 2 = 4k2 - 4, a family of
circles for lkl > 1, and the traces in z ·= k are 4y 2 - x 2 = 4 + k 2 , a family
of hyperbolas. Thus the surface is a hyperboloid of two sheets with
axis the y -axis.
17. 36x 2 + y 2 + 36z2 = 36. The traces in x = k are y 2 + 36z2 = 36(1 - k 2 ),
a family of ellipses for lkl < 1. (The traces are a single point for lkl = 1
and are empty for lkl > 1.) The traces in y = k'are the circles
36x 2 + 36z2 = 36- k 2
{"}
x 2 + z 2 = 1 - tok 2 , lkl < 6, and the
traces in z = k are the ellipses 36x 2 + y 2 = 36(1 - k 2 ), 1~1 < 1. The
graph is an ellipsoid centered at the origin with intercepts x = ± 1, y = ± 6,
z = ±1.
19. y = z 2 - x 2 • The traces in x = k are the parabolas y = z 2 -:- k 2 ;
the traces in y = k are k = z 2 - x 2 , which are hyperbolas (note the hyperbolas
are oriented differently fork > 0 than fork < 0); and the traces in z = k are
y
the parabolas y = k 2 .- x 2 • Thus, l
21. This is the equation of an ellipsoid: x
z2
x2
= 12 - ]2 is a hyperbolic paraboloid.
2
2
2
(1/2)
(1/3t
X
+ 4y2 + Dz2 = x 2 + ~ + ___!___, = 1, with x-intercepts ±1, y-intercepts ±i
0
-
and z-intercepts ±~. So the major axis is the x-axis and the only possible graph is Vll.
23. This is the equation of a hyperboloid of one sheet, with a
= b = c = 1. Since the coefficient of y 2 is negative, the axis of the
hyperboloid is the y-axis, hence the correct graph is II.
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138
D
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
25. There are no real values of x and z that satisfy this equation for y < 0, so this surface does not extend to the left of the
.
xz-plane. The surface intersects the plane_y = k > 0 in an ellipse. Notice that y occurs to the first power whereas x and z
.
occur to the second power. So the surface is an elliptic paraboloid with axis they-axis. Its graph is VI.
27. This surface is a cylinder because the variable y is missing from the equation.' The intersection of the surface and the xz-plane
is an ellipse. So the graph is VIII .
. z2
.
29. y 2 = x 2 + ~z 2 or y 2 = x 2 + g represents an elliptic
cone with vertex (0, 0, 0) and axis the y-axis.
•
2
31. x 2 + 2y - 2z 2 = 0 or 2y = 2z2 - x 2 or y = z 2 - ~
represents a hyperbolic paraboloid with center (0, 0, 0).
z
35. Completing squares in all three variables gives
33. Completing squares in y and z gives
4x + (y - 2)
2
2
x z + (y ~ )
22
(x - 2)2 - (y + 1? + (z- 1)2 = 0 or
+ 4(z - 3) = 4 or
2
+ (z - 3) 2 = 1, an ellipsoid with
center (0, 2, 3).
(y + 1? = (x - 2) 2 + (z -1?, a circular cone with
center (2, -1, 1) and axis the horizontal line x = 2,
z =l.
z
y
37. Solving the equation for z we get z. = ±)1 + 4x2 + y 2 , so we plot separately z = ) 1 + 4x 2 + y 2 and
z = - ) 1 + 4x2 + y2 .
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SECTION 12.6
CYLINDERS AND QUADRIC SURFACES
0
139
To restrict the z-range as in the second graph, we can use the option view= - 4 . . 4 in Maple's p l"ot3d command, or
Plo.t Range - > {- 4 , 4} in Mathematica's Pl ot 3D command.
39. Solving the equation for z we get z = ±y'4x2 + y 2 , so we plot separate ly z = y'4x2 + y2 and z = - y'4x2 + y2.
z
41.
z=2
43. The surface is a paraboloid of revolution (circular paraboloid) with vertex at
the origin, axis the y-axis and opens to the right. Thus the trace in the
yz-plane is a lso a parabola: y = z 2, x = 0. The equation is y = x 2
+ z 2.
The parabola
y=x2
X
45. Let P = (x, y, z) be an arbitrary point equidist ant fro m ( -1, 0, 0) and the plane x = 1. Then the distance from P to
( -1, 0, 0) is y'(x + 1) 2 + y 2 + z 2 and the distance from P to the plane x = 1 is lx- 11/Vf'i = lx - 11
(by Equation 12.5.9). So lx- 11 = y'(x + 1)2 + y 2 + z 2
x 2 - 2x + 1 = x 2 + 2x + 1 +. y 2 + z 2
¢:>
¢::>
(x- 1) 2 = (x + 1) 2 + v. 2 + z 2
¢:>
- 4x = y 2 + z 2 • Thus the collection of all such points P is a circular
paraboloid with vertex at the origin, axis the x -axis, which opens in the negative direction.
2
2
47. (a) An equation for an ellipsoid centered at the origin with intercepts x = ±a, y = ±b, and z = ±cis x 2 + Y2
a
b
2
+ .=.._2 = 1.
c
Here the poles of the model intersect the z-axis at z = ± 6356.523 and the 'e quator intersects the x- andy-axes at
x = ±6378.137, y = ±6378. 137, so an equation is
y2
z2 .
x2
1
(6378.137) 2 + (6378. 137) 2 + (6356.523)2 =
•
•
~
~ I
~
(b) Traces to z = k are the Circles (6378.137)2 + (6378.137)2 = 1 - (6356.523)2
2
k2
x + y = (6378.137)2- (6378.137)
~6.~
.
2
2
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1
140
D
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
(c) To identify the traces in y = mx we substitute y = m x into the equation of the ellipsoid:
2
x
(mx?
. z2
1
2
(6378.137) + (6378.137)2 + (6356.523) 2 =
(1 + m 2 )x2
z2
1
(6378.137) 2 + (6356.523)2 =
x2
z2
1
(6378.137)2/(1 + m2) + (6356.523)2 =
As expected, this is a family of ellipses.
49. If (a, b, c) satisfies z = y 2 - x 2 , then c
£2: x =a+ t, y = b- t, z = c- 2(b
= b2 - a 2 . L 1 : x = a + t, y = b + t, z = c + 2( b - a )t,
+ a)t. Substitute the parametric equations of £1 into the equation
of the hyperbolic paraboloid in order to find the points of intersection: z = y2 - x 2
.
'
I
c + 2(b - a)t = (b + t) 2 -
=>
(a+ t) 2 = b2 - a 2 + 2(b- a)t => c = b2 - a2 . As this is true for·all values of t,
L 1 lies on z = y2 - x 2. Performing similar operations with £2 gives: z = y 2 - x 2
c- 2(b + a)t = (b- t) 2 - (a+ t? = b2 - a2 - 2(b + a)t =>
=>
c = b2 - a 2 . This tells us that all of £2 also lies on
The curve of intersection looks like a bent ellipse. The projection
51.
of this curve onto the xy-plane is the set of points (x, y, 0) which
satisfy x
x2 +
12
2
+ y2 = 1 - y2
y2
{1 //2)2
<=>
x
2
+ 2y 2 = 1 <=>
= 1. This is an equation of an ellipse.
Review
CONCEPTCH ECK
1. A scalar is a real number, while a vector is a quantity that has both a real-valued magnitude and a direction.
2. To add two vectors geometrically, we can use either.the Triangle Law or the Parallelogram Law, as illustrated in Figures 3
and 4 in ·section 12.2. Algebraically, we add the corresponding components of the vectors.
3. For c > 0, c a is a vector with. the same direction as a and length c times the length of a. If c < 0, ca points in the opposite
direction as a and has length JcJ times the length of a. (See Figures 7 and IS in Section 12.2.) Algebraically, to find c a we
multiply each component of a by c.
4. See (I) in Section 12.2.
5. See Theorem 12.3.3 and Definition 12.3.1.
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CHAPTER 12 REVIEW
0
141
6. The dot product can be used to find the angle between two vectors and the scalar projection of one vector onto another. In
particular, the dot product can determine if two·vectors are orthogonal. A lso, the dot product can be used to determine the
work done moving an object given the force and di~placement vectors.
7. See the boxed equations as well as Figures 4 and 5 and the accompanying discussion on page 828 [ET 804].
B. See Theorem 12.4.9 and the preceding discussion; use either (4) or (7) in Section 12.4.
9. The cross product can be used to create a vector orthogonal to two given vectors as well as to determine if two vectors are
parallel. The cross product can also be used to find the area of a parallelogram determined by two vectors. In addition, the
cross product can be used to determine torque if the force and position vectors are known.
•
10. (a) The area of the parallelogram determined by a and b is the length of the cross product: Ia x bl.
(b) The volume of the parallelepiped determined by a, b, and cis the magnitude of their sca lar triple product: Ia . (b x c )l.
11. If an equation of the plane is known, it can be written as ax
+by + cz + d = 0. A normal vector, which is' perpendicular to the
plane, is (a , b, c) (or any scalar multiple of (a , b, c)). If an equation is not known, we can use points on the plane to find two
non-parallel vectors which lie in the plane. The cross,product of these vectors is a vector perpendicular to the plane.
12. The angle between two intersecting planes is defined as the acute angle between their normal vectors. We can find this angle
using Corollary 12.3.6.
13. See (1), (2), and (3) in Section 12.5.
14. See (5), (6), and (7) in Section 12.5.
15. (a) Two (nonzero) vectors are parallel if and only if one is a scalar multiple of the other. In addition, two nonzero vectors are
parallel if and only if their cross product is 0.
( b) Two vectors are perpendicular if and only if their dot product is 0.
(c) Two planes are parallel if and only if their normal vectors are parallel.
--+
--+
16. (a) Determine the vectors PQ = (a1 , a2 , a3 ) and P R = (b1, ~ . b3) . If there is a.scalar t such that .
(a 1 , a 2, a 3) = t (b1, ~ . b3 ), then the vectors are parallel and the points must all lie on the same line.
~
---+
-+
---+
Alternatively, if PQ x PR = 0, then PQ and PR are paralle l, ~o P, Q, and Rare collinear.
Thirdly, an algebraic method is to determine an equation of the line joining two o.f the points, and then check whether or
not the third point satisfies this equation.
--+
--+
--+
(b) Find the vectors PQ =a, P R = b , PS = c . a x b is normal to the plane formed by P, Q and R, and so S lies on this
plane if a x band care orthogonal, that is, if(a x b)· c = 0. (Or use the reasoning in Example 5 in Section 12.4.)
A lternatively, find an equation for the plane determined by three of the points and check whether or not the fourth point
satisfies this equation.
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142
D
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
17. (a) See Exercise 12.4.45.
(b) See Example 8 in Section 12.5.
(c) See Example 10 in Section 12.5.
18. The traces of a surface are the curves of intersection of the surface with planes parallel to the coordinate planes. We can find
the trace in the plane x = k (parallel to the yz-plane) by setting x = k and determining the curve represented by the resulting
equation. Traces in the planes y = k (parallel to the xz-plane) and z = k {parallel to the xy-plane) are found similarly.
19. See Table I in Section 12.6.
TRUE-FALSE QUIZ
1. This is false, as the dot product of two vectors is a scalar, not a vector.
3. False. For example, ifu = i and ·v = j then iu · v i= 101 = 0 but iu l lvl = 1 · 1 = 1. In fact, by Theorem 12.3.3,
lu · v i = liu llvl cos 91
.
5. True, by Theorem 12.3.2, property 2.
7. True. If 9 is the angl~ between u and v , then by Theorem 12.4.9,lu x vi = iu l lv l sin 9 = lvllul sin 9 = 1':' x ui.
(Or, by Theorem 12.4.1l,lu x v i = 1-v x ul = 1- lllv x ul = lv x ui.)
9. Theorem 12.4.11 , property 2 tells us that this is true.
11. This is true by Theorem 12.4.11, property 5.
13. This is true because u x vis orthogonal to u (see Theorem 12.4.8), and the dot product of two orthogonal vectors is 0.
15. This is false. A normal vector to the plane is n = (6, - 2, 4) . Because (3, - 1, 2) = ~n, the vector is parallel ton and hence
perpendicular to the plane.
17. This is fa lse. In IR2 , x 2 + y 2 = 1 represents a circle, but { (x, y, z) I x 2 + y 2 = 1} represents a three-dimensional surface,
namely, a circular cylinder with axis the z -axis.
19. False. For example, i · j = 0 but i i- 0 and j =f. 0 .
21. This is true. Ifu and v are both nonzero, then by (7) in s~ c tion 12.3, u · v = 0 implies that u and v are orthogonal. But
u x v = 0 implies that u and v are parallel (see Corollary 12.4.10). Two nonzero vectors can't be both parallel and
orthogonal, so at least one of u, v must be 0 .
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CHAPTER 12 REVIEW
0
1~
EXERCISES
1. (a) The radius of the sphere is the distance between the points ( - 1, 2, 1) and {6, -2, 3), namely,
J [6 - ( -1)]2 + ( - 2 - 2) 2 + (3 - 1) 2 = 0f!}. By the formula for an equation of a sphere (see page 813 [ET 789]),
an equation of the sphere with center ( -1, 2, 1) and radius J69 is (x + 1? + (y - 2) 2 + (z- 1) 2 = 69.
(b) The intersection of this sphere with the yz -plane is the set of points on the sphere whose x-coord inate is 0. Putting x = 0
into the equation, we have (y - 2) 2 + (z - 1) 2 = 68, x = 0 which represents a circle in the yz-plane with center (0 , 2, 1)
and radius .J68.
(c) Completing squares gives (x - 4) 2 + (y + 1) 2 + (z +
3? = - 1 + 16 + 1 +9 = 25. Thus the sphere is centered at
(4, -1, -3) and has radius 5.
3. u . v = Ju J.JvJ c~s45° = (2)(3)
'{!- = 3 .;2.
Ju x vJ = JuJJv Jsin45° = (2)(3)~ = 3 .;2.
By the right-hand rule, u x vis directed out of the page.
5. Forthetwo vectors to be orthogonal, we need (3, 2,x} · (2x, 4,x} = 0
~ + 6x + 8 = 0
<=?
(x + 2)(x + 4) = 0
<=?
#
(3)(2x) + (2){4) + (x )(x) = 0
#
x = - 2 or x = -4.
7. (a) (u x v ) · w = u · (v X w) = 2
(b) u · (w x v ) = u · [- (v X w )] = - u · (v X w) = - 2
(c) v · (u x w) = (v x u )· w =- (u x v) · w = - 2
(d) (u x v ) · v = u · (v X v) = u · 0 = 0
9. For simplicity, consider a unit cube positioned with its back left corner at the origin. Vector representations of the diagonals
joining the points (0, 0, 0) to (1, 1, 1) and (1, 0, 0) to (0, 1, 1) are (1 ,' 1, 1} and (-1, 1, 1). Let 8 be the angle between these
twovectors.
8 = cos- 1
---+
=> cosO = ~ =>
(1, 1, 1)·{- 1, 1, 1} = -1+1 + 1 = 1=\(1, 1, 1) \ \{- 1,1,1)\ cos8=3cos8
G) ~ 71°.
---+
11 . AB = (1, 0, -1}, AC = (0, 4, 3), so
---+
---+
(a) a vector perpendicular to the plane is AB x AC = (0 + 4, -(3 + 0) , 4 - 0} = {4 , - 3, 4}.
(b)
t lAB X AC I= h116 + 9 + 16 = 4J..
13. Let H be the magnitude of the force directed 20° away from the direction of shore, and let H be the magnitude qfthe other
force. Separating these forces into components parallel to the direction of the resultant force and perpendicular to it gives
H cos 20° + Fz cos 30° = 255 (1), and F1 sin 20° into (1) gives F2 (sin30° cot20° + cos 30°) = 255
F2 sin 30° = 0
=> A = F2 s~n
°
3 0
(2). Substituting (2)
s1n 2 0 °
=> H ~ 114 N. Substituting this into (2 ~ gives F 1 ~ 166 N.
I
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144
0
CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
15. The line has direction v = ( -3, 2, 3). Letting Po = (4, - 1, 2), parametric equation~ are
X
= 4 - 3t, y = -1 + 2t, Z = 2 + 3t.
17. A direction vector for the line is a normal vector for the plane, n = (2, -1, 5), and parametric equations for the line are
.
X
= -2 + 2t, y = 2 - t, Z = 4 + 5t.
'
19. Here the vectors a= (4- 3,0- (-1) ,2 - 1) = (1,1,1} and b = (6 - 3,3- (-1), 1 - 1) = {3,4, 0) lie in the plane,
so n = a x b = ( -4, 3, 1) is a normal vector to the plane and an equation of the plane is
- 4(x- 3) +3.(y - (- 1)) + 1(z -1) = 0 or - 4x+3y +z = - 14.
21 . Substitution of the parametric equations into the equation of the plane gives 2x- y + z = 2(2- t) - (1 + 3t) + 4t = 2
=?
.
- t + 3 = 2 => t = 1. When t = 1, the parametric equations give x = 2 - 1 = 1, y = 1 + 3 = 4 and z = 4. Therefore,
.
the point of intersection is (1,4,4).
23. Since the direction vectors (2, 3, 4) and (6, --;1, 2) aren't parallel, neither are the lines. For the lines to intersect, the three
+ 2t = -1 + 6s, 2 + 3t = 3 - s, 3 + 4t = -5 + 2s must be satisfied simultaneously. Solving the first two
equations gives t = k. s = ~ and checking we see these values don't satisfy the third equation. Thus the lines aren't parallel
equations 1
and they don't intersect, so they must be skew.
25. n 1 = (1, 0, -1) and n2 = (0, 1, 2). Setting z = 0, it is easy to see that (1, 3, 0) is a point on ,the line of intersection of
x - z = 1 andy+ 2z = 3. The direction of this line is v 1 = n 1 x n 2 = (1, -2, 1). A second vector parallel to the desired
plane is v 2 = (1,1, -2), since it is perpendicular to x + y- 2z = 1. Therefore, the normal of the plane in question is
n = v 1 x v 2 = (4 - 1, 1 + 2, 1 + 2) = 3 (1, 1, 1) . Taking (x0 , y 0 , z 0 ) = (1, 3, 0), the equation we are looking for is
(x- 1) + (y- 3) + z = 0
.
27. By Exercise 12.5.75, D -
¢*
x + y + z = 4.
l-2- (-24)1
y'32 + I2 + (-4)2
-
22 .
~·
v26
29. The equation x = z represents a plane perpendicular to
the xz-plane and intersecting the xz-plane in the line
31. The equation x 2 = y 2 + 4z2 represents a (right elliptical)
cone with vertex at the origin and axis the x -axis.
x = z,y = 0.
X
X
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CHAPTER 12 REVIEW
.
.
.
33. An equivalent equation IS - x 2 +
y2
2
4 -;- z = 1, a
0
145
35. Completing the squate in y gives
z2
hyperboloid of two sheets with axis the y-ax.is. For
4x 2 + 4(y - 1) 2 + z 2 = 4 or x 2 + (y - 1) 2 +4 =.1,
IYI > 2, traces parallel to the xz-plane are circles.
an ellipsoid centered at (0, 1, 0).
z
(0, 1,2)
(O, l , -2)
. x2
<=>
y2
2
i
2
2
4 + 16 = l. The equation of the ellipsoid is ~ + 6 + : 2 = l, since the horizontal trace in the
plane z = 0 must be the original ellipse. The traces of the ellipsoid in the yz -plane must be circles since the surface is obtained
2
by rotation about the x-axis. Therefore, c2 = 16 and the equation of the ellipsoid is ~ +
i + ~6 =
2
6
2
1
<=>
4x2 + y 2 + z 2 = 16.
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D
PROBLEMS PLUS
1. Since three-dimensional situations are often difficult to visualize and work with, let
us first try to find an analogous problem in two dimensions. The analogue of a cube
is a square and the analogue of a sphere is a circle. Thus a similar problem in two
dimensions is the following: if five circles with the same radius rare contained in a
square of side 1 m so that the circles touch each other and four of the circles touch
two sides ofthe square, find r.
The diagonal of the square is .J2. The diagonal is a lso 4r + 2x . But x is the diagonal of a smaller square of side r . Therefore
x = .J2r
~
.J2=4r+2x =4r+2.J2r= (4+2.J2)r
~ r = 4 /{,/2.
Let's use these ideas to solve the original three-dimensional problem. The diagonal of the cube is \/12 + 12 + 12 = v'a.
The diagonal of the cube is also 4r + 2x where x is the diagonal of a smaller cube with edger. Therefore
x = Jr 2 + 1·~ + r 2 = v'3 r
~
r;;
r;;
v'a = 4r+2x .= 4r+2v3r = (4+2v3)r.Thus r=
v'3 r;;= 2 v'a - 3
4 + 2v3
2
The radius of each ball is ( v'3 - ~) m.
3. (a) We find the line of intersection L as in Example 12.5. 7(b). Observe that the point ( - 1, c, c) lies on both planes. Now since
L lies in both planes, it is perpendicular to both of the normal vectors n1 and n 2, and thus parallel to their cross product
n1 x n2 =
c
j
k
1
1 = (2c, -c2 + 1, -c2 - 1). So symmetric equations of L can be written as
1 - c
c
x +1
y- c
z- c
- - = ~ = ~ , provided that c I 0, ±1.
- 2c
(..- - 1
c +1
If c = 0, then the two planes are given by y + z = 0 and x = - 1, so symmetric·equations of L are x = - 1, y = - z. If
c = - 1, then the two planes are given by -x + y + z = - 1 and x + y + z = -1, and they intersect in the line x = 0,
y = - z - 1. If c = 1, then the two planes are given by x + y + z = 1 and x - y + z = 1, and they intersect in the line
y = 0, X = 1- Z.
(b) If we set z = t in the symmet~ic equations and solve for x andy separately, we get x + 1 = (t - 2c)( - 2c)
c +1
y - c=
(t- c)(c2 - 1)
c2 +1
- 2ct + (c
2
~
x=
c2 +
1
-
1)
2
, Y=
(c
-
1)t + 2c El. . .
fr
th
.
. unmatmg c om ese equattons, we
&+ 1
have x 2 + y 2 = t 2 + 1. So the curve traced out by L in the plane z = tis a circle with center~~ (0, 0 , t) and
radius J fl + 1.
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147
148
0
CHAPTER 12 PROBLEMS PLUS
(c) The area of a horizontal cross-section of the solid is A (z) = 1r(z2
53
53
54
jv5 1 = 24 . 32 jv 1j = 23 . 32 . Similarly, jv5j = 24 .
+ 1), so V = J; A(z )dz = 1r ( ~ z3 + z] ~ = 43,..
50
33
, jv.71 :;= 25 . 34 , and in general, jv ,. j =
5"- 2
n_ 2 .
2
3
n_ 3
-2
= 3(ir .
Thus
f Jv ,.J = lv1 l + lv 2l + f 3(i)"- 2 = 2 + 3 + f 3(~)"
n= l
n=3
00
n=l
ll
1
= 5 + E .2.(!~)"= 5 + ___L_
2
n= l
1- ~
G
[sum of a geometric series]
= 5 + 15 = 20
7. (a) When 8 = 8., the block is not moving, so the sum of the forces on the block
I
must be 0, thus N + F + W = 0 . This re lationship is illustrated
geometrically in the figure. Since the vectors form a right triangle, we have
- J!l_ 11-.•nn --
tan(O.) - INI -
fl- s·
(b) We place the block at the origin and sketch the force vectors acting on the block, including the additional horizontal force
H , with initial points at the origin. We then rotate this system so that Flies along the positive x -axis and the inclined plane
is parallel to the x-axis. (See the follo~ing figure.)
N
F
w
IF! is maximal, so jFj = 11-. n for f) > B_,. Then the vectors, in terms of components parallel and perpendicular to the
inclined plane, are
N =n j
W = (- mg sin 0) i + (-mg cos fJ) j
F = (11-.n) i
H = (hmin cos B) i + (- hmin sin B) j
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l
CHAPTER 12 PROBLEMS PLUS
0
149
Equ!lting components, we have
JJ-,n- mg sin(}+ hm1n cosO= 0
,=?
hmin cos(}+ J-L•• n = mgsin(}
(1)
n- mg cos()- hrnin s infJ = 0
=?
hmin sin(}+ mgcos(} = n
(2)
(c) Since (2) is solved for n, we substitute into (1):
hmin cos() + JJ-~ (hmin sin(}+ mg cos 9) = mg sin 9
hmin cos fJ + hmin/1-s sin fJ = mg sin fJ -
- JJ-.• cos'(} )
1tmin = mg (sin8
cos (}+ 1-t. sin(}
.
= mg
=?
mgJJ-. cos(}
'*
( tau8 - /1-., )
1 + JJ-. tau(}
h
( tan8 - tanfJ8 )
d .
.
. .
.
. b
+tan (}, tau(} an usmg a tngonomctnc tdentlty,
From part ()
a we know JJ-~ = tan (} ., so th IS ecomes min = mg
1
this is mg tau(8- 8. ) as desired.
Note for fJ = 8., hmin = mgtau 0 = 0, which makes sense since the block is at rest for 8. , thus no additional force H
is necessary to prevent it from moving. As(} increases, the factor tau(8- 8.), and hence the value of hmin, increases
slowly for small values ofB- 8. but much more rapidly as 8 -
e. becomes significant. This seems reasonable, as the
steeper the inclined ~lane, the less the hori zont~l components of the various forces affect the movement of the block, so we
would need a much larger magnitude of horizontal force to keep the block motionless. If we allow f) --+ 90°, corresponding
to the inclined plane being placed vertically, the value of hmin is quite large; this is to be expected, as it takes a great
amount of horizontal force to keep an object from moving vertically. In fact, without friction (so (} 8 = 0), we would have
(} --+ 90°
:::?
hmin --+ oo, and it would be impossible to keep the block from slipping.
(d) Since hmo.x is the largest value of h that keeps the block from s lipping, the force of friction is keeping the block from ·
moving up the inclined plane; thus, F is directed down the plane. Our system of forces is similar to that in part {b), then,
except that we have F = -(JJ-.n) i. (Note that IFI is again maximal.) Following our procedure in parts (b) and (c), we
equate components:
- JJ-,n- mgsin(} + hmux cos(} = 0
n - mg cos (} - hm:l.X sin (} = 0
:::?
hme.x cos(}- /)-8 n = mgsin8
'* hmax sin (} + mg cos e = n
Then substituting,
hmax cos 8 -
/1-s ( hmax sin 8 + mg cos B) = mg sin 8
hmax cos(} -
'*
hmn.x/1-s sin(} = mg sin(} + mgJJ-. cos e
'*
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150
0
CHAPTER 12 PROBLEMS PLUS
· ( tan8 +J.£.)
_ mg (sin0+J.£.cos8) -_ mg
h m axcos 0 - J.£8 sin B
1 - J.£8 tan B
tanB +tan B. )
== mg ( 1- tan B. tanB
= mgtan(B +B.•)
We would expect hmax to increase as 8 increases, with similar behavior as we establ ished for hmin, but with hmax values
al~ays larger than hmin· We can see that this is. the case if we graph hmo.x as a function of 8, as the curve is the graph of
hmin translated 20. to the left, so the equation does seem reasonable. Notice that the equation predicts hmo.x --+ oo as
. 8--+ (90° - 0., ). In fact, as hmnx increases, the nonnal force increases as well. When (90° - Bs) :50 :S: 90°, the
horizontal force is completely counteracted by the sum of the f!Orlnal and frictional forces, so no part of the horizontal
force contributes to moving the block up the plane no matter how large its magnitude:
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VECTOR FUNCTIONS
D
13
13.1 Vector Functions and Space Curves
1. The component functions v'4- t 2 , e- 3 t, and ln(t + 1) are a ll defined wher4 - t 2 2: 0
=> - 2 ~ t ~ 2 and
t + 1 > 0 => t > - 1, so the domain of r is ( -1, 2].
1
1
1
_ I_ _ 1
3. lim e - 31 = e 0 = 1, lim~=
lim - - =
=
2
. t)2 - !2 - '
t-+O
t--+o sin t .
t.-o sin2 t
lim sin2 t
( tlim~
~
t--+0
t2
--+0
t
and lim cos 2t = cos 0 = 1. Thus
t-+0
lim
t-+O
(e-st i + 4
j +cos 2t k) = [lim e- t] i + [urn 4] j + [rim cos 2t] k = i + j + k.
sm t
sm t
3
t-+O
t-+O
t -+O
.
1 + t2
(1/ t 2 ) + 1
0+ 1
_ t
,. . 1 - e-2 t
1
1
.
.
.
]
1 tan 1 = 2 • t~
1
1
1
1- t 2 = t~
(1/t2)- 1 = 0 - 1 = - 1't2.~
t
= t2.~
t - -te_2_t = 0 - 0 = 0 · Thus
S. ~~
e
1+
1 - e-2
lim ( ,tan- 1 t,
t
2
t--+oo
1- t
t) =(-l,f,O).
.
7. The corresponding parametric equations for this curve are x = sin t, y = t.
We can make a table of values, or we can eliminate the parameter: t = y
=>
x = sin y, with y E JR. By comparing different values oft, we find tl)e direction in
which t increases as indicated in the graph.
X
9. The corresponding parametric equations are x = t, y = 2 - t, z = 2t, which are
parametric equations of a line through the point (0, 2, 0) and with direction vector
(1, -1, 2).
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151
152
D
CHAPTER 13 VECTOR FUNCTIONS
11. The corresponding parametric equations are x = 1, y = cos t , z = 2 sin t.
Eliminating the parameter in y and z gives ~ 2 + (z/ 2) 2 = cos2 t + sin2 t = 1
or y2 + z 2 / 4 = 1. Since x = 1, the curve is an ellipse centered at (1, 0, 0) in
the plane x = 1.
X
13. The parametric equations are X =
e. y = t z = t These are positive
4
•
6
.
I
I
for t =I 0 and 0 when t = 0. So the curve lies entirely in the first octant.
I
I
I
I
I
2
The projection of the graph onto the xy-plane is y = x , y > 0, a half parabola.
I
I
I
Onto the xz-plane z = x 3 , z > 0, a half cubic, and the yz-plane, y 3 = z 2 •
,I
,,,-
,'
y= xl
,,'
X
15. Tbe projection of the curve onto the xy-plane is given by r (t) = (t , sin t, 0} [we use 0 for the z-component] whose graph
is the curve y =sin x, z = 0. Similarly, the projection onto the xz-plane is r (t) = (t, 0, 2 cost), whose graph is the cosine
wave z = 2 cos x, y = 0, and the projection onto the yz-plane is r (t) = (0, sin t, 2 cost) whose graph is the ellipse
2
y
- I
-2
xy-plane
xz-plane
yz-plane
From the projection onto the yz-plane we see that the curve lies on an elliptical
cylinder with axis the x -axis . The other two projections show that the curve
oscillates both vertically and horizontally as we move in the x-direction,
suggesting that the curve is an elliptical helix that spirals along the cylinder.
17. Taking ro = (2, 0, 0) and r1 = (6, 2, -2), we have from Equation 12.5.4
r (t) = {1 - t)ro +t r1 = {1- t) {2,0, 0) +t{6, 2,-2), 0 ~ t ~ 1 or r {t) = (2 + 4t,2t, - 2t);O ~ t ~ 1.
Parametric equations are x = 2 + 4t, y = 2t, z = -2t, 0 ~ t ~ 1.
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SECTION 13.1 VECTOR FUNCTIONS AND SPACE CURVES
0
153
19. Taking r o = (0, - 1, 1) and r 1 = (~ . ~. i), we have
r (t) = (1- t) ro +t r1 = (1 - t) (0, - 1, 1) + tO,~.~), 0 ~ t ~ 1 or r (t) = (~t ,- 1 + 1t,1 .
.
tt), 0 ~ t ~ 1.
Parametric equations are x = ~t, y = - 1 + ~t, z = 1 - tt. 0 ~ t ~ 1.
21 . x = tcost, y = t, z = t ~in t, t ;:::: 0.
At any point (x, y, z) on the curve, x 2 + z 2 = t 2 cos2 t + t 2 sin2 t = t 2 = y 2 so the
curve Hes on the circular cone x 2 + z 2 = y 2 with axis they-axis. Also notice that y ;:::: 0; the graph is II.
23. x = t , y = 1/ (1 + t 2 )., z = t 2 •
At any point on the curve we have z = x 2 , so the curve lies on a parabolic cylinder parallel
to they-axis. Notice that 0 < y ~ 1 and z ;:::: 0. Also the curve passes through (0, 1, 0) when t = 0 and y --+ 0, z --+ oo as
t --+ ±oo, so the graph must be V.
25. x = cos Bt, y = sin Bt, z = e0 ·8 L, t ;:::: 0.
x 2 + y 2 = cos2 Bt + sin2 Bt = 1, so the curve lie~ on a circular cylinder with
axis the z-axis. A point (x, y, z) on the curve lies directly above the point (x, y, 0), which moves counterclockwise around the
unit circle in the xy-plane as t increases. The curve starts at (1-, 0, 1), when t = 0, and z --+ oo (at an increasing rate) as
t --+ oo, so the graph is IV.
27. Ifx = tcost, y = tsint, z = t , then x 2 + y 2 = t 2 cos 2 t + t 2 sin 2 t = t2 = z2 ,
so the curve lies on the cone z 2 = x 2
+ y 2. S ince z = t, the curve is a spiral on
this cone.
29. Parametric equations for the curve are x = t, y = 0, z = 2t - t 2 • Substituting into the equation of the paraboloid
gives 2t- t 2 = t 2
=> 2t = 2t2 => t = 0, 1. Since r(O) = 0 and r (1) = i + k , the points of intersection
are(O, O, O)and (1, 0, 1).
31. r (t) =(cost sin 2t,sint sin2t,cos 2t).
We include both a regu lar plot and a plot
showing a tube of radius 0.08 around the ·
·curve.
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154
0
CHAPTER 13 VECTOR FUNCTIONS
35. r (t) = (cos2t,cos3t ,cos4t)
33. r(t) = (t, t sin t, t cost)
X
:li =
37.
(1 + .cos 16t) cost, y = (1 +cos 16t) sin t, z = 1 +cos 16t. At any
point on the graph,
x 2 + y 2 = (1 +cos 16t) 2 cos2 t + (1 +cos 16t) 2 sin2 t
= (1 + cos 16t )2 = z2 , so the graph lies on the cone x 2 + y 2 = z2 •
From the graph at left, we see that this curve looks like the projection of a
X
leaved two-dimensional curve onto a cone.
39. Ift = - 1, then x = 1, y = 4, z = 0, so the curve passes through the point (1,4, 0). 1ft= 3, then x = 9, y = - 8, z = 28,
so the curve passes through the point {9, - 8, 28). For the point (4, 7, - 6) to be on the curve, we require y = 1 - 3t = 7 ==>
t = -2. But then z = 1 + ( - 2? = -7 # -6, so (4, 7, -6) is not on the curve.
41. Both equations are solved for z, s~ we ~an substitute to eliminate z :
x 2 = 1 + 2y ==>
J x2 + y2 = 1 + y ==> x 2 + y 2 = 1 + 2y + y 2 ==>
y = ~(x 2 - 1). We can form parametric equations for the curve C of intersection by choosing a
parameter x = t , then y = ~(t 2 - 1) and z = 1 + y = 1 + ~(t 2 - 1) = ~(t2 + 1). Thus a vector function representing C
is r (t) = t i + 1 (t2 - 1) j + ~(t 2 + 1) k.
"·
43. The projection of the curve C of intersection onto the xy-plane is the circle x 2 + y 2 = 1, z = 0, so we can write x = cost, ·
y = sin t, 0 $ t $ 27!'. Since C also lies on the surface z = x 2 - y 2 , we have z = x 2 - y 2 = cos2 t - sin2 tor cos 2t.
Thus parametric equations for Care x = cost, y = sin
t, ; = cos 2t, 0 $ t $ 27l', and the corresponding vector function
is r (t) = cost i + s in t j +cos 2t k , 0 $ t $ 27!'.
The projection of the curve C of intersection onto the
45.
xy-plane is the circle x 2 + y 2 = 4, z = 0. Then we can write
x = 2 cost, y = 2 sin t, 0 $ t $ 27l'. Since C also lies on
z
X
the surface z = x 2 , we have z = x 2 = (2 cost)~ = 4 cos2 t.
Then parametric equations for C are x = 2 cost, y = 2 sin t,
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SECTION 13.1
47. For the particles to collide, we require r1 (t)
= r 2(t)
¢:>
(
VECTOR FUNCTIONS AND SPACE CURVES
D
155
t 2, 7t - 12, t 2) = .(4t - 3, t2, 5t - 6). Equating components
gives t2 = 4t- 3, 7t - 12 = t2, and t 2 = 5t- 6. From the first equation, t 2 - 4t + 3 = 0
(t- 3)(t -1) = 0 sot = 1
¢:>
or t = 3. t = 1 does not satisfy the other two equations, butt = 3 does. The piuticles collide when t = 3, at the
point (9, 9, 9).
49. Let u (t) = (u1(t), u2(t ), u3(t)) and v (t) = (v1(t), V2(t), va(t)). In each part of this problem the basic procedure is to use
Equation l and then analyze the individual component functions using the limit properties we have already developed for
real-valued functions.
(a) lim u (t ) + lim v(t) = I lim u1(t), lim u2(t), lim ua(t)) + I Jim V1(t ), lim v2(t), lim v3(t) ) and the limits of these
t-+a.
t--+a
\ t -+a
t-+a
t--+a
\t--+a
t-a.
t--+a
component functions must each exist since the vector functions both possess limits as.t -+ a. Then adding the two vectors
and using the addition property oflimits for real-valued functions, we have that
lim u(t) + lim v (t) = I Jim u 1(t) + lim v1(t) , lim u2(t) + lim v2(t), lim ua(t) + lim v 3(t))
t --..o.
t -+ a
t-+a
\t--+<L
t -.a
t - (1.
= lim (u1(t) +VI (t), U2(t) + V2(t ), 'U3(t) + V3(t))
t-+a
t-~>a
t-+u
.
[using (I) backward]
= lim (u(t ) + v(t)]
t-a
(b) lim cu(t) = lim (cu1 (t), cu2(t), C'U3(t)) = I lim cu1(t) , lim cu2(t) , lim cua(t) )
t -+a
t --+ a
\t-
a
t -+ a
t -+a
= I c lim u 1(t), c lim u~(t), c lim u 3 (t) ) = c I lim u 1(t), lim u2 (t), lim ua(t))
\
t -a
t -+ a
\t__,.a.
t -+ a
t-+a
t-+a
= c lim (u1 (t), u2(t) , u3(t)) = c lim u (t)
t-a
t--+a
(c) lim u(t) · lim v(t) = I lim u 1(t), lim u2(t), lim u 3(t) ) ·I lim v1(t), lim v 2 (t) , lim v3(t))
t-+a
t--+a
\t-a
t -+ a
t-a
\t--.n. ·
t-+a
t - a.
= [lim
u1(t)] [u~
v1(t)] + [lim
u2(t)] [lim
v2(t)] _;_ .[lim
us (t)]
t --+ a
t -+ a
t -+ a
t --+ a
t -+ a
[urn v3(t)]
t -+a
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156
CHAPTER 13 VECTOR FUNCTIONS
0
(d) lim u (t) x ·lim v (t) = ( lim u1 {t), l.i m u2(t), lim u3(t)) x ( lim v1(t), lim v2(t), lim va(t))
t-a
t-a
t- a
=
t--+a
t.-a
t-+a
t-+o.
t-ta.
([lim u2(t)] [lim v3(t)] - [lim us(t)] [lim v2(t)] ,
[lim ua(t)] [lim v1(t)] - [lim u 1(t)] [tim vs(t)],
t-+a
t-+a
t -+a.
t-+n
t-+a
t-+a
t -+a
t-+a
[p~u1(t)) [~~ v2(t)) - [l~ u2(t)] [E~ vl(t)])
= (J~ [u2(t )v3(t)- u3(t)v2(t)] , E~ [ua(t)v1(t) - u1(t)va(t)] ,
~~ [u1(t)v2(t)- u2(t)v1(t)])
=
lim (·u2(t)va(t)- ua(t)v2(t),
t-+a
U3 (t) v1 (t) -
u1(t)v3(t), u1(t)v2(t)- tt2(t)v1 (t))
·
= t-<>
lim (u (t) x v (t)]
51 . Let r (t) = (! (t), g (t), h (t)) and b = {b1, b2, ba) . If lim r (t) = b, then lim r (t) exists, so by (1),
t-a.
b = lim r (t) ~ ( lim f(t), lim g(t),
t-+a
t-a
t-a
t-+a
lim h(t)). By the definition of equal vectors we have lim f(t) ·= b1, lim g(t) = b2
t -+ a
t-a
t -+a
and lim h(t) = b3 • But these are lim its of real-valued functions, so by the definition of limits, for every c > 0 there exists
t-Hl
<h > 0, 02 > 0, oa > 0 so that ifO < It - a l < 01 then lf(t) - b1 l < c/3, ifO < it- a l < 02 then lg(t) - b2l < c/ 3, and
ifO < It- al <·o3 then lh(t) - ba l < c/3. Letting o = minimumof{o1,02,6a }, then ifO < It - a l < 8we have
!f(t) - ~1 l + lg(t)- ~2 l + lh(t)- bal < c/3 + c/3 + c/3 =c. But
lr(t) - bl = l(f(t) - b1, g(t) - b2 , h(t) - ba) l = J(f(t)- b1) 2 + (g(t) - b2 )2 + (h(t) - ba) 2
::; J[J(t)- b1)2 + J[g(t)- b2]2 + J[h(t)- b3)2 = lf (t)- bll + lg(t)- b2 l + lh(t)- b3l
Thus for every c > 0 there exists
o > 0 such that if 0 < It - a l < othen
lr.(t)- b l ::; !f(t) - b1 l + lg(t) - b2 l + lh(t) - b3 l <c. Conversely, suppose for every e > 0, there exists
tharif O <It - a!< othen !r (t) - bl < c
¢:>
J [f(t) - b!) 2 + [g(t)- b2j2 + [h(t) - baj2 < c
l(f(t)- b1 ,g(t) - b2,h(t) - ba) l < c
¢:>
o> 0 such
*>
(J(t) - b1 ] 2 + [g(t)- b2] 2 + [h(t) - baf < c 2. But each term
on the left side of the last inequality is positive, so if 0 < It - a l <
o, then [! (t) - b1]2 < c2, [g( t) - b2]2 < c2 and
[h(t) - b3] 2 < c 2 or, taking the square root of both sides in each ofthe above, lf(t) - b1 l < c, lg(t) - b2 l < . c and
lh(t) - b3l <c. And by definition of limits of real-valued fu nctiQns we have lim f(t) = b1, lim g(t) = b2 and
t -+ a
lim h(t)
t -+a.
t- a
= b3. But by (1), t-+a
lim r (t) = I lim f (t) , lim g(t ), lim h(t)), so lim r (t) = (b1, ~. ba) = b.
\ t-a
t - •a
t- a
t-+c&
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SECTION 13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
13.2
0
157
Derivatives and Integrals of Vector Functions
1. (a)
r (4.5) - r(4)
Q
r(4.2)- r (4)
)(.
4
(b) r(4 ·5)- r ( ) = 2[r(4.5}- r(4}], so we draw a vector in the same
0.5
direction but with twice the length of the vector r(4.5}- r(4).
r (4·2) - r (4) = 5[r(4.2) - r (4)], so we draw a vector in the same
0.2
direction but with 5 times the (ength of the vector r( 4.2) - r( 4).
(c) By Definition I, r'(4) =
J..i~ r(4 + ~ - r (4). T (4) = ~~~!~I·
(d) T(4) is a unit vector .in the same direction as r' (4), that is, parallel to the
tangent line to the curve at r(4) with length 1.
3. Since (x
+ 2) 2 = t2 = y - 1 =>
(b) r'(t) = (1, 2t},
(a), (c)
y = (x + 2? + 1, the curve is a
r'( - 1) = (1,·-2)
parabola.
)(.
5. x = sin t, y = 2 cos t so
x
2
+ (y/ 2) = 1 and the curve is
(a), (c)
(b) r'(t) ~ cost i - 2sintj,
2
an e llipse.
r
'(71")v'2.
4
'2 .
-21 -y~ J
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158
0
CHAPTER 13 VECTOR FUNCTIONS
· 7. Since x = e2 t = (et) 2 = y 2 , the
(b) r '(t) = 2e2 t i + et j ,
y
(a), (c)
curve is part of a parabola. Note
r '(O)
= 2i +j
that here x > 0, y > 0.
X
9. r ' (t) =
\!
[t sin t],
:t [e) ·, !
[t cos 2t]) = (t
c~s t +sin t, 2t, t(- sin 2t) · 2 +cos 2t)
= (t cost+ sin t , 2t, cos 2t - 2t sin 2t)
11. r(t) =t i+j + 2Vt k
2
=> r'(t)=1 i + Oj +2(~t- 1 1 2 )k = i+ ~ k
13. r (t) = et i - j + ln(1 + 3t) k
2
3
=> r'(t) = 2tet i + - - k
1 +3t
15. r ' (t) = 0 + b + 2t c = b + 2t c by Formulas 1 and 3 of Theorem 3.
2
2
r '(O) = (1, 2,
. 2). So lr '(O) I = .JP + .2 + 2 = .J§ = 3 and
19. r '(t) = - sin ti + 3j + 4cos 2t k
=> r '(O) = 3j + 4k. Thus
T(~) = 1::~~~~ = .J02 +~2 +42 (3 j +4k) = t(3j +4k) = ~j + ~ k.
21. r(t) = (t, t 2 , t 3 )
=> r' (t) = ( 1, 2t, 3t 2 ). Then r ' (1) = (1, 2, 3) and lr' (1)1 = \,/1 2 + 22 + 3 2 = v'l4, so
1 (
)
1 1
2
a )
"( )
(
)
r ' ( 1)
T (1 ) = ir' ( 1 )i = V'i4 1,2,3 = \V'i4'V'i4 ' V'i4 . r t = 0,2,6t ,so
j
k
r '(t) x r"(t) = 1 2t 3t2 = 12t 3t21 i 2 6t
0 2 6t
I
11 3t2 1· + 11 2t k
0 6t J
0 2
= {12f - 6t~) i - (6t- O) j + (2 - 0) k = ( 6f , -6t, 2)
23. The vector equation for the curve is r(t) = ( 1 + 2 Jt, t 3 - t, t 3 + t ), so r '(t) = ( 1/ Vt, 3f - 1, 3t2 + 1). The point
(3, 0, 2) corresponds tot= 1, so the tangent vector there is r ' (1) = (1, 2, 4). Thus, the tangent line goes through the point
(3, 0, 2) and is parallel to the vector (1, 2, 4). Parametric equations are x = 3 + t, y = 2t, z ,;,. 2 + 4t.
25. The vector equation for the curve is r (t) = ( e- t cost, e- t s in t, e-t), so
r' (t) = (e-t(- sin t) +(cost)( -e-t), e-t cost+ (sin t)( -e-t), ( -e- t))
= ( - e-t(cos t +sin t), e- t(cos t - sin t), -e-')
The point (1, 0, 1) corresponds to t = 0, so the tangent vector there is
r '(O) = ( -e0 (cos 0 +sin 0), e 0 (cos0- sin 0), -e0 ) = (-1, 1, -1}. Thus, the tangent line is parallel to the.vector
(- 1, 1, - 1} and parametric equations are x = 1 + ( -1)t = 1 - t, y = 0 + 1 · t = t , z = 1 + ( - 1)t = 1 - t.
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SECTION 13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
0
159
27. First we parametrize the curve C of intersection. The projection of C onto the xy-plane is contained in the circle·
x 2 + y 2 = 25, z = 0, so we can write x = 5 cost, y = 5 sin t. C also lies on the cylinder y 2 • + z 2 = 20, and z ~ 0
near the point (3, 4, 2), so we can write z = )20 - y2 = V20 - 25 sin2 t. A vector equation then for C is
r (t) = (5c~st,5sint, )20- 25sin2 t)
=}
r '(t) = ( -5 sint,5cost, ~(20- 25sin2 t)- 1 12 (-50sintcost) ).
The point (3, 4, 2) corresponds tot = cos- 1 (~),so the tangent vector there is
The tangent line is parallel to this vector and passes through (3, 4, 2), so a vector equation for the line
is r (t) = (3 - 4t)i + (4 + 3t)j + (2 - 6t)k.
29. r(t) =(t,e- t , 2t - e )
=* r '(t)=(1,-e- t,2 - 2t). At(0,1,0),
t = 0 and r '(O) = {1, - 1, 2). Thus, parametric equations of the tangent
.line are x = t, y = 1 - t, z = 2t.
31 . r(t) = (tcost,t,t sint)
=* r'(t) = (cost-tsint, 1,tcost+sint).
At ( -1r, 1r, 0), t = 1r and r ' (1r) = (-1, 1, -1r). Thus, parametric equations
of the tangent line are x = -1r - t, y = 1r + t, z = -1rt.
33. The angle of intersection of the two curves is the angle between the two tangent vectors to the curves at the point of
intersection. Since r; (t) = (1, 2t, 3t2 ) and t = 0 at (0, 0, 0), r} (0) = (1, 0, 0) is a tangent vector to r 1 at (0, 0, 0). Similarly,
r~(t) = (cost, 2cos2t, 1) and since r 2(0) =
(0, 0, 0), r~ (0) = (1, 2, 1) is a tangent vectortor2 at (0, 0,0). IfO is the angle
between these two tangent vectors, then cos 0 = 7f7r, (1, 0, 0~ · (1, 2, 1) = -js and 9 = col:l- 1 ( -js) ~ 66°.
35.
J; (t i - t j + 3t k ) dt = (J; t dt). i - (J; t dt) j + (! 3t dt) k
3
5
3
=
2
0
5
(~t 2 ]~ i - [tt4 ]~ j + [~t 0 ] ~ k
= H4- O)i - t{16 - O)j + ~(64- O) k = .2i -4j +32 k
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160 . 0
CHAPTER 13 VECTOR FUNCTIONS
37. f~"' 12 (3sin2 t cost i + 3 sin t cos2 t j + 2 sin t cost k ) dt
= (I; 12 3 sin2 t cost dt) i + (.ro" 12 3 sin t cos2 t dt) j + (I; 12 2 sin t cost dt) k
= [sin3 t) ~
12
i + [- cos t] ~
3
12
12
j+ [sin2 t) ~ k = ( 1 - 0) i + (0 + 1) j + ( 1 - 0) k = i + j + k
2
2
2
39. f (sec2 t i + t(e + 1) 3 j + t ln t k ) dt = (f sec t dt) i + (f t(t,2 + 1) 3 dt) j + (f t ln t dt) k
=tan t i + t(t2 + 1)4 j + (tt3 ln t- ~t 3 )k + C ,
where C is a vector constant of integration. [For the z-cornponent,- integrate by pa~s with u = ln t, dv = t 2 dt.]
41. r'(t) = 2t i + 3t2 j + Vt k
=}
r (t) = t2.i +
e j + tt 12k + c , where cis a constant vector.
3
But i + j = r (1) = i + j + i k + C. Thus C = -tk and r (t) = ·t 2 i + t 3 j + ( tt 312 -
~) k.
Fo.r Exercises 43-46, 1et u (t) = (u1 (t), u 2 (t), u 3 (t)) and v (t) = (v 1 ( t) , v2(t), v 3 ( t)). In each of these exercises, the procedure is to apply
Theorem 2 so that the corresponding properties of derivatives of real-valued funciions can be used.
d
t
d
t
43. -d [u (t) + v (t)] = -d (u1(t) +v1(t),u2(t) +v2(t),ua(t) +v3 (t))
.
= ( ! [u1(t) +v1 (t)],! [u2(t) +v2(t)],
;t
[ua(t) + va(t)])
= M(t) + vW), u~(t) + vHt), u3(t) +vW))
= M (t), u~ (t ), u3(t)) + M(t), v~(t), v3(t)) = u ' (t) + v' (t)
d
d
45. dt [u(t) x v (t)] = dt (u2(t)va(t) - ua(t)v2(t),ua(t)v1(t) - u1(t)v3(t),u1(t)v2(t) - u2(t)v1(t))
= (u~va(t) + u2(t)v3(t) - u3(t)v2(t) - u3(t)v2(t),
u3(t)vl(t) + ua(t)v~ (t)- uW)va(t)- u 1(t)vW),
u~ (t)vz(t) .+ U1(t)vW) - u2(t)vl (t) - uz(tM (t))
= (uW)va(t) - u3(t)v2 (t), u3(t)vl (t)- u~(t)va(t), ui(t)'!!2(t)- u2(t)vl(t))
+ (uz(t)vW) - ua(tM(t), u3(t)vi (t)- ul(t)v3(t), u1 (t)vW) :- uz(t)v~ (t))
= u ' (t) x v (t) + u (t) x v' (t)
· Alternate solution: Let r(t) = u(t) x v(t). Then
r(t +h)- r (t)
= [u (t +h) X v(t +h)] - (u(t) x v(t)]
= [u(t + h) x v(t + h)] - [u(t) x v(t)] + [u (t + h) x v(t)] - [u (t t h) X v (t)]
= u (t + h) x [v (t +h) - v(t)] + [u (t +h) - u (t)] X v(t)
(Be careful of the order of the cross product.) Dividing through by hand taking the limit as h--+ 0 we have
r'(t) = lim u (t +h) x [v~ +h) - v (t)] + lim [u(t +h) - u (t)] X v(t) = u (t) x v'(t) + u'(t) X v (t)
h- o
.
h- o
h
by Exercise 13.1.49(a) and Definition 1.
® 2012 Cengogc Lcwning. All RighlS Rescn•!!d. May not be scnnncd. copied. or duplicalcd, or posted to u public-ly accessible website, in whol e or in parl.
SECTION 13.3 ARC LENGTH AND CURVATURE
47.
:t
[u (t) · v(t)] = u'(t) · v (t) + u (t) ·.v'(t)
D
[by Formula 4 of Theorem 3]
= (cost, - sin t , 1) · (t, cost, sin t) + (sin t, cost, t) · (1,- sin t, cost)
= t cos t - cos t sin t + sin t + sin t - cos t sin t + t cos t
= 2tcost + 2sint - 2cost sint
49. By Formula 4 of Theorem 3, f' (t) = u ' (t) · v (t) + u (t) · v' (t), and v' (t) = (1, 2t, 3t2 ), so
j'(2) = u'(2) · v{2) + u {2) · v'{2) = (3, 0, 4) · (2,4,8) + (1, 2, - 1) · (1, 4, 12) = 6 + 0 + 32 + 1 + 8- 12 = 35.
d
.
51 . - [r(t) x r ' (t)] = r ' (t) ~ r' (t) + r (t) x r" (t) by Formula 5 of Theorem 3. But r' (t) x r' (t) = 0 (by Example 2 in
dt
Section 12.4). Thus,! [r (t) x r '(t)] = r(t) x r "(t) .
53.
1
~
Jr (t)l = dd [r(t) · r(t)] 11 2 = ~ [r(t) · r (t)]- 1 1 2 [2r (t) · r '(t)] = - ( )I r(t) · r '(t)
dt
t·
1r t
55. Since u (t) = r (t) · [r '(t) x r" (t)),
u' (t) = r 1 (t) · [r' (t) X r" (t)] + r (t) · ! [r ' {t) X r 11 (t))
= 0 + r(t). · (r"(t) x r "(t) + r'(t) X r (t)]
= r(t) · (r' (t) X r (t)]
111
[since r' (t) ..l r ' (t) x r " (t)]
111
[since r"(t) x r "(t) = 0]
13.3 Arc Length and Curvature
1. r (t) = (t,3cos t,3sin t)
=?
r' (t) = (1,-3 sin t,3cost)
=?
Jr '(t)l = ) 12 + (- 3 sint) 2 + (3cost) 2 = ) 1 + 9(sin2 t + cos2 t) = v'IO.
Then using Formula3, we have L = f~ 5 lr'(t)l dt =
3. r(t) = v'2t i + etj + e- tk
lr'{t)l =
=?
ts v'Wdt = .fi0t] ~ 5 ~ 10 .fiO.
r '(t) = v'2 i + etj - e-tk
=?
J(v'2/ + (et) + (-e- t)2 =:= J2 + e t + e u .J(et + e- t)2 = et + e- t [since et + e- t > OJ.
2
2
=
Then L = }~1 lr ' (t)i dt = J~(et + e- t) dt = [et - e- t] ~ = e- e- 1 .
5. r(t) = i + t 2 j + t 3 k
Then L = J0 lr '(t) i dt =
1
7. r(t) = (t 2 , t 3 , t 4 )
L = J~ lr'(t)l dt =
=?
r'(t) = 2t j + 3t2 k
=?
=?
ir'(t)i = J4t 2 + 9t4 = t J4 + 9t2 . [since t 2': OJ.
J; tJ4 + 9t dt = fa· t(4 + 9t
2
r 1 (t) = (2t, 3t2 , 4t3 )
2 3 2
) 1
t=
3 2
1
2 7 (13 1 -
431 2 ) = 2\ (133 / 2 - 8).
=> lr ' (t) i = V(2t? + (3t2 )2 + (4t3 ) 2 = J4t2 + 9t4 + 1fit6, so
J; J4t + 9t + 16t dt ~ 18.6833.
2
4
6
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161
162
CHAPTER 13 VECTOR FUNCTIONS
0
9. r (t) = (sin t,cost,tant)
=}
r '(t) = (cost, - sin t,sec2 t)
=}
lr '(t)l = ..jcos2 t + (-sin t)2 + (sec2 t)2 = v'l + sec4 t and L = J0,.14 lr '(t)l dt = J~"' 14 v'1 +sec4 t dt ~ 1.2780.
11. The projection of the curve C onto the xy-plane is the c6rve x 2 = 2y or y = ~ x 2 , z = 0. Then we can choose the parameter
x =t
y = ~t2 . Since C also lies on the surface 3z = xy, we have z .= ~xy = ~(t)(~t2) = i t 3 . Then paramen:ic
=}
equations for C are x = t, y = ~t2, z = *t 3 and the corresponding vector equation is r (t) = (t, ~ t 2 , ~t3 ). The origin
corresponds to t = 0 and the point (6, 18,_36) corresponds tot = 6, so
6
6
2
L = f0 lr'(t)l dt = ]~ 1(1, t , lt )1dt = J~
V1 + t + (~t2 ) dt = f J1 + t + i t4. dt
2
2
2
6
0
2
= f 06 J(1 + ~t2 )2 dt = j~(1 + tt2 ) dt = [t + ~t3)~ = 6 + 36 = 42
13. r (t) = 2t i + (1 - 3t) j + (5 + 4t) k
=}
r '(t) = 2 i- 3j + 4 k and
"* = lr'(t) l :::::: v'4 +9 + 16 = .;29. Then
8 = 8(t) = J~ lr' (u) Idu = J; .;29 du = .;29 t. Therefore, t = ~ 8, and substituting for t in the original equation, we
have r (t(s)) = ~8 i + (1- ~8)j + ( 5+ ~8) k.
15. Here r (t) = (3 sin t , 4t, 3 cos t), so r ' (t) = (3 cos t, 4, -3 sin t) and lr '(t)l = ..)9cos2 t + 16 + 9 sin2 t = v'25 = 5.
The point (0, 0, 3) corresponds to t = 0, so the arc length function beginning at (0, 0, 3) and measuring in the positive
direction is given by 8( t) =
J; lr' (u) Idu = J; 5 du = 5t. 8(t) = 5
5t = 5 . =} t = 1, thus your location after
=}
moving 5 units along the curve is (3 sin 1, 4, 3 cos 1).
17. (a) r (t) = (t, 3 cost, 3 sin t)
=}
r' (t) = (1, - 3 sin t, 3 cost)
=}
lr' (t) l = ..)i + 9sin 2 t + 9 cos2 t = v'IO.
·( )
r' (t)
1 (
.
)
/ 1
;l
•
3
)
ThenT t = lr '{t)l = "710 1, - 3 sm t,3cost or \"Tlii,-'710smt,'711icost.
7io ..)o + 9cos t + 9sin t = #o· Thus
T' (t) =
fto (0, - 3cost, -3sin t)
N (t) =
~~:~~~~ = ~~~ (0, -3 cos t , -3 sin t) = (0,-cost,- sin t).
=}
IT '(t)l =
2
2
(b) ~t(t) = IT '(t) l = 3/v'W = ~
lr'(t)l
v'W 10
19. (a)r(t) = (v'2 t,et,e- t)
=}
r '(t) = (v'2,et,-e-t)
=}
Jr'(t)J=v'2 +e2L + e 2t =..J(et+e-t)2= et+e-t.
Then
[after multiplying by :: ]
and
T '(t) = _1_ (v'2e' 2e2t 0)2e2t
( V'fe' e2t - 1)
e2'+ 1
'
'
(e2 ' + 1)2
''
~ (e2' ~ 1)2 ((e2 ' + 1) (v'2e' ,2e2', 0) - 2e2 ' ( .J2 e',e2 ', - 1)) = (e2' ~ 1)2 (v'2e' (1 .:... e2 ') ,2e2 ',2e2')
® 20 12 Cengagc Lc11nting. All Righls Reserved. Muy no1be :scnnn<:d, copied, or duplicolcd, or poSicd 1o • publicly occcssiblc wcbsilc, in whole: ur In pa n.
SECTION 13.3 ARC LENGTH AND CURVATURE
d
Then
Therefore
2
e t +1
1
( rn t(
2t) 2t 2t)
T '(t)
N (t) = !T '(t)! = J2 et (e2t + 1)2 v2e 1 - e , 2e , 2e
=
!T'(t)!
1
(J2et(1- e2t) 2e2t 2e2t) = _1_ (1- e2t J2 et· J2 et)
J2et(e2t+1)
'
,
e2t+ 1
,
,
1
J2et
-
J2et
-
(b) ~~:(t) = lr'(t)l = e2t + 1 . et + e t - eSt+ 2et + e-t -
21. r (t) = t 3 j + e k
=}
e4t
J2e2t
.J2e2t
+ 2e2t + 1 -:- (e2t + 1)2
lr '(t) ! = y'o2 + (3t2)2 + (2t)2 = v'9t4 + 4t2,
r'(t) = 3e j + 2t k, r"(t) = 6t j + 2 k,
,
"
_ 2
_ lr '(t) X r "(t)i _
r ' (t) x r"(t) = -6t2 i, Jr (t) X r (t)J - 6t . Then ~~:(t)I () 3
r' t I
23. r (t) = 3t i + 4 sin t j + 4cost k
Jr '(t)! =
=?
·6t2
_
6t2
.
(
) .
9t '1 + 4t2 312
(v'9t4 + 4t2 ) 3
r '(t) = 3 i + 4costj - 4sin tk, r"(t) = -4sintj - 4cos~k,
Jg + 16 cos2 t + 16sin 2 t = .J9+T6 = 5, r '(t) x r"(t) = -16 i + 12cos t j - 12 sin t k,
Jr' (t) x r " (t)! = }256 + 144 cos2 t + 144sin2 t = J405 = 20. Then ~~:(t) = lr' (t) x r~ (t)J = 2 ~ = ..! .
,
Jr '(t)[ ·
5
25
25. r (t) = (t, t 2, t 3)
=?
r ' (t) = (1, 2t, 3t 2). The point (1, 1, 1) corresponds to t = 1, and 1)1 (1) = (1, 2, 3)
Jr '(1)1 = v'1 + 4 +9 = Jf4. r"(t) = (0, 2,6t )
=?
r"(1) = (0, 2,6). r '(1) x r"(1) = (6, - 6,2),so
·lr '(1) x r"(1)! = yf36 + 36 + 4 = J76. Then ~~:(1_} = lr'(~;,~ ~'~( )1 = ::;; = ~
1
4
,
3
"( )
27. f (x) = x ,
f (x) = 4x , f
29. f( x) = xe"',
J'(x) = xe"' + e"',
lf"(x) i
x
2
=?
( )
1
lf"(x) !
VSJ.
2
l12x 1
12x 2
= 12x • II: x = [l + (J'(x))2)3/2 = [1 + (4x3)2]3/ 2 = (1 + 16x6)3/ 2
f"(x) = xe"' + 2e"',
!xe"'+2e"' l
!x+2Je"'
~~:{x) = [1 + (/'(x))2J312 = [1 + (xe"' + e"') 2)3/2 = [1 + (xe"' + e"')2 j3/2
31. Since y' = y" = e"', the curvature is ~~:(x)
.
=
Jy"(x) J
_ :r
2x -3/2
= (1 + e"'
e2"' ) 312 - e (1 + e )
.
[1 + (y'(x))2]312
To find the maximum curvature, we first find the critical numbers of ~~:(x):
2:r
3 2x
1 2 2x
'( ) _ "'(
2x)-3/2
2x)- 5/2( 2 2x) _ x 1 + e - e _ x
e
x - e 1+ e
+ e "'(-;!)(
. 2 1+ e
e
- e (1 + e2"')5/2 - e (1 + e2x)5/2.
II:
~~:' (x) = 0 when 1 - 2e 2"' = 0,
so e2"' = ~ or x = - ~ ln 2. And since 1 - 2e2"' > 0 for x < - ~ ln 2 and 1 - 2e2"' < 0
© 2012 Cengogc Learning. All Righls Reserved. Mny not be scn.nncd. copied. or dupljc:1tcd. or posted to a publicly acces.coible website. in whole or in part
163
164
D
CHAPTER 13 VECTOR FUNCTIONS
for x > - ~ ln.2, the maximum curvature is attained at the point (- ~ ln 2, e(- In 2 ) 12 ) = (- ~ ln 2, ~).
Since lim e"'(1 + e2 x)- 3 12 = 0, ~~:(x) approaches 0 as x---+ oo.
:.~: - oo
33. (a) C appears to be changing direction more quickly at P than Q, so we would expect the curvature to be great~r, at P.
(b) First we sketch approximate osculating circles at P and Q. Using the
axes scale as a guide, we measure the radius of the osculating circle
at P to be approximately 0.8 units, thus p = .!.
K,
c
=>
1
"'= .!. ~ - ~ 1.3. Similar!y,'we estimate the radius of the
p
0 .8
osculating circle at Q to be 1.4 units, son, =
.!.p ~ ....!...
~ 0.7.
1. 4
j6x-'1 1
X
6
[1 + (-2x-3)z]3/2 = :z;4 (1 + 4x - 6)3/ 2'
The appearance of the two humps in this graph is perhaps a little surpris ing, but it is
explained by the fact that y = x - 2 increases asymptotically at the origin from both
-1
directions, and so its graph has very little bend there. [Note th~t r;,(O) is undefined.]
37. r(t) = (tet , e- 1 ,../2t)
=>
.r '(t) = ((t +l)et,-e-'·,../2),
r"(t) = ((t+2)et,e-t, O).
Then
r' (t) x r" (t) = (-.J2e-t, ../2(t + 2)e 1 , 2t + 3), · Jr ' (t) x r " (t)l = y'2e- 21 + 2(t + 2) 2 e 2t + (2t + 3)2,
r'(t)J = . l(t + 1)2e2t + e zt + 2,
l
v
( )
and
"' t
2
2
lr'(t) x r "(t) l
y'2e- t +'2(t + 2)2e t + (2t + 3)2
= lr'(t)13 =
((t + 1)2 e2t + e-2t + 2]3/2
We plot the space curve and its curvature function for - 5 :::; t :::; 5 below.
K(l)
0.6
y
-5
5
I
From the graph of r;,(t) we see that curvature is maximized fort = 0, so the curve bends most sharply at the point (0, 1, 0).
The curve bends more gradually as we move away from this point, becoming almost linear. This is reflected in the curvature
graph, where n,(t) becomes nearly 0 as /tl increases.
39. Notice that the curve b has two inflection points at which the graph appears alrnos,t straight. We would expect the curvature to
be 0 or nearly 0 at these values, bu't the curve a isn 't near 0 there. Th~s, a must be the graph of y = f (x) rather than the graph
of curvature, and b is the graph .of y = K-( x).
® 2012 Ccngngc l cnm ing. All Rights Reserved. Mny not be scanned, copied, or duplicated. or posted to t1 publicly accessible webs ite, in who le o r in part.
SECTION 13.3 ARC LENGTH AND CURVATURE
41. Using a CAS, we find (after simplifYing)
~~:(t) =
D
165
K(!)
6 v'4cos2 t - 12cost + 13
( _
cos t)3 12
• (To compute cross
17 12
products in Maple, use the VectorCa1cu1us or
Li nearA1gebra package and the Cross Produc t (a, b)
0
command; in Mathematica, use Cross (a, b].) Curvature is
21T
41T
61r
I
largest at integer multiples of2:rr.
43. X = t 2
=> :i; = 2t => X = 2, y = t 3 => iJ = 3t2 => ij = 6t.
j(2t)(6t) - (3t2 )(2) j
j12t2- 6t21
6t2
Then ~~:(t) = [:i;2 + y2)3/2 = [(2t)2 + (3t2)2j3/ 2 = (4t2 + W)3/2 = (4t2 + 9t4)3/2 ·
\xy- iJx\
45. x = et cos t
=>
:i;
= e1(cos t - sint) => x = e1( - sint- cost)+ et(cost- sin t) = - 2et sin t,
y = et sint => iJ = et (cos t + sin't )
=> y = e1(- sint +cost)+ et(cos t + sint) = 2et cost. Then
\:i;ij - 1ixl
jet(cos t - sin t)(2et cost) - e1 (cos t + sin t)( -2et sin t}j
.
.
= .:....__;..__
_ ___;_;_---:-:_-:---:--------'-:-'-;;-;;;--~
2
2 3 2
[x + y ] 1
([et(cos t- sin t)J2 + [et(cos t +sin t)j2) 3/ 2
x;(t) =
j2e21(cos 2t - sin t cost + sin t cos t + sin2 t) j
j2e2 t(1) j
2e2t
1
3
2
3
2
31
= [e21 (cos2 t - 2costsint +sin2 t + cos2 t + 2 cos t sint +sin2 t)] ' = [e2t(1 + 1)] 1 = e (2) 3 / 2 = -..fiet
2
)
_
47. ( 1, 3 ,1 correspondstot - 1.
2
_ r '(t) _
(2t,2t ,1)
T (t)-\r'(t)\ - v'4t 2 + 4t 4 +
1
T '(t) = - 4t(2t2 + 1)- 2 ( 2t, 2t2, 1) + (2t2 + 1)-1 (2,4t, 0)
_ (2t,2t\ 1)
_ (2 2 1
2t 2 + 1 ,soT(1) - 3•3•3)·
[by Formula 3 of Theorem 13.2.3]
= (2t2 + 1)- 2 ( -8t2 + 4t2 + 2, -8t 3 + 8t3 + 4t, -4t) = 2(2t2 + 1) - 2 (1- 2t2 , 2t, -2t)
( 1 - 2t2, 2t, -2t) - (1- 2t2, 2t, - 2t)
v'1 - 4t2 + 4t4 + 8t2 1 + 2t2
49. (0,7r,-?) corresponds tot ='lr. r(t) = (2sin 3t,t,2cos3t) =>
T(t) = r ' (t) =
(6 cos 3t,1,-6sin3t)
=: - 1- (6 cos 3t, 1,- 6 sin 3t ).
} 36 cos 2 3t + 1 + 36sin2 3t
V37
\r' (t) \
T(1r) = ~ (- 6, 1, 0) is a normal vector for the normal plane, and so (-6, 1, 0) is also normal. Thus an equation for the
plane ·is - 6(x - 0) + 1(y -1r) + O(z + 2) = 0 ory ·- 6x = ~·
18
ffi=>
N (t) = ~~:m, = (-sin3t, O, - cos 3t). So N(1r) = (0, 0, 1) and B (1r) = ~ (-6, 1, 0) x (0,0, 1) = ~ {1, 6, 0).
Since B(1r) is a normal to the osculating plane, so is (1, 6, 0) .
An equation for the plane is 1(x- 0) + 6(y -1r) + O(z + 2) = 0 or x + 6y = 61r.
© 2012 Ccngo&e Lcorning. All Rights Reserved. May not be scanned, copied, or duplicutcd, or posted to • publicly accessible wel><ite, in whole or in part.
166
0
CHAPTER 13 VECTOR FUNCTIONS
51. The ellipse is given by the parametric equations x = 2 cost, y = 3 sin t, so using the result from Exercise 42,
~~:(t) =
l:i:ii- xyl = l(-2sint}(-3sint)- (3cost)(- 2cost) 1 =
6
2
2
[:i:2 + 1?]3 / 2
( 4 sin t + 9 cos2 t~S/ 2
( 4 sin t + 9 cos2 t)3/ 2 •
s
At (2, 0}, t = 0. N.ow ~~:(0} = 267 = ~.so the radius of the osculating circle is
1/ ~(0} = ~ and its center is ( -~, 0). Its equation is therefore (x + ~ )
2
+ y 2 = ¥·
At (0, 3}, t = ~.and~~:(~) = ~ = ~· So the radius of the osculating circle is~ and
its center is (0, ~). Hence its equation is x 2 + (y - ~) =
2
lf.
53. The tangent vector is normal to the normal plane, and the ve~tor (6, 6, - 8) is normal to the given plane.
But T (t) II r '(t) and (6, 6, -8) II (3, 3, -4), so we need to find t such that r' (t) II (3, 3, - 4}.
r (t) = (t3 1 3t, t 4 )
r ' (t) = (3t 2 , 3, 4t3 ) II (3, 3, - 4) when t = - 1. So the planes are parallel at the point ( - 1, -3, 1).
:::::?
55. First we parametrize the curve of intersection. We can choose y = t; then x = y 2 = t 2 and z = x 2 = t 4 , and the curve is
given by r (t) = ( t 2 , t, t 4 ).
r ' (t) = (2t, 1, 4t3 ) and the point (1, 1, 1) corresponds tot= 1, so r ' (1) := (2, 1, 4) is a normal
vector for the norma l plane. Thus an equation of the normal plane is
r' (t)
1
3
2(x - 1) + 1(y - 1} + 4(z- 1) = 0 or 2x + y + 4z = 7. T (t) = - '( )I =
. ( 2t, 1, 4t ) and
2
·
J 4t + 1 + 16t6
1r t
T ' (t) = - H4t2 + 1 + 1?t6 ) - S/ 2 (8t + 96t5 ) (2t, 1, 4t3 ) + (4t2 + 1 + 16t6 ) - l/ 2 ( 2, 0, 12t 2 ). A normal vector for
the osculating plane is B (l) = T (1) x N (1), but r'(1 ) = (2, 1, 4) is parallel to T (1) and
T ' (1} = - ~ (21)- 3 : 2 ( 104} (2, 1, 4} + (21)- 1 12 (2, 0, 12) = 21
3TI (-31, -26, 22) is parallel to N (1) as is (-31, .:...26, 22),
so (2, 1, 4} x (- 31, - 26, 22} = (126, -168, -21) is normal to the osculating plane. Thus an equation for the osculating
plane is 126(x - 1) - 168(y - 1) - 21(z- 1) = 0
57
or 6x- 8y - z = -3.
dTI dT
- l dTI_IdT /dtl_ ldT / dtl d N - dT / dt
-N - ldt dt _dT /dt_dT b h Ch' R l
. ~~:- ds- ds/dt - ds/dt an
- ldT/dtl 'so~~: - ~dTids - ds / dt-ds yte am ue.
dt
59. (a) IB/ = 1
B ·B = 1
~
(b) B = T x N
:
=
=
~
dt
d
dB J. B
ds (B · B )= 0
ds
:::::?
!
(T x N) =
1
[(
T
X
:~ (T x N } ds~dt =
:t
(T x N }
T' )
(
')] 1 ,
T X N'
IT ' I + T X N
lr '(t)l = Tr'Tt)'l
lr'~t)l = [(T ' x N ) + (T X N'))lr'~t)l
:::::?
dB
-J. T
ds
(i) 2012 C.flll•S• !.=nina;. All RighiS Rc:sen'Cd. Moy not be scanned, copied. or duplicated. or posted to o publicly ucccssible website. in whole or in part.
SECTION 13.3 ARC LENGTH AND CURVATURE
=>
(c) B = T x N
0
167
T _L N , B _L T and B _LN. SoB, T and N form an orthogonal set of vectors in the three-
dimensional space !R3 . From parts (a) and (b), dB fds is perpendicular to both BandT, so dB / ds is parallel to N .
Therefore, dB / ds = -r(s)N , where r(s) is a scalar.
-
(d) Since B = T x N , T _L N and both T and N are unit vectors, B is a unit vector mutually perpendicular to both T and
N. For a plane curve, T and N always lie in the plane of the curve, so that B is a constant unit vector always
perpendicular to the plane. Thus dB / ds = 0, but dB f ds = - r(s)N and N =/= 0, so r(s) = 0.
=> r " = s" T + s' T ' = s" T + s' ~~ s' = s" T + ~(s') 2 N by the first Serret-Frenet formula.
61. (a) r ' = s' T
(b) Using part (a), we have
r ' X r" = (s' T ) X (s" T + ~cs') 2 N ]
= ((si T ) x (s" T )] + ((s'T ) x (~(s'? N))
[by Property 3 of Theorem 12.4. 11)
= (s' s")(T x T ) + ~~;(s') (T x N ) = 0 + ~~;(s'? B = ~~;(s') B
3
3
(c) Using part (a), we have
r 111 = [s" T + ~~;(s') 2 N ]' = s T + s" T ' ;t- ~'(s'? N + 2~~;s' s" N + ~~;(s'? N '
111
= s"' T + s"dT s' + ~'(s'? N + 2~s' s" N + ~(s') 2 dN s'
ds
ds
= s"' T + s" s'~ N + K1(s') 2 N + 2~s' s" N + ~~;(s') (-~ T + rB)
= [s"' - K2 {s'?J T + (3~s' s" + ~~;'(s') ] N + ~~;r(s') B
3
[by the second formu la]
3
2
(d) Using parts (b) and (c) and the facts that B · T = 0, B · N = 0, and B · B = 1, we get
(r ' x r ") . r 111 = ~~:(s') 3 B · { (s
111
-
~2 (s') 3 ] T + (3~s' s" + ~~ (s') ] N + KT(s') 3 B} = ~~:(s') 3 ~r(s')3 = T
2
2
Jr ' x r " J
·
63. r = (t, tt2, !t3 )
T -
-
(~~;(s')3] 2
J~~:(s')3 B J
2
=> r' = (1, t, t 2 ), r" = (0, 1, 2t), r 111 = (0, 0, 2) => r ' x r " = (t 2 , - 2t, 1)
•
=>
(r 'x r ")· r "'
(t2 , - 2t,1) ·(0,0,2)
2
- --,---....,..-Jr' x r 11 J2 t 4 + 4t2 + 1
- t 4 + 4t2 + 1
65. For one helix, the vector equation is r (t ) = (10 cost, 10 sin t, 34t/(2rr)) (measuring in angstroms), because the radius of each
helix is 10 angstroms, and z increases by 34 angstroms for each increase of 271' in t. Using the arc length formula, Jetting t go
from 0 to 2.9 x 108 x 21r, we find the appmximate length of each helix to be
L=
J;·
9
B
xlO x zrr Jr'(t)J dt
= j~ · x
9
10s
. /
x zrr y (- 10 sint) 2
8
] 2.9x l0 x2rr
+ {10cost) 2 + U!)2 dt = ..j100 + (~!)2 t
0
= 2.9 x 108 x 21r
J10o+ (g! ) ~ 2.07 x 10 A- more than two meters!
2
10
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168
CHAPTER 13 VECTOR FUNCTIONS
D
13.4 Motion in Space: Velocity and Acceleration
1: (a) Ifr(t) = x(t) i + y (t) j + z(t) k is the position vector of the particle at timet, then the average velocity over the time
.interval [0, 1] is
? k s· . I
_ r(l)- r (O) _ (4.5 i + 6.0j + 3.0k)- (2.7 i + 9.8j + 3.7k) _
.
.
h th
v ave - 1.8 1 - 3 . 8 J - 0 . . 1m11ar y, over t e o er
1
1-0
•
intervals we have
_ r(1)- r (0.5) _ (4.5 i + 6.0j + 3.0 k)- (3.5 i + i2j + 3.3 k ) _
._
k
._
1 - 0.5
0.5
- 2.0 1 2 . 4 J 0.6
[0.5, 1] :
Vave-
[1, 2] :
V ave -
[1, 1.5]:
_ r (2)- r (1) _ (7.3i+7.8 j+2.7k) - (4.5i+6.0j +3.0k) _
.
._
k
.
- 28
· 1 + 1·8 J 03
·
2_ 1
1
_ r(1.5) -r(1) _ (5.9i+6.4j+2.8k)-(4.5 i +6.0 j+3.0 k ) _
. ·
._
k
- 28
· 1 + 08
· J 04
•
1. 5 -1
0.5
.
.
Yavo-
(b) We can estimate the velocity at t = 1 by averaging the average velocities over the time intervals [0.5, 1] and [1, 1.5]:
v(1) ~ ~{(2 i - 2.4 j - 0.6k) + (2.8 i +0.8j- 0.4k)] = 2.4i - 0.8j - o·. 5k. Then the speed is
Jv (1)\ ~ )(2.4) 2 + ( - 0.8)2 + ( -0.5)2 ~ 2.58.
3. r(t)
= ( -. 21 t 2 ,t)
=>
Att = 2:
v{t) = r'(t) = (-t, i)
v{2) = (-2, 1)
a(t) = r"(t) = (-:- 1, 0)
a(2) .= (-1,0)
jv(t)\ = Jt2 + 1
5.r(t)=3costi+2sintj
=>
Att = 71'/3:
v (t) = -3sinti+2costj
v(~) = - ¥ i+j
a(t) = -3cost i- 2sintj
a(i) = -~ i - J3j
/
(3,0)
X
Jv (t) \ = )9sin 2 t + 4cos2 t;:::: .) 4 + 5sin2 t
Notice that x 2 /9 + y 2j4 = sin 2 t + cos2 t = 1, so the path is an ellipse.
z
Att = 1:
v(t) = i + 2tj
v (1) = i + 2 j
a(t) = 2j
a(1) = 2j
Jv(t)l=~
Here x = t, y = t2
Y
=> y = ~ and z = 2, so the path of the particle is a
parabola in the plane z = 2.
2
X
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.
SECTION 13.4 MOTION IN SPACE: VELOCITY AND ACCELERATION
9. r (t) = (t2 + t,
e - t ,t
3
0
169
=> v(t) = r ' (t) = (2t + 1, 2t - 1, 3t 2 ), a(t) = v' (t) = (2, 2, 6t),
)
lv (t)i = .j(2t + 1)2 + (2t- 1)2 + (3t 2 )2 = v'9t4 + 8t 2 + 2.
11. r (t) = v'2t i + e' j + e-t k
=> v (t) = r'(t) = v'2i + e' j - e-' k, a(t) = v'(t) = e' j + e-' k,
lv(t)l = v'2 + e2 ' + e zt = J(e' + e-') 2 = e' + e-' .
13. r (t) = e'(cost, sint, t)
=>
v (t) = r'(t) = e'(cost,.sin t, t~ + et (-sint,cost, 1) = et (cost - sint,sint +cost, t + 1)
a(t) = v'-(t) = 'e'(cost- sint - sin t - cost, sin t +cost+ cost - sin t , t + 1 + 1)
= et(- 2sin t , 2 cost, t + 2)
lv(t)i = etJcos2 t + sin2 t - 2cost sint + sin2 t + cos2 t + 2sin tcost + t 2 + 2t + 1
= e' .Jt2 + 2t + 3
15. a(t) = i + 2j
v(t) = J a(t) dt = J(i + 2j) dt = t i + 2tj + C and k = v {0) = C,
=>
.soC = k and v(t) = t i + 2tj + k. , r {t) = J v(t)"dt = J (t i + 2t j + k) dt = ~t 2 i + t2 j + t k + D .
But i = r (0) = D, soD= i and r(t) = (tt2 + 1) i + t 2 j + tk.
17.. (a) a(t) = 2t i + sintj +cos 2t k
=> ·
(b)
t
v(t) = J (2t i + sin t j + cos 2t k) dt = t2 i - cost j + sin 2t k + C
and i = v (O) = -j + C, so C = i + j
0.6
0.4
% 0.2
and v(t) = (t 2 + 1) i + (1- cost) j + t sin2tk.
0
r (t) = J[(t 2 + 1) i + (1- cost) j + t sin2tk)dt
= (tt3 + t) i + (t- sin t)j- ~cos 2t k + D
Butj = r(O) =
19. r (t) = (t 2 ' 5t,
- i k+ D , soD = j + {kand r (t) = (tt +t) i+ (t -sint + 1)j + (!- { cos2t) k.
e- 16t)
3
=}
v(t) = (2t, 5, 2t - ,16), lv(t)i = v'4t2 + 25 + 4t2 - 64t + 256 = v'8t2 - 64t + 281
and dd lv(t)i = t(se - 64t + 281)- 1 12 (16t - 64) . This is zero if and only if the numerator is zero, that is,
t
.
16t - 64 = 0 or t = 4. Since
:t
lv(t) l < 0 fort < 4 a~d
:t
lv(t)l > 0 fort > 4, the minimum speed of -/153 is atta.ined
at t = 4 units of time.
21 . IF (t)i = 20 N in the direction of the positive z-axis, so F (t) = 20 k. Also m = 4 kg, r{O) = 0 and v {O) = i - j .
Since 20k = F (t) = 4 a(t), a (t) = 5k. Then v (t) = 5t k + c1 where c1 = i - j so v(t) = i - j + 5t k and the
speed is lv (t)l = v 1 + 1 + 25t 2 = v'25t2 + 2. Also r (t) = t i - t j + ~t2 k + cz and 0 = r (O), so C 2 = 0
and r (t) = t i - t j + ~ t2 k.
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170
0
CHAPTER 13 VECTOR FUNCTIONS
23. lv(O) I = 200 m/s and, since the angle of elevation is 60°, a unit vector in the _direction of the velocity is
(cos60°)i+ (sin60°)j = ~i + 4!j. Thusv(O) = 2oo(~i + 4j) = lOOi+ 100J3j and ifwesetuptheaxessothatthe
projectile starts at the origin, then r(O) = 0. Ignoring air resistance, the only force is that due to gravity, so
F(t) = ma(t) = -m9 j where 9 ~ 9.8 m/s 2. Thus a(t) = -9.8j and, integrating, we have v (t) = - 9.8t j +C. But
100 i + 100 V3 j = v(O) = C , so v( t) = 100 i + (100 V3 - 9.8t) j and then (integrating again)
r (t) = lOOt i + (100 V3 t - 4.9t2 ) j + D where·O = r(O) = D. Thus the position function of the projectile is
'
r(t) = lOO t i + (100 V3t- 4.9t2 ) j .
(a) Parametric equations for the projectile are x(t) = lOOt, y(t) = 100 V3 t - 4.9t2. The projectile reaches the ground when
y(t) = O(and t > 0)
=> 100 V3t- 4.9t 2 = t(100 V3- 4.9t) = 0 => t = 104~f ~ 35.3 s. So the range is
x( 4°;0) = 100e 4~f3) ~ 3535 rn.
0
10
(b) The maximum height is reached when y(t) has a critical number (or equivalently, when the vertical component
of velocity is 0): y' (t) = 0
=> 100 J3 - 9.8t = 0 => t = 100~{3 ~ 17.7 s. Thus the maximum height is
v( 10~{!) = 100 V3 ( 10~{!) - 4.9 ( 109~{!r ~ 1531 m.
(c) From part (a), impact occurs at t =
J.;f s. Thus, the velocity at impact is
10
°{!) = 1~0 i + [100 J3 - 9.8 ( 104°f)] j = 100 i - 100 J3j and the speed is
v ( 104
1.f) I= JIO,OOO + 30,000 = 200 mjs.
lv ( 10
25. As in Example 5, r (t) = (vo cos45°)t i + [(vo sin45°)t- t9t2] j = t[voJ2 t i + (voJ2 t- gt2 ) j] . The ball lands when
y = 0 (and t > .0) ::::>:
voJ2
.
. Iands 90 m away,
. 90 = x = -1 vo vrn2'
vo J2
2
909 and the ..
. I
t=- s. Now smce
1t
L. - - or v 0 =
m1t1a
2
9
g
'
velocity is vo = J90g ~ 30 mjs.
27. Let a be the angle ,of elevation. Then v 0 = 150m/sand from Example 5, the horizontal distance traveled by the projectile is
1502
2
2
d = v5 s~ a. Thus
;in a = 800 => sin 2a = ~~~; ~ 0.3484 =>
2a ~ 20.4° or 180 - 20.4 = 159.6°.
Two angles of elevation then are a ~ 10.2° and a :=::: 79.8°.
29. Place the catapult at the origin and assume the catapult is 100 meters from the city, so the city lies between (100, 0)
and (600, 0). The initial speed is v 0 = 80 m/s and let 0 be the angle the catapult is set' at As in Example 5, the trajectory of
the catapulted rock is given byr(t) = (80cosO)ti + [(80sin0)t- 4.9t2] j. The topofthe near city wall is at (100, 15),
which the rock will hit when (80 cos 0) t = 100 =>
5
O and (80 sin O)t - 4.9t2 = 15
4 cos
t=-
=>
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D
SECTION 13.4 MOTION IN SPACE: VELOCin' AND ACCELERATION
171
2
80 sin 9 . -5- - 4.9 ( -5-)
4cos9
4 cosfJ
= 15
=> 100 t a n () - 7.65625 sec2 fJ = 15. Replacing sec2 9 with tan 2 8 + 1 gives
7.65625 tan2 9 - 100 tanfJ + 22.65625 = 0. Using the quadratic formula, we have tan 9 ~ 0.230635, 12.8306
=>
B ~ 13.0°, 85.5°. So for 13.0° < 9 < 85.5°, the rock will land beyond the near city wall. The base of the far wall is
located at (600, 0) which the rock hits if (80 cos fJ)t = 600 =>
t =
15
2
n and (80sin 9)t - 4.9t = 0
2 COSu
=>
2
.
15n - 4.9 ( - 15 (} )
80sm0
· -2cOS!7
2 COS
= 0
=> 600tan9- 275.625sec2 fJ = 0 =>
275.625 tan2 9 - 600 t an0 + 275.625 = 0. Solutions are tan(}~ 0.658678, 1.51819 =>
(} ~ 33.4°, 56.6°.· Thus the
rock lands beyond the enclosed city ground for 33.4° < (} < 56.6°, and the angles that allow the rock to land on city ground
are 13.0° < () < 33.4~, 56.6° < 9 < 85.5°. If you consider t~at the rock can hit the far wall and bounce back into the city, we
calculate the angles that cause the rock to hit the top of the wall at (600, 15): (80 cos 9)t = 600
(80sin fJ)t- 4.9t2 = 15 =>
600tan0 -275.625sec2 (} = 15 =>
Solutions are tan 9 :::::: 0. 727506, 1.44936
=> t = ~ and
2 cos(}
275.625 tan2 (} - 600 tan(}+ 290.625 = o.
=> (} :::::: 36.0°, 55.4°, so the catapult should be set with angle fJ where
13.0° < (} < 36.0°' 55.4° < (} < 85.5° .
31. Here a (t) = - 4 j - 32 k so v (t) = -4tj - 32t k + v o = - 4t j - 32t k + 50 i +80 k = 50 i- 4t j + (80 - 32t) k and
r (t) = 50t i- 2t2 j + (BOt- 16e) k (note that r o = 0). The ball lands when the z-component of r (t) is zero
and t > 0: SOt - 16t 2 = 16t{5 - t) = 0
=>
t = 5. The position of the ball then is
r (5) = 50(5) i - 2{5) 2 j + [80(5) - 16(5) 2 ] k = 250 i - 50 j or equivalently the point (250, - 50, 0). This is a distance of
J250 2 + ( -50)2 + 02 = )65,000 :::::: 255ft from the origin at an angle oftan- 1 (~) :::::: 11.3° from the eastern direction
toward the south. The speed of the ball is lv (5) l = I50 i - 20j - 80 k l = J50 2 + ( - 20)2 + ( -80)2 = v'9300:::::: 96.4 ftls.
33. (a) After t seconds, the boat will be 5t meters west of point A. The velo~ity
20
of the water at that location is 4~0 (5t)(40- 5t) j . The velocity of the
boat in still water is.5 i , so the resultant velocity of the boat is
v(t) = 5 i + 4~ 0 (5t)( 40- 5t) j = 5i + ( ~t- fG-t 2 ) j . Integrating, we obtain
r (t) = 5t i + (~tl - -fGt3 ) j +C. lfwe place the origin at A (and consider j
to coincide with the northern direction) then r (O) = 0
=> C = 0 and we have r (t) = 5t i + (~t 2 -
1
16
e) j . The boat
reaches the east bank after 8 s, and it is located at r(8) = 5(8)i + ( ~(8) 2 - -fG (8?) j = 40 i + 16j. Thus the boat is 16 m
downstream.
(b) Let Q be the angle north of east that the boat heads. Then the velocity of the boat in still water is given by
5(cos Q) i + 5(sin Q) j . At t seconds, the boat is 5(cos Q)t meters from the west bank, at which point the velocity
of the water is .1 ~ 0 [5(cos Q)t][40- 5(cos Q)t] j . The resultant velocity of the boat is given by
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172
D
CHAPTER 13 VECTOR FUNCTIONS
-fue cos 2 a:) j.
v(t) = 5(cos a:) i + [5 sin a:+ 4 g0 (5t cos a:)( 40 - 5t cos a:)] j = (5 cos a:) i + (5 sin a:+ ~t cos a: Integrating, r (t) = (5t cos a:) i + (5tsin a:+ ~t2 cos a:- -ftt3 cos2 a:) j (where we have again placed
the origin at A). The boat will reach the east bank ~hen 5t cos a: = 40
In order to land at point B ( 40, 0) we need 5t sin a: + ~
8
5(- - ) sin
cos a:
=?
t = ~ = - -.
5 cos a:
cos a:
8
e cos a: - ft t cos a: = 0 =?
3
2
8
8
1
a:+~ (cos
-- ) cos a:- -ft ( -- ) .cos a:= 0 =? - - (40sina: + 48 - 32) = 0 =?
a:
cos a:
cos a:
2
3
2
40 sin a: + 16 = 0
=?
sin a: = - %. Thus a: = sin - l (-%) ~ -23.6°, so the boat should head 23.6° south of
If
east (upstream). The path does seem realistic. The boat initially heads
upstream to counte~act the effect of the current. Near the center of the river,
0 1--=:::==::::::::;::>o-f'...::::::::=:=::::~ 40
the current is stronger and the boat is pushed downstream. When the boat
nears the eastern bank, the current is slower and the boat is able to progress
upstream to arrive at point B.
- 12
35. Ifr' (t) = c x r(t) then r' (t) is perpendicular to both c and r(t). Remember that r ' (t) points in the direction of motion, so if
r' (t) is always perpendicular to c, the path of the particle must lie in a plane perpendicular to c. But r ' (t) is also perpendicular
to the position vector r(t) which confines the path to a sphere centered at the origin. Considering both restrictions, the path
must be contained in a circle that lies in a plane perpendicular to c, and the circle is centered on a line through the origin in the
direction of c.
37. r(t) = (3t - t 3 ) i
+ 3t 2 j
r'(t) = (3 - 3t2 ) i + 6tj,
=?
2
lr' (t)i = J(3- 3t2 )2 + (6t)2 = v'9 + 18t2 + 9t4 = J(3 - 3t 2 ) 2 = 3 + 3t ,
r" (t) ·= -6t i + 6j, r' (t) x r" (t) = (18 + 18t2 ) k. Then Equation 9 gives
2
- r ' (t) . r" (t) - (3 - 3e)( -6t) + (6t)(6) - 18t + 18t 3 - 18t(1 + t ) - 6t
aTlr '(t)/
3 +3t2
3 + 3t2 - 3(1 + t 2 ) I
d [
2]
]
aT = v = dt 3 + 3t = 6t
39. r (t) = cost i +sin t j + t k
r" (t) = - cost i - sin tj,
an
=?
dE
[or by Equation 8,
2
. LO .
- lr'(t) X r "(t) i - 18 + 18e - 18(1 + t ) - 6
quat10n g1ves aN ir'(t)i
- 3 + 3 t 2 - 3 (1 + t 2) - ·
r '(t) = - sint i + costj + k ,
ir'(t)i = Vsin 2 t + cos2 t + 1 =
V'2,
r' (t) x r" (t) =sin t i - cos tj + k.
_ r'(t) · r"(t) _ sintcost - sintcost _
d
_lr'(t) x r"(t)l _ Vsin2 t+cos 2 t+ 1 = .J2 = l
0
lr'(t)i
- .
.j2
- an aN lr'(t)i
-/2
-/2
·
Then D.T 41. r (t)=et i +-/2t j +e-t k
=?
r"(t) = et i + e-t k. Then aT =
r'(t)=et i+-/2j -e- 1 k,
e2t _ e- 2t
et + e t
=
ir(t) i=v'e21· +2+ e 2 t=J(e1· +e t)2=e'· +e- t,
(et + e- t)(et _ e- t)
et + e t
.
= et - e:- t = 2sinht
® 2012 Ccngagc Learning. All Rights Reserved. May not be scanned. cop ied, or t.luplicatcd. or posted 10 a publicly nccessiblc website. in '"'hole or in p01rt.
CHAPTER 13 REVIEW D
173
43. The tangential component of a is the length of the projection of a onto T, so we sketch
the scalar projection of a in the tangential direction to the curve and estimate its length to
be 4.5 (using the fact that a has length 10 as a guide). Similarly, the normal component of
a is the length of the projection of a onto N, so we sketch the scalar projection of a in the
normal direction to the curve and estimate its lengttJ to be 9.0. Thus aT ::::::: 4.5 cmf s 2 and
X
45. _If the engines are turned off at timet, then the spacecraft wi ll continue to travel in the direction ofv(t) , so we need at such
that for some scalars > 0, r (t) + s v (t) = (6, 4, 9).
4
24t- 12t2 - 4
8(3- t)t
so 7- t2 + 1 + (t2 + 1)2 = 9
v(t) = r ' (t) = i + ~ j + (t 2 ~ ) 2 k
1
¢}
- -;-:-;;:
--:--::-.;--:- = 2
(t2 + 1)2
¢}
t
4
+
+ 8t2 - 12t + 3 = 0.
·a is easily seen that t = 1 is a root of this polynomial. Al~o 2 ln 1 + ~
13
=>
3
1
= 4, sot= 1 is the desired solution.
Review
CONCEPT CHECK
1. A vector function is a function whose domain is a set of real numbers and whose range is a set of vectors. To find the derivative
or integral, we can differentiate or integrate each component of the vector funct ion.
2. The tip of the moving vector r (t) of a continuous vector function traces out a space curve.
3. The tangent vector to a smooth curve at a point P with pos ition vector r(t) is the vector r ' (t). The tangent line at p ' is the line
through P parnllel to the tangent vector r ' (t). The unit tangent vector is T (t) =
,::m,.
4. (a) (a) - (f) See Theorem 13.2.3.
5. Use Formula 13.3.2, or equivalently, 13.3.3.
I
6. (a) TI1e curvature of a curve is"' = I~~ where Tis the unit tangent vector.
T '(t)
(b) K.(t) = . r '(t)
I
I
(c) K.(t) = lr' (t ) X r" (t)l
T' (t)
7. (a) The unit normal vector: N (t) = IT'(t) l'
[r'(t)[ 3
_
if"(x) l
·
(d) K-(x)- [1 + (f'(x))2j3/2
The binormal vector: B (t) = T (t) x N (t).
(b) See the discussion preceding Example 7 in Section 13.3.
8. (a) Ifr(t) is the position vector of the particle on the space curve, the velocity v (t) = r'(t), the speed is given by lv (t )i,
and the acceleration a (t) = v ' (t) = r " (t).
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174
D
CHAPTER 13 VECTOR FUNCTIONS
(b) a = a7:T +aN N where aT = v' and aN = 1w 2 .
9. See the statement of Kepler's Laws on page 892 [ET 868].
TRUE-FALSE QUIZ
1. True. If we reparametrize the curve by replacing u
= t 3 , we have r(u) = u i + 2u~ + 3uk, which is a line through the origin
with direction vector i + 2 j + 3 k.
3. False. The vector function represents a line, but the line does not pass through the origin; the x-component is 0 only for t ;, 0
which co~esponds to the point {0, 3, 0) not {0, 0, 0).
d
5. False. By Formula 5 ofTheorem 13.2.3, -d (u(t) x v(t)] = u '(t ) x v (t) + u (t) x v'(t).
.
t
7. False. K- is the magnitude of the rate of change of the unit tangent vector T with respect to arc length s, not with respect tot.
9. True. At an inflection point where f is twice continuously differentiable we must have !" (x) = 0, and by Equation 13.3.11,
the curvature is 0 there.
11. False. If r(t) is the position of a moving particle at timet and lr(t)l = 1 then the particle lies on the unit circle or the unit
sphere, but this does not mean that the speed lr'(t)l must be constant. As a counterexample, let r (t) = (t, Jl=t2), then
r' (t) = ( 1, -t/J1 - t 2 ) and lr(t) l = Jt 2 + 1 - t 2 = 1 but lr'(t)l
= yf1 + t 2 /(1- t 2 ) = 1/ ~ which is not
constant.
13. True. See the discussion preceding Example 7 in Section 13.3.
EXERCISES
1. (a) The corresponding parametric equations for the curve are x = t,
y = cos 1rt, z = sin -rrt. Since y 2 + z 2 = 1, the curve is contained in a
I
circular cylinder with axis the x-axis. Since x
(b) r(t) = t i + cos ri j +sin -rrt k
= t, the curve is a helix.
=>
r'(t) = i - -rrsin 1rt j + -rrcos -rrt k
~
X
r" ( t) = --rr2 cos 1rtj - 1r 2 siri 7rt k
3. The projection of the curve C of interSection onto the xy-plane is the circle x 2 + y 2 = 16, z = 0. So we can write
= 4 cost, y = 4 sin t, 0 S t S 21r. From the equation of the plane, w~ have z = 5 - x = 5 - 4 cos t, so parametric
equations for Care x = 4 cost, y = 4 sin t, z = 5 - 4 cost, 0 S t S 2-rr, and the corresponding vector function is
x
r(t) = 4 cost i + 4 sin t j + (5 - 4 cost) k , 0 S t S 21r.
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CHAPTER 13 REVIEW· 0
5. f~(t 2 i +tcos 1rtj +sin 1rt k ) dt =
(!1 t dt) i + (I~ tcos 1rtdt) j +(I; sin 1rtdt) k
2
0
I
= [~t3 ]~ i + (~sin 1rt]~ - f 01 ~sin 1rtdt) j +[-~cos 1rt]~ k
1 •
= ]j l +
2 k
t]l .
[ 1
;1fCOS1l" 0 J+ ;r
2 •
2 k
;= 31 1• - ;:lf
J+ ;r
where we integrated by parts in the y-component.
1. r(t)
= ( t 2 , t 3 , t 4) . =? r (t) = ( 2t, 3e,4t3 )
. L=
J; Jr (t)J dt = J; J4t2 + 9t4 + 16t6 dt. Using Simpson's Rule with f(t) = J4t 2 + 9t4 + 16t and n = 6 we
1
=?
Jr (t) J = J 4t 2 + 9t4 + 16t6 and
1
1
6
have tlt = 3(i0 = ~ and
L ~ ';' [f(O) + 4f(~) + 2f{1) + 4Ja) + 2/(2) + 4f(~) + /{3)]
=i
[vo
+o +o + 4 . ..j4(~/ + 9 (~) + 16(~) + 2 . .J4(1)2 + 9{1) 4 + 16(1)6
4
6
2
4
+ 4. J4G) + 9G) + 16 Gt + 2 . .J4(2)2 + 9(2)4 + 16{2)6
+4·
4(~)2 +9(~)4 +16(~t+.J4(3) 2 +9{3)4+ 16(3)6 ]
~ 86.631
9. The angle of intersection of the two curves, 8, is the angle between their respective tangents at the point of intersection.
For both curves the point {1, 0, 0) occurs when t = 0.
r W) = - sin t i +cos t j + k
=?
rHO) = j + k and r 2(t) = i + 2t j + 3t k
2
=}
r~(O) = i.
r i {0) · r~ (0) = (j + k ) · i = 0. Therefore, the curves intersect in a right angle, that is, 8 = ~.
(b) T'(t) = -Ht4 +t2 + 1)- 3 12 (4t3 +2t) (t2,t, 1) + (t4 + t 2 + 1)- 1 12 (2t, 1, 0)
- 2t3 - t
2
1
(t4 + t2 + 1)3/ 2 (t , t , 1) + (t4 + t2 + 1)1 / 2 (2t, 1, 0)
- ( -2t5 - t 3 , - 2t4 - t 2 , - 2t3 - t) + (2t5 + 2t3 + 2t, t 4 + t 2 + 1, 0) (t4 + t2 + 1)3/ 2
1
IT (t)l =
Jt 6 + 4t4 + 4t2 + t 8 - 2t4 + 1 + 4t6 + 4t4 + t 2
(t4 + t2 + 1)3/2
=
(t 3 + 2t, - t 4 + 1, -2t 3 - t)
(t4 + t2 + 1)3/2
JtB + 5t6 + 6t4 + 5t2 + 1
(t4 + t2 + 1)3/ 2
and
(t 3 + 2t 1 - t 4 - 2t3 - t )
'
'
- Jt!!. + 5t6 + 6t4 + 5t2 + 1 .
N (t) -
JT 1 (t) \
J t!!. + 5t6 + 6t4 + 5t2 + 1
(c) "'(t) = Jrl(t) l =
(t4 + t2 + 1)2
or
Jt 4 + 4t2 + 1
(t4 + t2 + 1)3/ 2
13. y I = 4X 3 , y II = 12 X 2 and It( X ) = ( + lv"l
1 (y')2)3/ 2
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175
176
0
CHAPTER 13 VECTO'R FUNCTIONS
15. r (t) = (sin2t, t,cos2t)
=> r '(t) = (2cos2t, 1, -2 sin2t) => T (t) = -js (2 cos2t, 1, - 2sin2t) =>
T '(t) = ~ (- 4sin 2t, 0, -4cos 2t)
=> N (t) = (-sin 2t, 0,- cos2t). SoN = N ('rr) = (0, 0, - 1) and
B = T x N = -js (-1, 2, 0}. So a normal to the osculating plane is (- 1, 2, 0) and an equation is
- l (x- 0) + 2(y- 1r) + O(z - 1) = 0 or x - 2y + 27r = 0.
17. r (t) = tlnt i +tj + e-t k , v (t) = r '(t) = (1 + lnt) i + j - e-~ k,
lv (t)i =
J(l + ln t)2 + 12 + (-e-:-t)2 = vf2 +2 ln t + (ln t)2 +e- t, a(t) = v'(t) = t i + e-t k
2
19. We set up the axes so that the shot leaves the ath lete's hand 7 ft above the origin. Then we are given r (O) = 7j,
lv (O)I = 43 ft/ s, and v (O) has direction given by a 45° angle of elevation. Then a unit vector in the direction ofv(O) is
~(i + j )
=> v (O) = ~(i + j ). Assuming air resistance is negligible, the only external force is due to gravity, so .as in
Example 13.4.5 we have a = - g j where here g ~ 32 ft/s 2 . Since v ' (t) = a (t), we integrate, giving v (t) = - gt j + C
where C = v (O) = ~(i + j )
=> v (t) = ~ i + ( ~ - gt) j . Since r '(t) = v (t) we integrate again, SO·
r(t) = ~t i + ( ~t - ~gt2) j + D. ButD = r (O) = 7j
=> r (t) = ~t i + ( ~t- ~gt2 + 7)j.
(a) At 2 seconds, the shot is at r(2) = ~(2) i + ( ~ (2) - ~g(2) 2 + 7) j ~ 60.8 i + 3.8j, so the shot is about 3.8 ft. above
the ground, at a horizontal distance of 60.8 ft from the athlete.
(b) The shot reaches its maximum height when the vertical component of velocity is 0: ~ - gt = 0
=>
t = ;: ~ 0.95 s. Then r (0.95) ~ 28.9 i + 21.4j , so the maximum height is approximately 21.4 ft.
v2g
.
(c) The shot hits the ground when the vertical component ofr(t) is 0, so ~t - ~gt 2 + 7 = 0 =>
-16t2 + ~t + 7 = 0 => t ~ 2.11 s.
r (2. 11) ~ 64.2 i - 0.08j , thus the shot lands approximately 64.2 ft from the
athlete.
21. (a) Instead of proceeding directly, we use formula 3 ofTheorem 13.2.3: r (t) = t R (t)
=>
v = r'(t) = R (t) + t R' (t) = coswt i + sinwtj +t v a.
(b) Using the same method as in part (a) and starting with v = R (t) + t R '(t), we have
a= v ' = R '(t) + R '(t) + t R "(t) = 2 R '(t) + t R "(t) = 2 v d + tad.
(c) Here we have r(t) = e-' coswt i + e- t sin wtj ,; e-t R (t). So, as in parts (a) and (b),
v = r '(t) = e-t R '(t)- e-t R (t) = e-t[R '(t) - R (t)) =>
a = v' = e-t [R"(t)- R '(t))- e-:t[R '(t) - R (t)) = e-~ [R"(t)- 2 R ' (t) + R (t)J
Thus, the Coriolis acceleration (the sum of the "extra" terms not involving aa) is -2e-~ v .i + e-t R.
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CHAPTER 13 REVIEW
23. (a) r (t) = R coswt i + R sin wtj
0
177
=> v = r ' (t) = -wR sinwt i + wR coswt j , so r = R(coswt i + sinwtj) and
v = wR(- sin wti + cos wtj). v · r = wR ( - coswt sin wt + sinwtcoswt) = 0, so v ..L r . Since r points along a
2
radius of the circle, and v ..L r , vis tangent to the circle. Because it is a velocity vector, v points in the direction of motion.
(b) In (a), we wrote v in the form wR u, where u is the unit vector - sin wt i + cos wt j . Clearly Ivi = wR lui = wR . At
. Ie campI etes one revo I ut10n,
.
. T = -2·nR
speed wR , the part1c
a d.1stance 21r R,.m t1me
R = -27r .
w
. w
(c) a = dv = - w2 R coswt i - w2 Rsinwtj = - w2 R(coswt i + sinwt j ), so a = - w2 r. This shows that a is proportional
dt
to rand points in the opposite direction (toward the origin). Also, ja j = w 2 jr j = w 2 R .
(d) By Newton's Second Law (see Section 13.4), F = ma, so jF j = m jaj = mRw2 = m (~R) = m ~~
2
2
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0
PROBLEMS PLUS
1. (a) The projectile reaches maximum he~ght when 0 = ~~ =
!
[(Vo sina)t- ~gt2] = vo sin a - gt; that is, when
vosina
. )(vosiiJ.a)
1 (vosina)
t= --an.d y = (vosma
- - - - -g
---
g
g
2
2
2
°
Vo sm a . Th'1s IS
. the max1mum
.
'ght attamed
.
11e1
when
2g
g
2
.
the projecti le is fired with an angle of elevation a. This maximum height is largest when a = ~ . In that case, sin a = 1
.
. . vfi
and the maximum he1ght IS - .
2g
(b) Let R = vfi
/g. We are asked to consider the parabola x 2 + 2Ry - R 2 = 0 which can be rewritten as y = - __!._ x 2 + !!:. .
2R
The points on or inside this parabola are those for which - R ~ x ~ Rand 0 ~ y ~ ;~ x 2 + ~.
2
When the projectile is
fired at angle of elevation a, the points (x, y) along its path satisfy the relations x = (v 0 cos a) t and
y = (v0 sina)t - tyt 2 , whereO ~ t ~ (2v0 sin a)/g(as inExan1ple 13.4.5). Thus
lxl
~ lvo cos aCvo ;ina) I= I~ sin 2al ~ I~; I = IRI. This shows that -R ~ x ~ R.
vo sin a- ~ gt ) = t gt Cvo ;in a- t) ;: : 0 and
For t in the specified range, we also have y = t (
y
,
X
9
X
= (vosma)
--- -- - -vo cos a
2 vo cos a
0
(
- 1 2
R)
y- ( 2Rx + 2
)2= (tana)x - v
g
2 02 cos2 a
1
2
x =
.
2
R
x +(tan a) x. Thus
2 cos 2 a
R
-1
1 2 (
2
= 2Rcos2 ax + 2Rx + tan a) x- 2
= ~ ( 1 - _1_ ) + (tana)x _ R = x (1- sec a) + 2R{tana) x - R
2R
cos2 a
2
2R
2
_ -{tan 2 a) x 2 + 2R (tan a) x - R 2 ·_
2R
-
2
:- [{tana)
x - R]2
. 2R
2
~0
We have shown that every target that can be hit by the projecti le lies on or inside the parabola y = _ __!._ x 2 + R.
2R
2
Now let (a, b) be any point on or inside the parabola y = -
2~ x + ~· Then·-R ~a~ Rand 0 ~ b ~ - 2~ a + ~2
2
We seek an angle a such that (a, b) lies in the path of the projectile; that is, we wish to find an angle a such that
b= -
2~c~s2 a a +(tan a) a or equivalently b = ~ (tan a ~1- 1)a +(tan a) a . Rearranging this equation we get
2
2
2
;~ tan2 a- a tan a+(;~+ b) = 0 or a 2 (tana? - 2aR(tana) + (a2 + 2bR) = 0 (*) . This quadratic equation
for tan a has real solutions exactly when the discriminant is nonnegative. N~w B 2 - 4AC ;::: 0
<=>
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179
180
0
CHAPTER 13 PROBLEMS PLUS
(-2aR?- 4a2 (a 2 + 2bR);:::: 0
b:::;
2~ (R
2
y ·= -
-
a2 )
<=?
<=?
4a2 (R 2 - a 2 - 2bR);:::: 0 <=?
- a2 - 2bR+ R 2 ;:::: 0 <=?
b:::; ;~ a2 + ~- This C?ndition is satisfied since (a, b) is on or inside the parabola
2~ x + ~- It follows that (a, b) lies in the path of the projectile when tan a satisfies(*), that is, when
tan a =
2
2aR ± J4a2(R2 - a 2 - 2bR)
2a2
R ± ,jR2- 2bR- a2
= ----''--- - - -a
If the gun is pointed at a target with height hat a distanceD downrange, then
(c)
tan a = h/ D. When the projectile reaches a distanceD downrange (remember
we are assuming that it do.esn' t hit the ground first), we have D = x = (vo cos a)t,
D
D
sot= --- andy=(vosina)t-~gt2 = Dtana-
2
gD 2
2
2
s=so + 2t i -~gt 2 j=3.5 j +2t i- tgt 2 j
=>
~~Q
.
~~Q
Meanwhile, the target, whose x-coprdinate is also D, has fallen from height h to height
2
h-
tge = D tan a - 2 v{D
. Thus the projectile hits the target. ·
cos2 a .
0
3.(a)a=-gj
=> v=vo-gt j =2 i -gtj =>
s = 2t i + (3.5- tgt2 ) j . Therefore y = 0 when t =
J7l9 seconds. At that instant, the ball is 2 J7l9 r::::: 0.94 ft to the
right of the table top. Its coordinates (relative to an ori'gi n on the floor direct ly under the table's edge) are (0.94, 0). At
impact, the velocity is v = 2 i - ,f7gj, so the speed is !v i= .J4+79 r::::: 15 ft./s.
(b) The slope ofthe curve when t =
dy
dy jdt
-gt
-g J779
- ,j7g
,;rg
Vf!_g.'s. dx
= dx jdt = 2 =
= - 2- . Thus cot B = - 22
and () r::::: 7.6°.
(c) From (a), lvl = vl4+7g. So the ball rebounds with speed 0.8 .J4+79 r::::: 12.08 ft/s at angle of inclination
2 . 2
90°- () r::::: 82.3886°. By Example 13.4.5, the horizontal distance traveled between bounc~s is d = Vo sm a, where
g
v 0 r::::: 12.08 ft./s and a~ 82.3886°. Therefore, d ~ 1.197 ft. So the ball strikes the floor at about
2
J779 + 1.197 ~ 2.13 ft to the right of the table's edge.
· 5. The trajectory oftbe projectile is given by r (t) = (vcosa)t i + [(v sin a)t- ~gt 2 ] j , so
v(t) = r '(t) = vcosa i + (vsina- gt) j and
lv (t)i = J(v cosa) 2 + (vsin a- gt) 2 = Jv 2 - (2vgsin a) t + g2t2 =
=g
( 't -
v .
g
- sm a )
2
2
2
v
. 2
+ -g2
- -vg2 sm
a =g
.
( t - 9~sino:)
2
g2
(
t 2 - -2v (sin a) t
g
2
+ -vg2 )
2
2a
+~cos
g2
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CHAPTER 13 PROBLEMS PLUS
The projectile hits the ground when (v sina)t- ~gt2 = 0
(2v/g)sin o
L(a) =
.9
10
=g
[
0
2
;
t =
181
sin a, so the distance traveled by the projectile is
2
1 (211/g)sin<>
lv(t)l dt =
=>
0
v
)
v2
2
( t - -.9 sin a + 29 cos a dt
t - (v / g) sin o
2
+ [(vl g) cosa]2 ln
2
(t-~sino +
g
[using Formula 2 I in the Table of lntegrals]
[v. (~.rna)' + (~ oo.o)' +(~ '"'")' m( ~ •ina+ (~ •mo)' +(~=aY)
= g -gsma
2
(~ •ina)' + (~=o)'- (~oooo)' m(-~•ina+ (~•ina)' + (~=oY)]
+~sin o
g
[~sino · ~+
v cos2 aln(~ sino+~) +~ sino·~ - v cos2 a ln(-~ sino+~)]
g
g2
g
g
g
g
g2
g
g
2
2
= g_
2 g
2
2
+ -cos
v
2
.
= -v stno
a
g
2f1
1n( - (vl(vIg)sin
a+v/g) = -sm
v
v
a + -cos
an
1 (1+sino)
g) sin a +vI g
g
2g
1 - sin a
2
2
.
2
We want to maximize L(a) for 0 s; a s; 1r/ 2.
vg
. (
. a ln ( 1 + sin
= -v cos a + -v cos a · -2- - 2 cos a sm
. a)]
g
2g
coso ,
1 -sma
2
L'(a) = - coso+ -v
2
9
a
a
a)]
2 cos
1 + sin
- 2cosa smo ln
cos2 a · 1 - sin
·
.
.
.
1 +sma (1-smo) 2
. 1 -sm o
2
•
-v
[
2g
2
=
2
[
2
2
v COSO! [ 1- SlDO!
•
] =
.
COSO!+ln (l+sina)
1 - SlDO!
g
2
-v
9
.
•ln (1+ sin
COSO! [ 2- SIDQ
. a) ]
1 - SJDO! ,
L( a ) has critical points for 0 < a < 1r12 when L' (a) = 0 => 2 - sin a ln ( ~ + s~ a) = 0 [since cos a =1= 0].
-
SIDQ
Solving by graphing (or using a CAS) gives a~ 0.9855. Compare values at the critical point and the endpoints:
L(O) = 0, L(1r12) = v 2 I g, and £(0.9855) ~ 1.20v 2 I g . Thus the distance traveled by the projectile is maximized
for a ~ 0.9855 or ~ 56°.
7. We can write the vector equation as r (t) = ·at2 + bt + c where a= (a1, a2 , a a}, b = (b1 , b2 , ba }, and c = (c1 , c 2, c3).
Then r ' (t) = 2t a+ b which says that each tangent vector is the sum of a scalar multiple of a and the vector b . Thus the
tangent vectors are all parallel to the plane determined by a and b so the curve must be parallel to this plane. [Here we assume
that a and bare nonparallel. Otherwise the tangent vectors are all parallel and the curve lies along a single line.] A normal
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182
0
CHAPTER 13 PROBLEMS PLUS
vector for the plane is a x b = (a2b3- a3~, a3b1 - a1b3, a1b2 - a2b1). The point (c1. c2, c3) lies on the plane (when
t = 0), so an equation of the plane is
(a2b3- a31h)(x- c1) + (a3b1 - a1 ~)(y - c2) + (al~ - a2bl)(z - c3) = 0
or
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PARTIAL DERIVATIVES
14
0
14.1
Functions of Several Variables
1. (a) From Table 1, f( - 15, 40) = -27, which means that if the temperature is - 15°C and the wind speed is 40 km/ h, then the
air would feel equivalent to approximately -27°C without wind.
(b) The question is asking: when the temperature is -2!tC, what wind speed gives a wind-chill index of - 30°C? From
Table 1, the speed is 20 km/h.
(c) The question is asking: when the wind speed is 20 km/ h, what tem·perature gives a wind-chill index of -49°C? From
Table I, the temperature is -35°C.
(d) The function W = j( -5, v) means that we fix T at -5 and allow v to vary, resulting in a function of one variable. In
other wor~s, the function gives wind-chill index values ~or different wind speeds when the temperature is -5° C. Fro111
Table 1 (look at the row corresponding toT = - 5), the function decreases and appears to approach a constant value as v
increases.
(e) _The function W = j(T, 50) means that we fix vat 50 and allow T to vary, again giving a function of one variable. In
other words, the function gives wind-chill index values for different temperatures when the wind speed is 50 km/h . From
Table 1 (look at the column corresponding to v = 50), the function increases almost linearly as T increases.
3. P(120, 20) = 1.47(120) 0 ' 65 (20) 0 ·35 ~ 94.2, so when the manufacturer invests $20 million in capital and 120,000 hours of
labor are completed yearly, the monetary value of the production is about $94.2 million.
5. (a) !(160, 70) = 0.1091(160) 0 .425 (70) 0 · 725 ~ 20.5, which means that the surface area of a person 70 inches (5 feet 10
inches) tall who weighs 160 pounds is approximately 20.5 square feet.
(b) Answers will vary depending on the height and weight of the reader.
7. (a). According to Table 4, f( 4·0, 15) = 25, which means that if a 40-knot wind has been blowing in the open sea for 15 hours,
it will create waves with estimated heights of25 feet.
'
(b) h = f(30, t) means we fix vat 30 and allow t to vary, resulting in a function of one variable: Thus here, h = f(30, t)
gives the wave heights produced by 30-knot winds blowing for t.hours. From the table (look at the row corresponding to
v = 30), the function increases but at a declining rate as t increases. In fact, the function values appear to be approaching a
limiting value of approximately 19, which suggests that 30-knot winds cannot produce waves higher than about 19 feet.
(c) h = f(v , 30) means we fix tat 30, again giving a function of one variable. So, h = f(v, 30) gives the wave heights
produced by winds of speed "! blowing for 30 hours. From the table (look at the column corresponding to t = 30), the
function appears to increase at an increasing rate, with no apparent limiting value. This suggests that faster winds (lasting
30 hours) always create higher waves.
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183
184
0
CHAPTER 14 PARTIAL DERIVATIVES
9. (a) g(2, - 1) = cos(2 + 2( - 1)) = cos(O) = 1
(b) x + 2y is defined for all choices of values for x andy and the cos ine function is defined for all input values, so the domain
of g is JR 2 .
·(c) The range of the cosine function is [- 1, 1] and x + 2y generates all possible input values for the cosine function, so the
range ofcos(x + 2y) is (-1, 1].
(b) ft, .,fij,
vz are defined o~ly when x ~ 0, y ~ 0, z ~ 0, and ln(4 - x
4 - x 2 - y 2 - z2 > 0
<=}
2
-
y 2 - z 2 ) is defined when
x 2 + il + z 2 < 4, thus the domain is
{ ( x, y, z) 1 x 2 + y 2 + z 2 < 4, x ~ 0, y ~ 0, z ~ 0} , the portion of the interior of a sphere of radius 2, centered at the
origin, that is in the first octant.
13. ,.f2X=Y is defined only when 2x - y ~ O,or y::; 2x.
So the domain off is { (x, y) I y ::; 2x}.
15. lll(9- x 2 - 9y2 ) is defined only when
9- x 2 - 9y 2 > 0, or ix 2 + y 2 < 1. So the domain off
is { (x, y) j ix2 + y2 < 1}, the interior of an ellipse.
y
, ,.,~,..- - - - -
--- ..........,
-----
,,~
17. J1 - x 2 is defined only when 1 - x 2 ~ 0, or
x 2 ::; 1
¢:}
-1 ::; x ::; .1, and \./1 - y 2 is defined
only when 1 - y 2 ~ 0, or y 2 ::; 1
-1 ::; y::; 1.
-¢*
19. Jy - x 2 is defi~ed only when y - x
2
X
~ 0, or y ~ x 2 .
ln addition, f is not defined if 1 - x 2 = 0
<=}
x = ±1. Thus the domain off is
{(x, y) I y ~ x 2 , x "I ±1}.
Thus the domain off is
{(x,y)l-1::;x::;1, - 1 ::; y ::; 1}.
y
1
-1
0
I
X
- I
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SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES
21 . We need 1 - x 2 - y 2 - z 2 ~ 0 or x 2 + y 2 + z 2 ~ 1,
soD= { (x, y , z) I x + y
2
2
+ z ~ 1} (the points inside
2
or on the sphere of radius 1, center the origin).
185
23. z = 1 + y, a plane which intersects the yz-plane in the
line z = 1 + y, x = 0. The portion of this plane for
x ~ 0, z ~ 0 is shown.
X
25. z = 10 - 4x - Sy or 4x + Sy + z = 10, a plane with
0
y
27. z = y 2 + 1, a parabolic cylinder
intercepts 2.5, 2, and 10.
(0, 0, 10)
X
29. z = 9 - x 2 - 9y 2 , an ell iptic paraboloid opening
downward with vertex at (0, 0, 9) .
31. z = )4- 4x2 - y 2 so 4x2 + y 2 + z 2 = 4 or
x
2
y2
z2
+ 4 + 4 = 1 and z ~ 0, the top half of an
ellipsoid .
.r
33. The point ( - 3, 3) lies between the level curves with z-values 50 and 60. Since the point is a little closer to the level curve with
z = 60, we estimate that f( -3, 3) ~56. The point (3, -2) appears to be j ust about halfway between the level curves with
z-values 30 and 40, so we estimate !(3, - 2) ~ 35. The graph rises as we approach the origin, gradually from above, steeply
from below.
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186
0
CHAPTER 14
PARTIAL DERIVATIVES
35. The point (160, 10), corresponding to day 160 and a depth of 10m, lies between the isotherrnals with temperature values
of8 and 12°C. Since the point appears to be located about three-fourths the distance from the 8° C isothermal to the l2°C
isothermal, we estimate the temperature at.that p~int to be approximately 11° C. The point ( 180, 5) lies betwe~n the 16 and
20°C isothermals, very close to the 20°C level curve, so we estimate the temperature there to be about 19.5° C.
37. Near A, the level curves are very close together, indicating that the terrain is quite steep. At B, the level curves are much
farther apart, so we would expect the terrain to be much tess steep than near A, perhaps almost flat.
41.
39.
5
X
y
X
43. The level curves ~re (y - 2x) = k or y = 2x ± ..Jk,
2
k ~ 0, a fami ly of pairs of parallel lines.
45. The level curves are vfx + y = k or y = -vlx + k, a'
family of vertical translations of the graph of the root
function y =
-vx.
y
2
0
-1
432 1
0
1234
-2
47. The level curves are ye~
= k or y = ke- x, a family of
49. The level curves are
J y 2 - x 2 = k or y 2 - x 2 = k2 ,
k ~ 0. When k = 0 the level curve is the pair of lines
exponential curves.
y = ±x. Fork > 0, the level curves are hyperbolas
with axis the y-axis.
0
X
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SECTION 14.1
FUNCTIONS OF SEVERAL VARIABLES
0
187
51. The contour map consists of the level curves k = x 2 + 9y2 , a family of
ellipses with major axis. the x -axis. (Or, if k = 0, the origin.)
The graph of f(x, y) is the surface z = x 2 + 9y2 , an elliptic paraboloid.
X
If we visualize lifting each eUipse k = x 2 + 9y2 of the contour map to the plane
z = k, we have horizontal traces that indicate the shape of the graph of f .
X
53. The isothermals arc given by k = 100/(1 + x 2 + 2y2 ) or
x 2 + 2y2 = (100- k)/k [0 < k ~ 100], a family of ellipses.
55. f(x , y) = xy 2 - x 3
The traces parallel to the yz-plane (such as the left-front trace in the graph above) are parabolas; those parallel to the xz-plane
(such as the right-front trace) are cubic curves . The surface is called a monkey saddle because a monkey sitting on the surface
near the origin has places fo r both legs and tail to res~.
-4
X
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4
188
0
CHAPTER 14
59. z = sin(xy)
PARTIAL DERIVATIVES
(a) C
(b) ll
Reasons: This function is periodic in both x andy, and the function is the same when x is interchanged withy, so its graph is
symmetric about the plane y = x. ln addition, the function is 0 along tl1e x- andy-axes. These conditions are satisfied only by
Can~ II.
61. z = sin(x- y)
(a) F
(b) 1
Reasons: This function is periodic in both X andy but is constant along the lines y = X + k, a condition satisfied only
by F and I .
(a) B
(b) VI
Reasons: This function is 0 a long the lines x
= ±1 andy = ±1. The only contour map in which this could occur is VI. Also
note that the trace in the x z-plane is the parabola z = 1 - x and the trace in the yz-plane is the parabola z = 1 - y , so the
2
2
graph is B.
65. k = x + 3y + 5z is a family of parallel plane$ w ith normal vecto r (1, 3, 5).
67. Equations for the level surfaces are k = y 2 + z 2 . For k > 0, we have a family of circular cyl inders with axis the x-axis and
radius ,fk. When k = 0 the level surface is the x-axis. (fhere are no level surfaces fork < 0.)
69. (a) The graph of g is 'the graph off shifted upward 2 units.
(~)The graph of g is the graph off stretched vertically by a factor of2.
(c) The graph of g i~ the graph off reflected about the xy-plane.
(d) T he graph of g(x, y) = - f(x , y) + 2 is the graph off reflected about the xy-plane and then shifted upward 2 units.
20r-----------------.
71. f(x,y) = 3x - x 4 - 4y 2 - lOxy
10
0
y
Three-dimensional view
y
Front view
It does appear that tlle function bas a maximum value, at the higher of the two "hilltops." From the front view graph, the
maximum value appears to be approximately 15. Bolli h illtop~ could be considered local maximum points, as the values off
there are larger than at the neighboring points. There doe~ not appear to be any local minimum point; although tlle valley sh(\pe
between the two peaks looks like a minimum of some kind, some neig hboring points have lower function values.
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SECTION 14.1
73.
FUNCTIONS OF SEVERAL VARIABLES
D
189
x+y
+y
10-------------
f (x, y) = - - • As both x and y become large, the function values
2
2
5
appear to approach 0, regardless of which direction is considered. As
0
(x, y) approaches the ~rigin, the graph exhibits asymptotic behavior.
-5
From some directions, f(x , y)--+ oo, while in others f(x , y) --+ -oo.
X
(These are the vertical spikes visible in the graph.) If the graph is
examined carefully, however, one can see that f(x, y) approaches 0
along the line y = - x.
y
2
2
75. f( x, y) = ecx +v • First, if c = 0, the graph is the cylindrical surface
2
z = eY (whose level curves are parallel lines). When c > 0, the vertical trace
above they-axis remains fixed whi.le the sides of the surface in the x -direction
z
"curl" upward, giving the graph a shape resembling an elliptic paraboloid. The
level curves of the surface are ellipses centered at the origin.
c=O
For 0 < c < 1, the ellipses have major axis the x -axis and the eccentricity increases as c --+ 0.
y
-1.2
c = 0.5 (level curves in increments of 1)
For c = 1 the level curves are circles centered at the origin.
-1.2
c = 1 (level curves in increments of 1)
[continued)
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190
0
CHAPTER 14
PARTIAL DERIVATIVES
When c > 1, the level curves are ellipses with major axis they-axis, and the eccentricity increases as c increases.
-1.2
c = 2 (level curves in increments of 4)
For values of c < 0, the sides of the surface in the x-direction curl downward and approach the xy-plane (while the vertical
trace x = 0 remains fixed), giving a saddle-shaped appearance to the graph near the point (0, 0, 1). The level curves consist of
a family of hyperbolas. As c decreases, the surface becomes flatter in the x -direction and the surface's approach to the curve in
the trace x = 0 becomes steeper, as the graphs demonstrate.
-1.2
c = - 0.5 (level curves in increments of0.25)
-1.2
c = -2 (level curves in increments of0.25)
77. z = x 2 + y 2 + =Y· When c < -2, the surface intersects the plane z = k I= 0 in a hyperbola. (See the following graph.)
It intersects the plane x =yin the parabola z = (2 + c)x 2 , and the plane x = - y in the parabola z = (2 - c)x2 . These
parabolas open in opposite directions, so the surface is a hyperbolic paraboloid.
When c = - 2 the surface is z = x 2 + y 2 - 2xy = (x- y) . So the surface is constant a long each line x- y = k . That
2
is, the surface is a cylinder with axis x - y = 0, z = 0. The shape of the cylinder is determined by its intersection with the
plane x + y
= 0, where z = 4x2 , and hence the cylinder is parabolic with mini.ma ofO on the line y = x.
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SECTION 14.1
c = - 5, z = 2
FUNCTIONS OF SEVERAL VARIABLES
c = - 10
0
191
c= - 2
When -2 < c:::; 0, z ~ 0 for all x andy. Ifx andy have the same sign, then
x 2 + y 2 + cxy ~ x 2 + 1i - 2xy = (x- y) 2 ~ 0. IJthey have opposite signs, then cxy ~ 0. The intersection with the
surface and the plane z = k > 0 is an ellipse (see graph below). The intersection with the s urface and the planes x = 0 and
y = 0 are parabolas z = y 2 and z = x 2 respectively, so the surface is an elliptic paraboloid.
When c > 0 the graphs have the same shape, but are reflected in the plane x = 0, because
x2 + y 2 + cxy = (- x) 2 + y 2 + (- c)( - x)y. That is, the value of z is the same for c at (x, y) as it is for- cat ( -x, y).
c = -1, z = 2'
c= O
c = 10
So the surface is an elliptic paraboloid for 0 < c < 2, a parabolic cylinder for c = 2, and a-hyperbolic paraboloid for c > 2.
79. (a) P = bLo: K 1 - o:
=?
= bL"' J<- <>
p
K
::}
p = b(JL( ) "'
J(
ln : = lnb+ aln(~)
(b) We list the values for ln(L/ K) and ln(P/ K) for the years 1899-1922. (Historically, these values were rounded to
2 decimal places.)
Year
x = ln(L( K )
y = ln(P/ I<.)
Year
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
0
- 0.02
-0.04
-0.04
-0.07
-0.13
- 0.18
- 0.20
-0.23
- 0.41
- 0.33
-0.35
0
-0.06
- 0.02
0
- 0.05
- 0.12
- 0.04
- 0.07
- 0.15
-0.38
- 0.24
-0.27
19 11
1912
19 13
19 14
19 15
19 16
1917
1918
1919
1920
1921
1922
X =
ln(L/ K)
- 0.38
- 0.38
- 0.41
-0.47
-0.53
-0.49
-0.53
- 0.60
- 0.68
- 0.74
- 1.05
- 0.98
y = ln(P/ K)
-0.34
- 0.24
- 0.25
- 0.37
- 0.34
- 0.28
- 0.39
-0.50
-0.57
- 0.57
- 0.85
-0.59
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192
0
CHAPTER 14
PARTIAL DERIVATIVES
After entering the (x, y) pairs into a calculator or CAS, the resulting least squares regression line through the points is
approximately y = 0.75136x + 0.01053, which we round toy= 0. 75x + 0.01.
(c) Comparing the regression line from part (b) to the equation y = ln b +ax with x = ln(L/ I<) andy= ln(P/ K), we have
a= 0.75 and lnb = 0.01
=> b = e0 ·0 1 ~ 1.01. Thus, the Cobb-Douglas production function is
p = bL"' ](1-a = 1.01£0.75 ](0.25.
14.2 Limits and Continuity
1. In general, we can't say anything about /(3, 1) I
lim
{x,y)-•(3, 1)
f(x, y) = 6 means that the values of f(x, y) approach 6 as
·
(x, y) approaches, but is not equal to, (3, 1). Iff is continuous, we know that
lim
(x,y)-(3,1)
lim
(x,y)--<(a,b)
f(x, y) = f(a, b), so
f(x, y) = f(3, 1) = 6.
.
3. We make a table of values of
x2y3 + x3y2 _ 5
f(x,y) =
·
fora set
2 - xy
of (x, y) points near the origin.
l>z
- 0.2
-0.1
- 0.05
0
0.05
0.1
0.2
-0.2
-2.551
-2.525
-2.5 13
-2.500
-2.488
-2.475
-2.451
-0.1
-2.525
-2.513
- 2.506
-2.500
-2.494
-2.488
- 2.475
-0.05
-2.513
-2.506
-2.503
-2.500
-2.497
-2.494
-2.488
0
-2.500
-2.500
- 2.500
-2.500
-2.500
- 2.500
0.05
-2.488
- 2.494
- 2.497
-2.500
-2.503
- 2.506
- 2.5 13
0.1
-2.475
- 2.488
- 2.494
-2.500
-2.506
-2.513
- 2.525
0.2
- 2.451
-2A75
-2.488
- 2.500
- 2.513
- 2.525
- 2.551
As the table shows, the values off (x, y) seem to approach - 2.5 as (x, y) approaches the origin from a variety of different
directions. This suggests that
lim
(:t:,!J) - {0,0)
f(x, y) = -2. 5. Since f is a rational function, it is continuous on its domain. f is
defi ne d at (0, 0 ) , so we can use d1rect substltutwn to esta bl'ISI1 th at
o
o
o
l'
1m
(:r.,.y)-->{0,0)
!( x, y ) = 0203 + 0302 - 5 = - 5 , ven'fy'mg
2
2- 0 0
0
our guess.
5. f(x, y) = 5x3 - x 2 y 2 is a poly nomial, and hence continuous, so
.
7. f( x, y ) =
4 - xy
X 2 +3y 2
' lim
(x,y) -{1 ,2)
2
f(x, y) = f(l, 2) = 5{1) 3 - {1) (2)
2
= 1.
.tS a ratwna
.
..,
.
dh
.
. d
.
1 ;unctiOn an encc contmuous on tts omam.
.
(2, 1) is in the domain off, so f is continuous there and
9. f(x, y) = (x4 - 4y2 )/(x2 + 2y 2).
lim
{x, y) --< (2,1)
f(x, y) = !(2, 1) = (~)~ ( ~~~~ 2
2
+
4
First approach (0, 0) along the x-axis. Then.f(x, 0) = x /x
2
2
7
= x 2 for x =f. 0, so
f(x, y) --+ 0. Now approach (0, 0) along they-axis. For y =f. 0, f(O, y) = -4y2 /2y2 = -2, so f(x , y)--+ -2. Since f has
two different limits along two different lines, the limit does not exist.
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SECTION 14.2
11. j(x , y) = (y 2 sin 2 x)j(x4 + y 4 ).
LIMITS AND CONTINUITY
0
193
On the x -axis, f(x, 0) = 0 for x '# 0, so f(x, y) --+ 0 as (x, y) --+ (0, 0) along the
2
.
2
x sin x
x -axis. Approaching (0, 0) along the hne y = x, f(x, x) = x 4 + x 4 =
2
2
sin x
1 ( sinx)
x 2 = 2 --;-- for x :/: 0 and
2
lim sin x = 1, so f(x, y)--+ ~·Since f has two different limits along two different lines, the limit does not exist.
:c- o
x
·
·
13. f(x, y) = ~·
We can see that the limit along any line through (0, 0) is 0, as well as along other paths through
xz +yz
(0, O) such as x = y 2 andy= x 2 . So we suspect that the limit exists and equals 0; we use the Squeeze Theorem to prove our
assertion. 0
~ I ~~~
lxl since lvl ~ )x + y and lxl --+ 0 as (x, y)--+ (0, 0). So
lim
f(x, y) = o:
x2 + y2
.
(x,y)-(o,o)
15. Let f(x, y) =
2
x2yeY
X 4 + 4y 2
. Then f(x, 0) =
2
,
0 for x '# 0, so f(x, y) --+ 0 as (x , y)--+ (0, 0) along the x -axis. Approaching
2
2
2
(0, 0) along they-axis or the line y = x also gives a limit ofO. But f (x, x ) = x::
4
::r2
4
tx)2 = x5:
2
::c2
4
T
a:2
=
for x '# 0, so
f(x , y) -+ e0 /5 = t as (x,y)--+ (0, 0) along theparabola y = x 2 . .Thus the limit doesn' t exist.
2
19. eY is a composition of continuous functions and hence continuous . xz is a continl.!ous function and tan t is continuous for
t '# ~ + mr (nan integer), so the composition tan(a<z ) is continuous for xz '# ~ + mr. Thus the product
f(x , y, z ) = ev~ tan{xz) is a continuous function for x z '# ~ + m r. Ifx = 1r and z = i then xz :j: ~ + nw, so
lim
(:r:,y ,z)- (,..,0 ,1/ 3)
21. f(x, y, z) =
f(x, y, z ) = f (w, 0, 1/3) = e
xy + yz
x2
+y
2
+ xz2 .
2 +z4
02
tan(w · 1/ 3) = 1 · tan{w / 3) = .;3.
2
·
Then f(x, 0, 0) = 0/x = 0 for x '# 0, so as (x, y, z) --+ {0, 0, 0) along the x-ax1s,
f(x, y, z) --+ 0. But f(x , x, 0) = x 2 /(2x 2 ) = t for x '# 0, so as {x, y, z) --+ {0, 0, 0) along the line y = x, z = 0,
f( x,y,z) --+ ~· Thus the limitdoesn'texist.
From the ridges on the graph, we see that as (x, y) --+ (0, 0) along the
23.
lines under the two ridges, f(x, y) approaches different values. So the
limit does not exist.
0.5
y
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194
CHAPTER 14
0
PARTIAL DERIVATIVES
25. h(x , y ) = g(f(x, y)) = (2x + 3y - 6) 2 + y'2x + 3y - 6. Since f is a. polynomial, it is continuo~s on JR2 and g·is
continuous on its domain { t I t ;::: 0}. Thus h is continuous on its domain.
D = {(x, y) I 2x + 3y- 6;::: 0} = { (x, y) I y;::: - ~x + 2}, which consists of all points on or above the line y = - ~x + 2.
From the graph, it appears that f is discontinuous along the line y = x.
27.
If we consider f(x, y) = e 1 /(a:-y) as a composition offun.ctions,
3
2
z
g(x, y) = 1/(x- y) is a rational function and therefore continuous except
1
where x- y = 0
0
-1
y = x. Since the function h(t) = et is continuous
=}
everywhere, the composition h(g(x,y)) = e 11(x-v) = f(x,y) is
2
continuous except along the line y = x, as we suspected.
29. The functions xy and 1 + ex -v are continuous everywhere, and 1 + ex- v is never zero, so F(x, y) =
xy
is continuous
.
1 + ex-y
on its domain JR2 .
31. F(x, y) =
l +x2 +y2
1 - x 2 -y2
is a rational function and thus is continuous on its domain
33. G(x, y) = ln(x 2 + y 2 - 4) = g(f(x, y)) where f(x, y) = x 2 + y 2 - 4, continuous on ]~_2, and g(t) = In t, continuous on its
domain {t ['t > 0}. Thus G is continuous on its domain { (x,_y) I x 2 + y 2 - 4 > 0} = { (x , y ) I x 2 + y 2 > 4 }, the_exterior
ofthe circle x 2 + y 2 = 4.
35. f(x, y, z ) = h(g(x, y, z )) where g(x, y , z) = x 2 + y 2 + z 2 , a polynomial that is continuous
everywhere, and h(t) = arcsint, continuous on [- 1, 1). Thus f is continuous on its domain
{(x,y, z) l-1 S x 2 + y 2 + z 2 S 1} = {(x,y,z)! x 2 + y 2 + z 2 S 1}, so f is continuous on the unit ball.
if (x , y) =I= (0, 0)
The first piece off is a rational function defined everywhere except at the
if (x,y) = (0,0)
origin, so f is continu~us on JR2 except possibly at the origin. Since x 2
s 2x + y we have jx y (2x + y 2 ) I.s jy j. We
2
2
2
,
3
/
.
2
3
2 3
know that jy3 \ --+ 0 as (x, y) --+ (0, 0). So, by the Squeeze Theorem,
lim
(a;,y) - (0,0)
f(x, y) =
lim
~ y
(x ,y) - (0,0) 2X
+ Y 2 = 0.
But f(O, 0) = 1, so f is discontinuous at (0, 0). Therefore, f is continuous on the set { (x, y) I (x, y) =/= (0, 0) }.
39 .
3
3
3
3
x + y
(r cos 8) + (1' sin8)
lim (
3
. 3 B)
.
- 2 - -2 = 1tm
=
r· cos 8 + rsm
= 0
2
( x ,y)-(0,0) X -f- y
r--> O+
1'
r~o+
. 41.
e -x2 - y 2 - 1
x2 +y2
(a;,y)--> (0,0)
lim
2
lim e- r ( - 2r·)
r - o+
2r
lim
[using !'Hospital's Rule]
2
= lim - e- r = - e 0 = - 1
r·-o+
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SECTION 14.3
43. f( x, y) =
{
sin(xy)
xy
if (x, y) :f. (0, 0)
1
if (x, y) = (0, 0)
PARTIAL DERIVATIVES
0
195
From the graph, it appears that f is continuous everywhere. We know
xy is continuous on R2 .and sin t is continuous everywhere, so
-1
d sin(:cy) .
.
R2
.
R
sin (xy ) is contmuous on 2 an - - - IS contmuous on
xy
except possibly where x y = 0. To show that f is continuous at those points, consider any point (a, b) in R 2 where ab = 0.
Because x y is continuous, x y --+ ab == 6 as {x, y ) --+ (a, b). If we lett= x y , then t--+ 0 as (x , y) --+ (a, b) and
lim
(x,y)-(a, b)
sin(xy). = Jim sin(t) = l by Equation 2.4.2 [ET 3.3.2). Thus
lim
f (x, y) = f(a , b) and f is continuous
xy
t-o t
(:z:,y)-(a, b)
on R 2 •
2
45. since Jx - aJ = lxl + J
al . 2
2
t > 0 be given and set
2JxJ ial cos O ~ Jxl2 + Jal2 - 2Jx.JJal = (lxJ -Ial) 2 , we have llxl - lall $ lx- aJ. Let
o= t. Then ifO < lx - al < o, llxJ- IaJI $ Jx - aJ < o= t . Hence limx-a Jxl = Jal and
f (x) = Jxl is continuous on R ".
14.3 Partial Derivatives
1. (a) 8TI 8x represents the rate of change ofT when we fi x y and t and consider T as a function of the single variable x, which
describes how q uickly the temperature changes when longitude changes but latitude and time are constant. 8T18y
represents the rate of change of T when we fix x and t and consider T as a function of.y, which describes how quickly the
temperature changes when latitude changes but longitude and time are constant. 8TI 8t represents the rate of change ofT
when we fix x andy and consider T as a function oft , which describes how quickly the temperature changes over time for
a constant longitude and latitude.
(b) f .,(158, 21, 9t represents the rate of change of temperature at longitude 158°W, latitude 21 °N at 9:00AM when o nly
longitude varies. Since the air is warmer to the west than to the east, increasing longitude results in an increased air
temperature, so we would expect .f:r: (158, 21, 9) to be positive. / 11 (158, 21 , 9) represents the rate of change oftemperature
at the same time and location when only latitude varies. Since the air is warmer to the south and cooler to the north,
increasing latitude results in a decreased air temperature, so we would expect / 11 (158, 21, 9) to be negative. ft(158 , 21, 9)
represents the rate of change of temperature at the same time and location when onJy time varies. Since typically air
temperature increases from the morning to the afternoon as the sun warms it, we would expect f t(158, 21, 9) to be
positive.
. 4 , f T (- 15 , 30)
3. ()
a By Definitlon
lim f( - 15 + h,30)
.
by cons1'dermg
. h=5
= h-0
h - f( - 15,30) , w h'1c h we can approx 1mate
and h = -5 and using the va lues given in the table:
© 201 2 Ccngoge J.cnming. All Righ t" Rt:scr,.cd. May not~ .sc<tMed. copied, o r duplicutcd. o r posted to a publicly accessible website, in whole or in prut,
196
D
CHAPTER 14
PARTIAL DERIVATIVES
! 7'(-15 ' 30) ~ f (-10, 30) -5 f (- 15, 30) = - 20 - 5(- 26) = ~5 = 1.2'
"
f{ - 20, 30)- f (-15,30)
- 33 - {- 26)
-7
. = _ = 1.4. Averaging these values, we estimate
h( - 10, 30) ~ ,
_
=
_
5
5
5
fr( -15, 30) to be approximately 1.3. Thus, when tl1e actual temperature is - 15°C and t he wind speed is 30 km/ h, the
apparent temperature rises by about 1.3°C for every degree iliat the actual temperature rises.
. . I f (
.
.
)
. f ( - 15, 30 +h) - f( -15, 30)
Stml 1ar y, v - 15 , 30 = 11m
h
wh1ch we can approxunate by considering h = 10
l• - 0
and h = - lO: f .., (- 15, 30) ~ f (- 15, 40) - f( - 15, 30) = - 27 - (-26) = - 1 = - 0. ,
10
f v{-15, 30) ~
10
10
1
f( -15 20)-f(- 15 30)
- 24 - (-26)
' _
'
=
_10
= _210 =. - 0.2. Averaging these values, we estimate
10
f..,( - 15, 30) to be appr~ximately - 0.15. Thus, when the actual temperature is - 15°C and ilie wind speed is 30 km/ h, ilie
apparent temperature decreases by about 0.15°C for every km/ h that the wind speed increases.
(b) For a fixed wind speed v, the values of the wind-chill index W increase as temperature T increases (look at a column of
the table), so
a;
is positive. For a fixed temperature 1', the values of W decrease (or remain constant) as v increases
(look at a row of the table), so ~: is negative (or perhaps 0).
(c) For fixed values of T , the function values f (T , v) appear to become constant (or nearly constant) as v increases, so the
corresponding rate of change is 0 or near 0 as v increases. This suggests that lim ( awI ov) = 0.
V-> 00
5. (a) If we start at (1, 2) and mo~e in the positive x-direction, the graph off increases. Thus f ,(1, 2) is positive.
(b) If we start at (1, 2) and move in the positive y-direction, tbe graph off decreases . Thus fv (1, 2) is negative.
7. (a) f zx
= g, (!,), so f xx is the rate of change of f , in tl1e x -direction. fx is negative at ( - 1, 2) and if we move in the
positive x -direction, the surface becomes less steep. Thus the values of f., are increasing and !:ex ( - 1, 2) is positive.
(b) f 1111 is the rate of change of f 11 in tlle y-direction. / 11 is negative at ( - 1, 2) and if we move in the positive y-d.i rection, the
surface becomes steeper. Thus the values of f 11 are decreasing, and / 1111 ( - 1, 2) is negative.
9. F irst of all, if we start at the point (3, - 3) and move in the positive y-direction, we see that both b and c decrease, while a
increases. Both b and c have a low point at about (3, - 1.5), while a is 0 at this point. So a is defi nite ly the graph of j 11 , and
one of b and c is the graph off . To see which is which, we start at the point ( - 3, - 1.5) and move in the positive x-direction.
b traces out a line with negative s lope, while c traces out a parabola opening downward. This tells us that b is tlle x-derivative
of c. Soc is the graph of f, b is the, graph of f ,, and a is tbe graph of f 11 •
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SECTION 14.3- PARTIAL DERIVATIVES
11. f(x, y) = 16 - 4x2 - y 2
0
197
=> f ,., (x , y) = - 8x and fv(x , y) = -2y => f ,., (1, 2) = - 8 and fv (1, 2) = - 4. The graph
off is the paraboloid z = 16 - 4x2 - y 2 and the vertical plane y = 2 intersects it in the parabola z = 12 - 4x 2 , y = 2
(the curve C 1 in the first figure). The slope of the tangent line
to this parabola at (1, 2, 8) is f x(1, 2) = - 8. Similarly the
plane x = 1 intersects the paraboloid in the parabola
z ~ 12 - y 2 , X = 1 (the curve C2 in the second figure) and
the slope of the tangent line at (1 , 2, 8) is f v(1, 2) = - 4.
X
JC
20
0
- 20
2
2
Note that traces off in planes parallel to·the xz-plane. are parabolas which open downwru:d for y < 0 and upward for y > O,
and the traces of ! r:: in these p lanes are straight lines, which have negative slopes for y < 0 and positive slopes for y > 0. The
traces off in planes parallel to the yz-plane are cubi_c curves, and the traces of fu in these planes are parabolas.
15. f( x, y) = y 5 ...:.. 3xy
19.
=>
f x(x, y) = 0 - 3y = - 3y, / y(x, y) = 5y4 - 3x
z = (2x + 3y) 10 => aazX = 10{2a: + 3y) 0 2 = 20{2x + 3y)9 , azy = 10(2x + 3y) 0 3 = 30(2x
+ 3y) 0
.
0
21. f(x , y) = x f y = x y- 1
=> fx( x, y ) = y - 1 = 1/ y, fv (x , y ) = -xy- 2 = - x / y 2
by
23· f (x,y) = ax+
ex +dy
f - (x • ) = (ex + dy )(a) - (ax+ by)(c) = (ad ·- bc)y '
- ,y
(cx +dy) 2
(ex + dy )2
x
f Y ( ' 1Y
)
=>
0
= (ex+ dy)(b} - (ax+ by)(d} = (be - ad)x
(ex+ dy) 2
(ex + dy} 2
© 201 2 Ccngnge Lcwning.. All Rights R(.SCn•..:U. M:ly not be scanned. copied. or duplic.olc:d. or posted to a publicly acc::t ssiblc website, in whole or in part.
198
D
PARTIAL DERIVATIVES
CHAPTER 14
25. g(u,v) = (u2 v- v3 ) 5
gu(u,v) = 5(u2 v - v 3 ) 4 • 2uv = 10u·u(u2 v - v 3 ) \
=?
gu(u, v) = 5(u2 v- v 3 ) 4 (u2 - 3v2 ) = 5(u2 - 3v2 )(u2 v- v 3 ) 4
.
29. F(x,y) = l " cos(et)dt
0
F11 (x,y) = o·y
'
1
Fx(x,y)= :xl" cos (e ) dt=cos(e"' ) by theFundamentaiTheoremofCalculus, Partl;
=?
1"' cos(e)
. dt = oy0 [- I.( cos(e) dt = - oy0 frv
.. cos(et) dt = - cos(e
11
.
t
t
11
]
).
11
33. w
= ln(x + 2y + 3z) =>
35. u = xysin- ~(yz)
-ou = xy .
az
ow
1
ow
2
ow
3
ox = x + 2y + 3z' oy = x + 2y + 3z' oz = x + 2y + 3z
=>
1
J 1- (yz)2
xy2
(y) = --;:::.:=:::::::::==:::
J 1- y2z2
37. h(x, y, z, t) = x 2 y cos(z/t)
=> h.,(x; y, z, t) = 2xy cos(z/t), h11 (x, y, z, t) = x 2 cos(z/t),
hz(x, y, z, t) = -x2 y sin(z/t)(1/ t) = (- x 2 y/t) sin(z/t), ht(x, y, z, t) = -x2 y sin(z/t)( -zC 2 ) = (x2 yz/t2 ) sin.(z/t)
39. u =
Vx 21 + x22 + · · · + Xn·2
For each t· = 1, ..., n, u,, = 21 ( x 2 + X22 + · · · + Xn2) -1/2 ( 2x; ) =
1
Xi
y X~ + X~ + · · · + x?,
= 1n(x+ Jx2 + y2) '*
41 . f(x, y)
f., ( x,y) -_
1
[ 1+21 ( x 2 +y2) - 1/ 2(2x)] -_
~
x + Y x2 + y2
.
so f.,(,
34=
)
1
3
(
)
1
( 1+
~
x + Y x2 + y2
X
)
~
,
Y x2
y2
+
.
l .
s ( + !s)! = s·
3 + ~ 1 + ...f32+42 =11
y
43. f(x,y , z) = x+y + z
=?
fv(x,y,z) =
2+(-1)
soj11 (2,1, - 1) = (2 + 1 +(- 1 )) 2
45. f(x, y) = x1l - x 3 y
l(x + y + z)- y(l )
x +z
(x + y + z )2
= (x+y+ z )2'
1
4·
=>
_ . f(x + h,y) - f(x,y) _ 1. (x +h)y2 - (x + h) 3 y- (xy 2 - x 3 y)
liD
f "' (x, y ) - ,.1liD
.....o
h
h-0
h .
= lim
h(y2 - 3x2 y - 3xyh- yh2 )
h
,...... o
f (
11
) _ li
x , y - h5
f(x, y + h) - f(x;y) _ U:U x(y + h) 2 - x 3 (y + h)- (xy 2 - x 3 y) _ 1'
h
-
h ->0
· = lim {2xy + xh- x ) = 2xy - x
3
/a.-+0
= lim (y 2 - 3x 2 y- 3xyh- yh 2 ) = y 2 - 3x2 y
h ..... o
h
- "~
h(2xy + xh- x 3 )
h
3
.
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.
SECTION 14.3
47. x 2 + 2y2 + 3z2 = 1
=?
-
8
ax
8
(x 2 + 2y2 + 3z 2) = - (1)
ox
8z
2x + 0 + 6z ox = 0
8z
-8z = -- 2x = - -x , and -8 (x 2 + 2y 2 + 3z 2) = -8 (1)
6z
8y
0y
3z
8x
ez '-8 z = y ( X -8z + Z · 1)
= xyz =>
49. ez
0 + 4y + 6z 8y = 0
ax
ax
PARTIAL DERIVATIVES
D
8z
6z ax = - 2x
=?
=?
=?
8z
6z - = -4y
0y
_ 8z
8z
e- - - xy - = yz
OX
ox
=?
=?
=>
•
8z
8z
yz
(e- - xy) - = yz, so - = - _
-.
e- - x:y
8x
8x
• 8z
8z
e· oy - xy oy = xz => ( e-- - xy ) -8 z = xz , so
8y
=>
8z
xz
ay = e=- xy
51. (a) z = f( x ) + g(y)
(b) z = f(x + y).
=>
~: = /'(x), ~ = ,q'(y)
.
8z
df QU
df
I
1
Let u = x + y. Then ox = du ox = du (1) = f (u) = f (x + y) ,
. az = df au= df (1) = J'(u) = f'(x + y).
8y
du 8y
du
53. f( x , y) = x 3 y 5 + 2x 4 y
f,(x, y) = 3x2 y 5 + Bx3 y , fu(x, y) = 5x 3 y 4 + 2x 4 . Then l rr::c (x, y) = 6xy 5 + 24x2 y,
=?
f xu(x, y) = 15x 2 y 4 + 8x 3 , fux(x , y) = 15x2 y 4 + Bx 3 , and f uu (x, y) = 20x3 y 3 •
55. w = Ju2 + v2
uv
(u2 + v 2)3/ 2 '
2
1 · ~ - v · ~(u + v 2 ) - 1 (2v)
2
W uv
=
(Ju2 + v2)2
x+y
57. z = arctan - -
1 - xy
1
'
W uu
= v (-21) ( u 2 +v2) - 3/ 2 ( 2u) =. (u 2 +uvv2)3I 2' .
Juz + v 1 - v 2 j .,Ju2 + v 2
u 2 + v2 - v2
u2
=
u2 + v2
= (u2 + v2)3/2 = (u2 + v2)3/2.
=?
_
1
(1)(1-xy)-(x+y)(- y)_
l+y2
1+y2
Z x = 1 + (..:.±1L..) 2 .
(1 - x y)2
- (1 - x y)2 + (x +y)2 - 1 + x2 + y2 + x2y2
1 - xy
1 + y2
1
= (1 + x2)(1 + y2) = 1 + x2 '
© 201 2 Cengogc Learning. All Righls R=rved. Moy nol be scanned, copied. or duplicnlcd. or posted loa publicly accessible wcbsilc. in whole or in part.
199
200
CHAPTER 14
0
PARTIAL DERIVATIVES
(1)(1-xy)-(x+y)(-x) _
1+x2
zu = 1 + ( x+u )2 .
(1- xy)2
(1- xy)2 +(x + y)2
1
1
1 +x 2
(1 + x 2)(1 + y2) = 1 + y2 ·
1-xy
(
+2xx 2 )2, Zxy = 0, Zyx = 0, Zyy = -(1 + y 2) -2 . 2y =
1
2y
Thus U:cy = Uy:c.
61. u = cos(x2 y)
=>
U:c
= -sin(x2 y) · 2xy = -2xysin(x2y),
= -2xy · cos(x2 y) · x 2 +sin(x 2y) · (-2x) = -2x 3 ycos(x2y)- 2xsin(x2 y) and
Uxy
uu = -sin(x2y) ·x2 = - x 2 sin(x 2y),
1t11 x
= - x 2 ·cos(x2y) ·2xy+sin(x2 y) · (- 2x) = -2x 3 ycos(x2 y) -2xsin(x 2 y).
Thus Uxy = Uux.
63. f(x, y) = x 4 y 2 -
fx = 4x3 y 2 - 3x2 y, fxx = 12x2 y 2 - 6xy, fxxx = 24xy2 - 6y and '
x3 y =>
2
2
f xv = 8x3 y- 3x , fxvx = 24x y- 6x.
67. u=er8 sin(J
2
;
;
0
=> :~ . =er 8 cosfJ+sinfJ·er 8 (r)=e'. 0 (cosO+rsinO),
= er8 (sinfJ) +(cosO+ rsinO) erO (0) = ero (sinO+ OcosB + rOsinO),
{)~:~O = ero (0 sin 0) +(sinO+ 0 cosO+ rOsinO) · e''8 (0) = Oero (2sin0 + Ocos 0 + rO sinO).
X
69.w=y+2z=x(y+2z)-1
=>
OW
82 w
.
2
2
ax=(y+2z) - I, 8yax = -(y+2z)- (1)=-(y+2z)- ,
83w
. .
4
ow
2
2
{)z{)y{)x = -(-2)(y + 2z)- 3 (2) = 4(y + 2z)- 3 = (y + z)3 and {)y = x(-1)(y + 2z)- (1) = -x(y + 2z)- ,
2
71. Assuming that the third partial derivatives off are continuous (easily verified), we can write fxzy = fyxz· Then
vz) =?
f(x, y, z) = :J;y 2 z3 + arCSin ( X
73. By Definition 4, fx(3, 2) = lim !(3 + h,
l-
2
h -0
= 2xyz3 + 0, fvx = 2yz3 , and fyxz = 6yz 2 = fxzy·
fu
2
!(3 , ) which we can approximate by considering h = 0.5 and h = -0.5:
f (3 2 ) ~ !(3.5, 2) - f(3, 2) _ 22.4 - 17.5 _ 9 8 f (3 2 ) ~ !(2.5, 2) - f(3, 2) _ 10.2- 17.5 _ 14 6 A
"' '
~
0.5
-
0.5
-
· ' "' '
~
-0.5
-
-0.5
-
.
· · veragmg
22
3 22
3
these values, we estimate f,.(3, 2) to be approximately 12.2. Similarly, f x (3, 2.2) = lim !( + h , · }- !( , · ) which
_
h"""""+O
L
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SECTION 14.3
·
· ·
. . '
dI
we can approximate by cons1denng h = 0.5 an
f x(3, 2.~) ~
1 =
PARTIAL DERIVATIVES
0
201
f (
26.1- 15.9
) !(3.5, 2.2) - !(3, 2.2)
- 0.5: "' 3, 2. 2 ::::::
=
= 20.4,
0 .5
0.5
f(2.5, 2.2) - f(3, 2.2)
9.3 - 15.9
.
I
I
h
f (
)'
_ _
=
_ _
= 13. 2. Averagmg
t 1ese va ues, we ave "' 3, 2.2 ~ 16.8.
05
05
To estimat~ f, 11 (3, 2), we first need an estimate for .fx(3, 1.8):
~ /(3.5, 1.8) - /(3, 1.8) - 20.0 - 18.1 - 3 8 f (3 1 8) ~ /(2:5, 1.8)- !(3, 1.8) - 12.5 - 18.1 -
.fx ( 3 • 1.8 ) ~
0.5
-
0.5
-
.
. ' x ' .
.
~
-0.5
-
- 0.5
- 11. 2.
a
Averaging these values, we get f,(3, 1.8) ~ 7.5. Now fx 11 (x, y) = -;::;- [f.,(x, y)) and fx(x, y) is itself a function of two
uy
..
()a[
variables, so Defimtton 4 says that fx 11 x, y
.
( )) li f,(x,y+h)-fx (x,y )
= -;::;fx x, Y = m
I
·
=>
uy
h-o
a
.
I· · 1
·
·
k ·11
f , 11 (3, 2) = lim f x(3, 2 +h)- fx(3, 2) . We can estimate
t liS va ue usmg our prev1ous wor Wit 1 a = 0.2 and h = -0.2:
h
h- o
) ~ j,(3, 2.2)- f x(3, 2) = 16.8- 12.2 =
f ( 2 ) ~ f x (3, 1.8)- J,(3, 2) = 7.5 - 12.2 = 23
23
0.2
0.2
' xy 3 '
-0.2
-0 .2
·5 ·
f xy (3 • 2 ~
Av~raging these values, we estimate f, 11 (3, 2) to be appr~ximately 23.25.
U:r:x = -(x2 + y2 + z2 ) - 3/2 -
x( -~) (x2 + y2 + z2) - 5f2(2x) =
.l
2z 2 - x2 - y2
2y2 - x2 - z·2
By symmetry, u uu
= .(x2 + y2 + z2 )5/2 and U ::z = (x2 + y 2 + z2 )5/2 .
Thus Uxo: ·+ u,,y + Uzz =
.
2x2 - y2- z2 .
(x2 + y2 + z2) 5/ 2
.
2x2 - y2 - z2 + 2y2 - x2 - z2 + 2z2 - x2 - y2
(X 2 +y-? + Z 2)"/2
= 0.
"
79. Let v = x +at, w = x - at.
Then Ut =
8[f(v) + g(w)]
f)t
df(v) 8v
dg(w) 8w
·
= ~ ot +~7ft = af'(v) - ag'(w) and
uu = o[af'(v)a~ ag'(w)J = a[af"(v) + ag"(w) ) = a 2 [f"(v) + y"(w) ). Similarly, by using the Chain Rule we have
u, = f'(v) + g'(w) and Ux x = f"(v) + g"(w) . Thus Utt = a2 Ux:r.·
81. z = in(e"' + e!l)
8z
8x
e"'
e"' + e!l
8z
eY
8z
8z
e"
e·u
e"' + eY
=-- , so - +- = - - - + - - - = - - - = 1.
8y
e"' + eY
Dx
8y
e"' + e11
e"' + eY
e"' + eY
=> -'- = - - - and -
D2z
e"'(e"' + eY) .- e"'(e'")
e"'+ 11
82z
0 - e11 (e"')
8x 2 =
(e +eY) 2
= (e"' + e11 )2' 8x8y=(e"' + ev) 2 =
82 z
8y2
e11 (e"' + e 11 ) - e 11 (e11 )
(e"' + eY )2
(
e" +eY) 2
ex+y
(e"' +eY)2' and
• T hus
e"'+Y
(e"' + eu) 2
)2
=
(e"'+u?
(e"'+Y?
(e"' + eY)1 - (ex + eu )1 = 0
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202
0
CHAPTER 14
PARTIAL DERIVATIVES
.83. By the C hain Rule, taking the partial derivative of both sides with respect to R 1 gives
8R- 1 8R
8[(1/Rl) + (1/R2) + (1/R3)]
- 2 8R
_2
8R
R2
8R 8R1 =
8R1
or -R 8R = - R 1 . Thus 8R = _Rf'
1
1
85. If we fix K = Ko , P.(L,-Ko) is a function of a single variable L, and ~~ =a
dP = aL
dL =>p
Jp = Ja L
dP
dL
I
is a separable differentia; equation. Then
=>- In IPI = a In 1-41 + C (Ko), where G_(Ko) can depend on K 0 • Then
.
-
IPI = e<> lniLI-f;C(Ko), andsinceP > 0 andL >0, we have P = ec:dnLeC(Ko) = eC(T<o)elnL"' = C 1 (Ko)L"' where
C1 (Ko) = ~C(I<o) .
2
2
n a)
87. ( P+ V2 (~ .:...nb) = nRT
_
n2a
1 (
n a)
8T
1
V- nb
=> T = nR
P+ "'if'2 (V -nb),so p = nR (1)(V - nb) = ~8
nRT
n RT
n2 a
=>- P = V _ nb - V2 = nRT(V - nb) - 1 - n 2 av- 2 , so
We can also wnte P + V 2 = V _ nb
2
8P
(
)_ 2 ( )
2n a
nRT
_3
2
1 +2naV = _V 3 - (V-n'b)2'
8 v= - nRTV- _nb
89. By Exercise 88, PV = mRT
.
PV
=>- P =
8P 8V
mRT
8P
mR
__.._ . V _ mRT d 8V _ mR
----v-•
so fJT = v· Also, PV = mRT --..p
an 8T
p .
PV mR mR
Smce T .= mR' we have T fJT fJT = mR · V
8K
·
8K
82 K
·p
8K &2 K
91. --;:;--- = 21 v 2 , = mv, ~
= m. Thus .-8 m · -8v 2
um
uv
8v
93. f:x:(x, y) = x + 4y
= mR.
= 21 v 2 m = K.
=>- f xv(x, y) = 4 and fv(x, y) = 3x- y =>- fvx(x, y) = 3. Since f xy and f vx are continuous
everywhere but f:x:v(x, y) f- f 11.,(x, y), Clairaut's Theorem implies that such a function f(x, y) does not exist.
95. By the geometry of partial derivatives, the slope of the tangent line is f x(l , 2). By implicit differentiation of
4x2 + 2y2 + z 2 = 16, we get8x + 2z (8zj8x) = 0
=> 8zf8x = - 4x/z, so when x = 1 and z = 2 we have ·
8zj8x = -2. So the slope is fx(1, 2) = -2. Thus the tangent line is given by z - 2 = -2(x - 1), y = 2. Taking the
parameter to bet = x - 1, we can write parametric equations for this line: x = 1 + t, y = 2, z = 2 - 2t.
97. By C lairaut's Theorem, f xvv = (f:x:y) 11 = (/11x}y = fv xv = Uv).,'ll = Uv)yx = fvv x·
99. Let g(x) = f(x, 0) = x(x 2 ) - 3 12 e0 = ·x ix J'- 3 • But we are using the point (1, 0), so near (1, 0), g(x) = x - 2 • Then
g'(x) = - 2x- 3 and g'(1) = -2, so using (l) we have fx(l, 0) = g'(l) = - 2.
101. (a)
(b) For (x, y) f- (0, 0),
f x(x,y) =
(3x2y- y3)(x2 + y2) - (x3y- xy3)(2x)
(x2+y2)2
x4y + 4x2y3 _ ys
(x2 + y2)2
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SECTION 14.4
TANGENT PLANES AND LINEAR APPROXIMATIONS
D
203
(c) f (0 0) = lim f(h, 0) - f(O, 0) = lim {O/ h2) - 0 = 0 and jy(O, 0) = lim j(O, h) - f(O, 0) = '0.
"' '
J•-O
h
h- o
h
h-o
h
5
Y
4
(0 0) = of, = lim f.,(O, h)- f , (O , O) = lim ( - h - O) / h = -1 while by (2),
' "'Y '
oy
h- o
h
h- o
h
(d) B (3) f
f ,(0, 0) = ofy = lim jy(h, 0) - jy(O, 0) = lim h5/ h4 = 1.
!I
ox
h
h .....o
,,..... o
h
(e) For (x,·y) i= (0, 0), we use a CAS to compute
x6 + gx4y2 _ gx2y4 _ y6 ·
f xy(x, y)=
(x2+y2)3
Now as (x , y) -> (0, 0) along the x-axis, fxv(x, y)-> 1 while~
(x, y) -> (0, 0) along they-axis, fxy(X , y) -> - 1. Thus fx 11 isn't
continuous at (0, 0) and Clairaut's Theorem doesn't apply, so there is
no contradiction. The graphs of f, 11 and fy x are identical except at the
origin, where we observe the discontinuity.
14.4 Tangent Planes and Linear Approximations
1. z = f(x, y) = 3y 2 - 2x2 + x
=?
f .,(x, y) = -4x + 1, / 11 (x, y) = 6y, so f,(2, -1) = ~7, / y(2, -1) = -6.
By Equation 2, an equation of the tangent plane is z - (-3) = f, (2 , - 1) (x- 2) + f v(2, -1)[y- ( -:-1)]
=:-
z+ 3= - 7(x - 2)-6(y + 1) or z= - 7x - 6y + 5.
3. z = f(x , y) = .,fXY =>
f,(x, y) = ~(xy) - 112 · y = ~gx, fu(x , y) = ~(xy)- 1 1 2 · x = ~.../XlY, so f.,(1 , 1) = ~
and fv(1 , 1) = ~-Thus an equation of the tangent plane is z - 1 = f ,(1, 1)(x- 1) + jy(1, 1)(y- 1)
=?
z- 1 = ~(x -1) + Hv - 1) or x + y - 2z = 0.
5. z = f( x, y) = xsin(x + y)
=?
f , (x , y) = x · cos(x + y) + sin (x + y) · 1 = x cos(x + y) + sin(x + y),
fv( x, y) = xcos(x + y), so f,( -1, 1) = (-1) cosO+ sin 0 = - 1, / 11 ( -1, 1) = ( -1) cosO = - 1 and an equation of the
tangent plane is z - 0 = (-1) ( x + 1) + (-1) (y - 1) or x + y + z = 0.
7. z = f(x , y) = x 2 + xy + 3y2, so f,.(x , y) = 2x
+y => f,(1, 1) = 3, fv(x, y) = x + 6y
=?
fv(1, 1) = 7 and an
equation of the tangent plane is z - 5 = 3(x- 1) + 1(y- 1) or z = 3x + 7y - 5. After zooming in, the surface and the
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204
D
CHAPTER 14
PARTIAL DERIVATIVES
tangent plane become almost indistinguishable. (Here, the tangent plane is below the surface.) If we zoom in farther, the
surface and the tangent plane will appear to coincide.
_ xy sin {X - y)
2
2
1+ x +y
9· !( x,y) -
fv(x, y) =
2
.
f ( ' ) ysin(x - y) + xycos(x-y) - 2x ysin {x- y) and
ACAS g1ves "' x,y =
1 + x2 + y2
( 1 + x2 + y2)2
x sin (x - y) - xy cos ( x - y)
2xy2 sin (x - y)
·
(
2 . We use the CAS to evaluate these at (1 , 1), and then
1 + x 2 + y2
1 + x2 + y2)
substitute the results into Equation 2 to compute an equation oqhe tangent plane:
z= tx - tY· The surface and tangent
plane are shown in the first graph below. After zooming in, the surface and the tangent plane become almost indistinguishable,
as shown in the second graph. (Here, the tangent plane is shown with fewer traces than the surface.) If we zoom in fart her, the
surface and the tangent plane will appear to coincide.
z 0
-1
1
11. f(x, y) = 1 +x Jn(xy - 5) : The partlal derivatives are f , (x , y) = x · - - (y) + Jn (xy - 5) · 1 = xy +ln(xy- 5)
.
xy - 5
xy - 5
and f v (x, y) = x · -
1
-
x y-5
(x) = __£_ , so f x(2 , 3) = 6 and / 11 (2 , 3) = 4. Both f , and / 11 are continuous functions for
xy- 5
,
xy > 5, so by Theorem 8, f is differentiable at (2, 3). By Equation 3, the linearization off at (2, 3) is given·by
L(x, y) = /(2, 3) + f x (2 , 3)(x- 2) + f v (2, 3)(y - 3) = 1 + 6( x - 2) + 4(y - 3) = 6x + 4y - 23.
13. f(x, y) ~ _ x _ .
x+y
The partial !;lerivatives are f x(x, y) = l(x
t
y) )~'t( ) = y/ (x + y) 2 and
x +y
1
fv(x , y) = x ( -1){x + y)- 2 • 1 = -x/ (x + y) 2 , so f x(2, 1) = ~and / 11 (2; 1) = -~. Both f, and j 11 are continuous
functions for y =I - x , so f is differentiable at (2, 1) by Theorem 8. The linearization of f at (2, 1) is given by
L(x, y) = !(2, 1) + f x(2 , 1)(x - 2) + / 11 (2;1)(y - 1) =
1+ ~ (x - 2) - ~(y - 1) = ~ X - ~ y + ~·
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SECTION 14.4
15. f(x , y) = e-"' 11 cosy.
TANGENT PLANES AND LINEAR APPROXIMATIONS
0
205
The partial derivatives are f,.(x , y) = e-"' 11 ( -y) cosy .= -ye-"'!1 cosy and
J11 (x , y) = e - "'V(- siny) + (cosy)e-"' 11 ( - x) = -e-"'11 (siny + xcosy), so f.,(1T, 0) = 0 and / 11 (1T, 0) = -1r.
Both f., and fv are continuous func;tions, so I is differentiable at (1r, 0), and the linearization of I at (1r, 0) is
L(x, y) = j(1r, 0) + /:r(1r, O)(x- 1r) + j 11 (1r, O)(y - 0) = 1 + O(x -1r) - 1r(y - 0) = 1 -1ry.
2
2x + 3
17.Letf(x,y) = --.Thenj.,(x,y)=
- 2 . andj11 (x,y) = (2x+3)( -1 )(4y+1) _ (4)=(- 8x - 12
)2 .Bothf.,andl11
4y + 1
4y + 1
4y + 1
are continuous functions for y =I= - ~. so by Theorem 8, f is differentiable at (0, 0) . We have f:,,(O, 0) = 2, f 11 (0, 0) = -12
and the linear approximation off at (0, 0) is f(x , y) ~ f(O, 0) + f:,(O, O)(x - 0) + / 11 (0, O)(y - 0) = 3 + 2x- 12y.
19. We can estimate /(2.2, 4.9) using a linear approximation off at (2, 5), given by.
f(x, y) ~ /(2, 5) + f,.(2, 5)(x - 2) + / 11 (2, 5)(y - 5) = 6 + 1(x - 2) + ( -1 )(y- 5) = x- y + 9. Thus
!(2.2, 4.9) ~ 2.2- 4.9 + 9 = 6.3.
21 .j(x,y,z)"=Jx2 +y2 +z2 =>
f:(x, y, z) =
z
,jx2 +y2 +z2
f.,(x,y,z)= J 2 +x 2 + 2 ,f11 (x,y,z)= J 2 y 2
,and
X
y
Z
X +y + Z 2
, so f.,(3, 2, 6) = ~. fv(3, 2, 6) = ~; f,(3, 2, 6) = ~·Then the linear approximation off
at (3, 2, 6) is given by
l(x, y, z) ~ /(3, 2, 6) + j .,(3, 2, 6)(x - 3) + / 11 (3, 2, 6) (y- 2) + f z(3, 2, 6)(z' - 6)
= 7 + ~(x- 3) + ~(y- 2) +
¥<z - 6) = tx + ~y + ~z
¥
Thus J(3.02) 2 + (1.97)2 + (5.99) 2 = /(3.02, 1.97, 5.99) ~ (3.02) + ~ (1.97) +
¥(5.99) ~ 6.9914.
23. From the table, /(94, 80) = 127. To estimate /T(9~, 80) and /H(94, 80) we follow the procedure used in Section 14.3. Sine~
/(94 + h, 80) - /(94, 80)
.
.
h"
.
. hh
li
Jr (94, 80) = "~
. h
, we approximate t IS quantity w1t
= ± 2 and use the values g1.ven .m the
table:
fr(94, 80) ~ /(96, 80) - /(94", 80) = 135- 127 = 4,
2
2
fr(94, 80) :::::: /(92, 80) - /(94, 80) = 119- 127 = 4
- 2
- 2
Averaging these values gives /T(94, 80) ~ 4. Similarly, /J.I(94, 80) = lim /(
h~ o
/H(94 , 80) ~ /(94, 85) ~ /(94, 80) = 132 ~ 127 = 1,
94 80
94 80
• + h~- /( • ), so we use h = ±5:
!u(94, BO) ~ /(94, 75)- /(94, 80) = 122- 127 = 1
-5
-5
Averaging these values gives /H(94, 80) ~ 1. The linear approximation, then, is
f(T, H) ~ /(94, 80) + fr(94, 80)(T - 94) + /H(94, 80)(H- 80)
~ 127
+ 4(T - 94) + 1(H- 80)
[or 4T + H - 329]
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206
0
CHAPTER 14
PARTIAL DERIVATIVES
Thus when T = 95 and H = 78, !(95, 78) ~ 127 + 4(95- 94) + 1(78 - 80) = 129, so we estimate the heat index to be
approximately 129°F.
25. z = e- 2 "' cos 27rt
=>
.
dz = -aaz dx + -f)z dt = e -2"' ( -2 ) cos .21rt dx + e -2"' (- sm2?rt)(27r)
dt = -2e- 2"'cos 27rt dx- 21re- 2 x sin 21rt dt.
.
X
0t
"3
27. m = p0 q ·
=> dm = -am dp + -om dq = 5p43
q dp + 3pa"2
q dq
ap
29. R = o:(3 cos 1
2
=>
oq
aR
oR
oR
2
2 .
dR = oo: do: + (3 d(3 + O"f d"f = (3 cos '"Y do: + 2o:(3 cos 1 d(3 - o:(3 sm 1 d1
0
•
31. dx = ll.x = 0.05, dy = ll.y = 0.1, z = 5x
2
+ y 2 , z., = lOx, Zy = 2y . .Thus when x. = 1 and y = 2,
dz = zx(1, 2) dx + z 11 (1, 2) dy = (10)(0.05) + (4)(0.1) = 0.9 while
ll.z = ! ,(1.05, 2.1)- /(1 , 2) = 5(1.05? + (2.1) 2 - 5 -4 = 0.9225.
8A
8A
.
33. dA = ox dx + oy dy = y dx + x dy and Jll.xJ :::; 0.1, Jll.yJ :::; ~.1. We use dx = 0.1, dy = 0.1 w1th x = 30, y = 24; then
the maximum error in the area is about dA = 24(0.1) + 30(0.1) = 5.4 cm2 •
35. The volume of a can is V = 1rr 2 h and ll. V ~ dV is an estimate of the amount of tin. Here dV = 27rr h dr + 1rr~ dh, so put .
dr = 0.04, dh = 0.08 (0.04 on top, 0.04 on bottom) and then ll. V ~ dV = 27r( 48)(0.04) + 7r(16)(0.08) ~ 16.08 cm3 .
Thus the amount of tin is about 16 cm3 .
37. T =
";'gR
2r +R2
, so the differential ofT is
2
2
2
2
dT = aT dR 8T d = (2r + R )(mg) - mgR(2R) dR + (2r +R )(0) - mgR(4r) d
oR
+ or . r
(2r 2 + R 2 ) 2
(2r2 + R2)2
r
mg(2r2 - R 2 )
4mgRr
= (2r2 + R2)2 dR- (2r2 + R2)2 dr
Here we have ll.R = 0.1 and ll.r = 0.1, so we take dR = 0.1, dr = 0.1 with R = 3, r = 0.7. Then the change in the
tension T is approximately
mg [2(0.7?- (3) 2 ]
dT
4mg(3)(0.7)
= (2(0.7) 2 + (3) 2 ]2 (O.l) - (2(0.7) 2 + (3)2)2 (0. 1)
0.84mg
1.642
0.802mg
(9.98)2 - (9.98) 2 = - 99.6004 mg ~ - 0.0 165mg
Because the change is negative, tension· decreases.
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SECTION 14.5 THE CHAIN RULE
0
207
39. First we find :~ implicitly by taking partial derivatives of both sides with respect to R 1 :
oR . R 2
oR = Ri. Then by symmetry,
1
2
2
1
17
oR
R
oR = R2 . When RI = 25, R2 = 40 and Ra = 50, -R = 2 00
oR2 - ~· oR3
R3
¢::}
R = 21°7° n. Since the possible error
for each Ri is 0.5%, the maximum error of R is attained by setting t:J.R, = 0.005R.. So
oR
oR
oR
2( 1
1
1 )
1
!:l.R ~=:::: dR = oR !:l.R1 + oR !:l.R2 + oR !:l.R3 = (0.005)R R + R + Ra = (0.005)R = l7 ~=:::: 0.059 st.
2
1
2
3
1
~ so l!:l.wl $ 0..02 and l!:l.hl $ 0.02. The relative
. . error m
. the calculated surface
41 . The errors in measurement are at ~ost 27o,
w
T
area is
0 425 - 1 )h0·725 dw + 0.1091w0.4 25 (0.725h0·725 - 1 } dh
dw
dh
0.1091w0.426h0.725
.
= 0.425-:;:; + 0.725/i:
dS 0.1091(0.425w ·
s!:l.S s=
1'::::
To estimate the maximum relative error, we use :
I~w I= 0.02 and ~~i = I~hI = 0.02 =>
=
dS = 0.425 (0.02) + 0. 725 (0.02} = 0.023. Thus the max imum percentage error is approximately 2.3%.
s
43. !:l.z = f(a + !:l.x, b + !:l.y) - f(a, b) =(a+ !:l.x? + (b + !:l.y) 2 - (a2 + b2 )
= a 2 + 2a!:l.x + (!:l.x) 2 + b2 + 2b!:l.y + (!:l.y?- a 2 - b2 = 2a !:l.x + (!:l.x? + 2b!:l.y + (!:l.y?
But f,(a, b) = 2a and /y(a, b) = 2b and so !:l.z = f x(a, b) !:l.x + /y(a, b) !:l.y + !:l.x !:l.x + !:l.y !:l.y, which is Definition 7
with e: 1 = !:l.x and e: 2 = !:l.y. Hence f is differentiable.
45. To show that f is continuous at (a, b) we need to show that
equivalently
lim
(6x ,6y)- (O,O )
lim
(z,y)~(a,b)
f(x, y) = f(a, b) or
f(a + !:l.x, b + !:l.y) = f(a , b). Since f is differentiable at (a, b),
f(a + !:l.x, b + !:l.y) - f(a, b)= !:l.z = f,(a, b) 6-x + /y(a, b) !:l.y + e:1 !:l.x + e:2 !:l.y, where €1 and €2 ---+ 0 as
(!:l.x, !:l.y} --> (0, 0). Thus f(a + !:l.x, b + !:l.y) = f(a, b)+ f,(a, b) !:l.x + / 11 (a, b) !:l.y + e:1 !:l.x + e:2 !:l.y. Taking the limit of
both sides as (.C:..x, .C:..y) ___.. (0 , 0) gives
lim
( 6 x,61J)- (0 ,0)
f (a + .C:..x, b + .C:..y) = f (a , b). Thus f is continuous at (a, b).
.
14.5 The Chain Rule
1. z = x? + y 2 + xy, x =sin t, y = et
~
&~
&~
=> -d =,.--d +,.- -d =(2x+ y}cost+(2y +x)e
t
uX
t
uy
t
t
3. z =·.J1+ x 2 + y 2 , x = In t, y = cost =>
8z-dy= 1 (1+x2 + y)
2 -1/ 2
.
1
(x
. )
2 -1/2
-dz =oz
- -dx + (2x)·-1 + 21 (1+x2 + y)
(2y)(-smt)
=
2
dt
ox dt
8y dt
t
·
.J1 + x2 + y2 -t - ysmt
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208
D
CHAPTER 14
PARTIAL DERIVATIVES
5. w=xevl =, x = t2, y = l-t, z=1+2t =>
(.!) .(- 1) +
dw = OW dx + ow dy + ow dz = ey/ z. 2t + xevl=
dt
ox dt
/Jy dt
oz dt
z
(-J!...)
.2 = evl=(2t - ~z - 2xy)
z
z2
xeYI=
2
7. z = x 2 y 3 , x = scost, y = ssint =>
az
us
oz 8y
3
2 2 •
= 2xy cost+ 3x y sm t
y s
.
oz ax
uxus
!i"'" = !i"'"' <> + -a -a
az . 8z Ox
az 8y
s .
.
2 2
3
2 2
ot = ox at+ oy
= (2xy )(- ssmt) + (3x y )(scost) = -2sxy smt + 3sx y cost
at
9. z = sin() cos</>, () = st2 , ¢ = s 2 t
=>
~; = ~; ~~ + ~~ ~! =(cos 8 cos q))(t~) + (- sinO sin q.,)(2st) = t 2 cosO cos¢ - 2stsin0 sin <P
~; = ~; ~~ + ~~-~~ =(cos() cos ¢>)(2st) + (-sinO sin </>)(s2 ) = 2stcos0 cos¢- s 2 sinO sin <P
11. z = er cos 0, r = st, 8 = J s 2 + t 2
=?
13. When t = 3, x = g(3) = 2 andy = h(3) = 7. By the Chain Rule (2),
dz
/Jf dx /Jf dy
I
I
•
dt = OX dt + oy dt = f ,., (2, 7)g (3) + fv(2, 7) h (3) = (6)(5) + ( - 8)( -4) = 62.
'
15. g(u, v) = f(x(u ,v) , y(u, v)) wherex = eu + sinv, y = eu + cosv
=>
&
u&
8y
u8y
·
·
~
M& M8y
au = e ' au = cos v, au = e ' av = -Sill v. By the Cham Rule (3), au = ox au + /Jy au. . Then
Yu(O, 0) = f ,., (x(O , 0), y(O, 0)) Xu(O, 0} + fv(x(O, 0), y(O, 0)} y,.(O, 0) = / :c (1, 2)(e0 ) + fv(l, 2)(~ 0 ) = 2(1) + 5_{1) = 7.
a
. .
ag
1 ax a 1
Similarly, OV = OX
+ oy
av
av
av . Then
g..,(O, 0) = f ., (x(O, 0), y(O, 0)) x.,(O, 0) + fv (x(O , 0), y·(o, 0)) Yv (0, 0) = f,. (1, 2)(cos O) + fv(l, 2)(- sin 0)
= 2(1} + 5(0) = 2
17.
u = f(x , y), x = x (r, s , t), y = y(r, s , t)
II
/~y
=>
X
/ 1\ /1\
r
s
I
r
s
I
© 2012 Ccng.:~gc Lcuning. All Right. Ro= rvc:d. May not_be scaruu:d, copied, or dupllcntcd , or posted too publicly accessible wcb:ritc, in whole or in p:lrt.
SECTION 14o5
w
19o
r
/Is ~
J
X
y
X
w = f(r, s, t), r = r(x, y), s = s(x, y), t = t(x, y)
=>
ow
aw as
ow or
ow as
ow at
aw
aw ar
D
209
ow at
fu = &ofu+asax+8tax·~=aroy+asay + Bt~
r
/\ /\ 1\
X
THE CHAIN RULE
J
2 =>
210 z = x 4 + x 2 y, x = s + 2t - u, y = stu
oz = az ox+ az.ay = ( 4xa + 2 xy)( 1) + (x2)(tu2),
OS
OX OS
ay as
'
0
oz = oz ax+ az ay = (4x3 + 2xy)(2) + (x2)(su2)
m &m
~m
·
oz = az ax+ az ay = (4x3 + 2xy)( - 1} + (x2)(2stu).
au ax au ay a·u
- When s = 4, t = 2, and u = 1 we have x = 7 and y = 8,
so a z = (1484)(1} + {49)(2} = 1582, aaz = {1484) (2) + (49){4) = 3164, aaz = (1484)(-1) + {49)(16} = -700.
as
t
u
23o w = xy + yz + zx, x = r cos 0, y = r sin 8, z = r8 =>
aw = awax + aawaay + aawaaz ~(y +z}(cos8} + (x+z)(sin8}+(y+x)(O), .
ar
axar
yr
zr
aw = aw ax+ aaw aay + aaw aaz8 = (y + z)( - rsinO) + (~ + z )(rcosO) + (y + x}(r·)o
ao
ox o8 . y 8
z
aw
When ro = 2 and()= ?r/2 we have x = 0, y = 2, and z = 1r, so ar = (2 + rr}(O) + (0 + rr){1) + (2 + 0)(1r / 2) = 27r and
~~ = (2 + 1r) ( - 2) + (0 + 1r)(O) + (2 + 0)(2) = -2rro
..
25o N = p + q, p = u + vw, q = v + uw, r = w + uv =9
p+r
aN = aN ap + aN aq + aN ar .
au
ap au
aq au
8r au
'
= (p + r){1)- (p + q){1) (1) + (p + r)(1) - (p + q)(O) (w) + (p + r) (O) - (p + q)(1) (v)
(p+r)2
(p + r)2
(p + r)2
=
(r- q) + (p+r)w - (p+q)v
(p+r)2
oN = aN ap + aN oq + aN or = r - q (w) + p + ro (1) + -(p + q) (u) = (r - q)w + (p + r} - (p + q)u '
av
op ov
8q 8v
or ov
(p+r)2
(p +r) 2
(p+ r)2
. (p+1')2
0
aN _ aN ap + 8N aq + 8N !!!.._ = ro - q (v) + p+r (u)+ -(p +q) (1) = (r- q)v +(p+r)u- (p + q)
ow - ap ow
aq ow
or OW
(p + r )2
(p + r) 2
(p + r) 2
(p + r )2
·
aN
-1 + (24)(4) - (25)(3)
20
5
When u = 2, v = 3, and w = 4 we have p = 14, q = 11, and r = ~0, so ou = .
( )2 .
=
=
,
24
576
144
aN _ (- 1)(4)+24-(25)(2) _ -30 __ ~
d 8N = ·(- 1)(3) + (24)(2)-25 =~ = ~
av (24)2
- 576 96' an aw
(24)2
576
144 °
© 2012 Ceng;~gc LCJming. AJI Rights Reserved. Mn)' nol be scanned. co ph.-d, or duplicated, or posted lo o publicly accessible websitt:, in whole or in p;~rt.
l
210
0
CHAPTER 14
PARTIAL DERIVATIVES
27. y cosx = x 2 + y 2 , so let F(x, y) = ycosx - x 2 - y 2 = 0. Then by Equation 6
dy
-y sin x - 2x
dx
cos x - 2y
2x + y sin x
= --:-----'"---cosx- 2y .
+
[2xy - (1 + y 2 ) (1 x-iy 2 )]/(1 + x 4 y 2 ) _ (1 + y 2 )(1 + x 4 y 2 ) - 2xy
(x2 - 2xy(1 + x4y 2 ))/(1 + x4y 2 )
x 2 - 2xy(1+ x4y2)
and
- 1 + x4y2 + y2 + x4y4 - 2xy
x 2 - 2xy - 2x 5 y3
31. x 2 + 2y2 + 3z 2 = 1, so let F(x, y, z) = x 2 + 2y 2 + 3z2 - 1 = 0. Then by Equations 7
f)z = .:_ Fx = _ 2x = -~
ox
Fz
6z
3z
and
oz = _ Fy = _ 4y = _ 2y
oy
Fz
QZ
3z .
_
•
oz
F. .
-yz
yz
33. e· = x yz, so let F(x, y , z) = e· - xyz = 0. Then~ = - -F = --.-- = - -.
ux
'z
e· - xy
ez - xy
8z
oy
- xz
F11
and
xz
- Fz = --e=---x-y = -ez___x_y
35. Since x andy are eac~ functions oft, T( x, y) is a function oft, so by the Chain Rule, ~ = ~: + ~~~;.After
dx
1
= 2 .J1+t
t
1 +t
3 seconds, x = .Jl+t = .,;r+3 = 2, y = 2 + ~ t = 2 + H 3) = 3, -d
-.!. and dy -.!.
1
2 v'1 + 3 - 4,
dt - 3 .
Then~~ = Tx(2, 3) ~~ + Ty(2 , 3) ~; = 4(t) + 3(~) = 2. Thus the temperature is rising at a rate of2°C/ s.
ac
ac
.
37. C = 1449.2 + 4.6T- 0.055T 2 + 0.00029T 3 + 0.016D, so 8T = 4.6 - O.llT + 0.00087T and oD = 0.016.
2
According to the graph, the diver is experiencing a temperature of approximately 12.5°C at t = 20 minutes, so
.c:;, = 4.6 - 0.11 (12.5) + 0.00087(12.5) ~ 3.36. By sketching tangent lines_at t = 20 to the graphs given, we est~ate
2
dD
1
dT
1
.
dC
80 dT
80 dD
1
1
.
dt ~ '2 and (jj ~ - 10 . Then, by the Cham Rule, (jj = &T (jj + &Ddt ~ (3.36) (-Iii) + (0.016) ( 2) r::::: -0.33.
Thus the speed of sound experienced by the diver is decreasing at a rate of approximately 0.33 m/s per minute.
39. (a) V = l!wh, so by the Chain Rule,
dV = 8V dl! + 8V dw + oV dh = wh dl! + l!h dw + l!w dh = 2 . 2 . 2 + 1 . 2 . 2 + 1 . 2 . ( _ 3 ) = 6 m3/s.
dt
81! dt
aw dt
&h dt
dt
dt
dt
© 2012 Ccngagc Learning. All Rights RescrYcd. May not be scanned. copied, or duplicated. or, posted toP publicly accessible website, in whole or in part.
SECTION 14.5
THE CHAIN RULE
D
211
(b) S = 2(£w + £h + wh), so by the Chain Rule,
~~ = ~~: + ~! :~ + ~~ ~~ = 2(w +h):+ 2(£ + h):~ + 2(£+ w) ~~
= 2(2 + 2)2 + 2(1 + 2)2 + 2(1 + 2)( - 3) = 10 m2fs
2
2
2
(c) L = £ + w + h
2
::::?
2£ ~~ = 2f. ~ + 2w :~ + 2h ~: = 2(1)(2) + 2(2)(2) + 2(2)(-3) = 0
::::?
dL/ dt = 0 mjs.
dP
dT
T
dV
8.31 dT
T dP
= 8.31? and dt = Pdt - 8.31 p 2 Yt· Thus when P = 20 andT = 320,
41. dt = 0.05, dt = 0.15, V
dV = 8 31 [ 0.15 _ (0.05)(320)] ~ _ 0 .27 Lf s.
dt
.
20
400
43. Let x be the length of the first side of the triangle andy the lengtl1 of the second side. The area A of the triangle is given by
A= ~xy sin B where 0 is the angle between the two sides. Thus A is a function of x, y, and (}, and x, y, and 0 are each in tum
functions of timet. We are given that ~~ = 3, ~~ ·= -2, and because A is constant, :~ = 0. By the Chain Rule,
dA
8A dx
8A dy
8A dO
dA
dt = ax dt + ay dt + ao dt
::::?
.
dx
.
dy
d8
1
1
1
0 · dt + 2xsrn
0 · dt + 2xycos(}
dt = 2YSI.Il
· dt. When x = 20, y = 30,
and 0 = 1r/6 we have
0 = ~(30)(sin ~)(3) + ~(20)(sin ~)(-2) + ~(20)(30)(cos ~) ~~·
= 45 · l2 - 20 · l2 + 300 · vf3
· dO
= ll2 + 150 v,)
'3 d(}
2
dt
dt
. .or
., -d
dO g1.ves -ddO = ---;;;
- 25/2 = -------:r.i·
1
· at a rate o f
So Ivmg
sot he angIe between the s1·des 1·s decreasmg
t
t
150 v 3 . 12 v 3
1/ (12 V3) ~ 0.048 radj s.
45. (a) By the Chain Rule, :
(
~= cos(} + ~; sin B, ~; = ~= - r sin B) + ~~ r cos 0.
.
( {)z )
or = ({)z
ax ) cos 0 +.2 8{)zx {)z
({Jz)
8 y cosO ~mO + By sm 0,
2
(b)
=
2
2
2
2
• 2
2
8 z ) + r2
1 (az
80 )
( 8r
8z
2
2
2
2
2
2
.
= [(az
8x ) + (az)
8y ] (cos 0 + sm
0) = (az
8x ) + (oz
8y ) .
dz 8u
dz
8z
dz
8z
8z
47. Letu = x - y.Then - = d - = -d and ay = -d U (-1). Thus -aX + -8 y = 0.
8X
U8X
U
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212
D
CHAPTER 14
PARTIAL DERIVATIVES
49. Let u =X + at, v =X - at. Then z = f(u) + g(v), so az jau ~ f'(u) and azjav ~ g1 (v) .
az
f}z aU az av
1
1
Thus at = au at + av at = af (u)- ag (v) and
2
8 z = 2._ [/1( ) a at
[)t2
u
1
I( )] = (df'(u) 8u - dg (v ) au ) = 2/"( )
2 "( )
g v
a
du 8t
dv 8t
a
u +a g v .
2
2
2
- uX
ut
ux
. ,arly .,-8z = fl( u ) + g I(v ) and <:12
8 z = !"( u) + g "(v )' . Thus ";5'2
8 z =a2 <:12·
8 z
S.
tmt
uX
51 .
8z
8z
.,-=-aazX 2s ·+ .,-2r. Then
uS
vy
2
2
a2
2
. .
.
8 z
8 z
8 z (
z
8z
)
By the contmutty of the parttals,
"' = 4rs - 2 + 4rs - 2 + 4r 2 + 4s 2 - "' + 2- ·
8y
8 xuy
8 rus
,8x
8y
az .
8z
az
[}z
8z
'
8z
53. - = .,-- cos (} +-a sm9and {)(} = -.,-r smO +-a rcosB. Then
8r
uX
y
vX
y
2
z
a
iJr2 = cos 0
(8
2
tPz
z
.
8:r:2 cos (} + ay 8x sm ()
)
.
+ sm ()
(8
2
2
z .
8 z
)
f):y2 sm (} + ax 8y cos ()
.
~z
.
~z
= cos2 (} 8~Xz2 + 2 cos 8 sm
(J + sm 2 8 -a 2
8X 8y
1J
and
2
Dz
~
a(J
a."'-,. + (- rsinO) (a<:12z (-r sin(J) + ~a
8 z
)
= - rcos(J uX
rcos9
uX
uy X
2
2
az + r cos f)
-r s~ (J .,-vy
(8
-a
y
2
2
z r cos 8 + ~
8 z ( -r sin 9))
2
uxuy
·
8z
. 8z
8 z
~z
8 2z
.
.
= - r cos (J .,-- - r sm (} .,-- + r 2 sm 2 f) -a2 - 27·2 cos 8 sm (} ~ + r 2 cos 29 <:12
2
vx
uy
,
X
v X vy
uy
Thus
a2
2
2
2
.
z
1 a z
1 fJz
a z
( .
2 ) 8 z
2
<>2 + 2 <:i2 + - .,-- = (cos 0 + sm2 0) - 2 + sm 2 (} + cos (J - 2
vr
r 'v8
r v1·
8y
8X
.
1
.
8z
1(
az
. az )
-cos(} -{)z - -1 sm
7'
ax r B- + -r cos(} - + sm (J -ay
ay
az
2
az
2
ax
.
= -aX2 + -a
as destred.
1J2
55. (a) Since f is a polynomial, it has continuous second-order partial derivatives, and
f(tx, ty) = (tx) 2 (ty ) + 2(tx)(tv? + 5(ty) 3 = t 3 x 2 y + 2t3 xy2 + 5t3 y3 = e(x2 y + 2xy2 + 5y3 ) = t 3 f (x, y).
Thus, f is homogeneous of degree 3.
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SECTION 14.6
DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
0
213
(b) Differentiating both sides of f(tx , ty) = t" f(x , y) with 'respect tot using the Chain Rule, we get
8
8
8
t f(tx , ty) =
at [t" f(x,y) ]
8
¢:>
8
8(tx)
8
8(ty)
8
a
-1
(tx) f(tx , ty) · {j't + 8 (ty) f(tx, ty) ·
= x a(tx ) f(tx, ty) + y 8 (ty) f(tx , ty) = nt" f(x , y).
at
Setting t = 1: x
8
8
f(x , y) + y
f (x, y) = nf(x, y).
.8 y
.
8X
57. Differentiating both sides of f(tx , ty) = tn f(x , y) with respect to x using the Chain Rule, we get
8
8
ax f(tx,ty) = 8x [t"'f(x , y)]
¢:>
8
8(tx)
8
8(ty)
n 8
= t 8x f(x, y)
a (tx) f.(tx , ty) . a;;- + 8 (ty) f(tx , ty).
a;:-
¢:>
tfx(tx, ty) = tn f,.(x, y).
Thus f ,. (tx , ty) = t"·- 1 J,.(x , y) .
59. Given a function defined implicitly by F(x, y) = 0, where F is differentiable and F11 =J: 0, we know that ~~ = - F,. . Let
.
~
~
G(x , y) = - F,. so ddy = G(x, y). Differentiating both sides with respect to x and using the Chain Rule gives
Fy
X
2
d y = 8G dx + 8G dy where 8G = ,E_ ( - Fx) = _ F11 Fxx dx2
8x dx
ay dx
8x
8x
Fy
FJ
F,.F
11 ., ,
8G = ,E_
ay
8y
(-F.,)
=
Fy
T~us
2
d y= (
dx 2
FuF:.,,. - F,.Fyx) (l) + (- FyFxll - FxF,m)
F112
F112
(-Fa;)
F11
Fxx F; - FyxFxFy - F,.uF11Fx + FyyF;
FJ
But F has continuous second derivatives, so by Clauraut's Theorem, F11 ,
= F.,11 and we have
d211
F.,.,F; - 2F.,11 F, Fu + FuuF;
d . ed
dx2 = F.3
as es1r .
II
14.6 Directional Derivatives and the Gradient Vector
1. We can approximate the directional derivative of the pressure function at K in the direction of S by the average rate of change
of pressure between the points where the red line intersects the contour lines closest to K (extend the red line slightly at the
left). In the direction ofS, the pressure changes from 1000 millibars to 996 millibars and we estimate the distance between
these two points to be approximately 50 km (using the fact that U1e distance from K to S is 300 km). Then the rate of change of
pressure in the direction given is approximately 996 ; 01000 = -0.08 millibar/km.
3. Du /{ -20, 30) = 'V j { -20, 30) · U = /T{ -20, 30) (
/T( - 20, 30) ~ lim !( h -+0
20
7z) + /v{ - 20, 30) ( ?z).
+ h, 30)t - !( - 20 • 30), so we can approximate /T( -20, 30) by considering h = ±5 and
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214
0
CHAPTER 14
PARTIAL DERIVATIVES
using the values given in the table·: h (- 20, 30) ~ f (-
fr( -20, 30) ~
!( -25 30) - f( -20 30)
'
_
= -39 -_5( - 33) = 1.2. Averaging these values gives fr( - 20, 30) ~ 1.3.
'
5
Similarly, f,( -20, 30) = lim /( h-0
15 30
• ) ~· f (- 20 • 30) = - 26 - ( - 33) = 1. 4,
5
20 30
20
• + h~- !( - • JO), so we can approximate f,( --120, 30) with h = ±10:
- J( - 20, 30) = -34 - ( - 33) = - 0 1
f (-20 30) ~ !( - 20, 40) 10
10
. '
1J
,
f,( - 20, 30) ~
!( - 20 20) - J( -20 30)
'
_
'
10
= - 30-_10(- 33) = -0.3. Averaging these values gives f ,( -20, 30) ~ -0.2.
Then Duf(- 20,30) ~ 1.3(-}z) + (-0.2>(-}z) ~ 0.778.
5. f(x, y) = ye- "'
:::>
fx (x, y) = -ye_., and fu(x, y) = e-"'. If u is a unit vector in the direction of 8 = 21r / 3, then
from Equation 6, Du /(0, 4) = f x (0, 4) co~
7. f(x, y) = sin(2x
en + f v(O, en
4) s in
= -4 · (- ~) + 1 · {1 = 2 +
4.
+ 3y).
• I
8
(a) 'il f (x, y) = . 1 i + ~! j = [cos(2x + 3y) · 2] i + [cos(2x + 3y) · 3]j = 2 cos {2x + 3y) i + 3 cos (2x + 3y)j
8X
~J
•
(b) 'il f( -6,4) = (2cos0) i + (3cosO)j = 2 i + 3j
(c) By Equation 9, Du f( - 6, 4) = 'il f( -6, 4) · u = (2 i + 3j ) ·
Hv'3 i - j ) = ~ (2v'3- 3) = v'3 - ~·
= x 2 yz- xyz 3
3
2
2
2
(a) 'il f( x, y, z ) = (f,. (x, y , z), fv(x, y, z), f:(x, y, z)) = (2xyz - yz , x z- xz 3 , x y - 3xyz )
9. f(x , y, z)
(b) 'il/(2, -1, 1) = (-4 + 1, 4- 2,-4 + 6) = (-3, 2, 2)
(c) By Equation 14, Duf(2, - 1, 1) = 'il /(2, - 1, 1) · u = (-3, 2, 2) · ( 0, ~. - ~) = 0 + ~ - ~ = ~·
11. f(x , y)
= e"' si.ny =>
'il f(x, y) = (e"' s iny, e"' cosy}, '\1 f(O , Tr/3) = (
unit vector in the direction.ofv is u =
4, ~ ),and a
J<-:>+8 (- 6, 8) = 1~ (-6, 8) = ( -i, V. so
2
2
-v
'<"7f(O , 1r. / 3) · U _D u f(o , Tr / 3) -
(:.2 1) (- 5•5
3 4) _ M + 10-
vector in the direction ofv is u =
Vl;H2 (i + 3 j ) = 'ftu Ci + 3j), so
2 •2
·
-
-10
il _
4-313
10
·
Dug(.2, 1) = 'ilg(2, 1) · u = (28 i - 12 j ) · ~(i + 3j) = ~ (28 - 36) = - ?to or- 4 ~.
15. f(x, y, z) = xeV + ye= + ze"'
:::>
'\1 f(x, y, z) = (ell + ze"', xeV + e=, ye= + e"'), 'ilf (O, 0, 0) = (1, 1, 1), and a unit
vector in the direction ofv is u = ..;2s.!.I+4 (5, 1, -2) =
7:io (5, 1, -2), so
D u f(O, 0, 0) = "V/(0, 0,0) · u = (1, 1, 1) · ~ (5, 1, - 2) = ~·
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SECTION 14.6
17. h(r, s, t) = ln(37· + 6s + 9t)
'ilh(1, 1, 1) = (k,
is u = .;16
DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
0
215
=> 'ilh(r, s, t) = (3/(3r + 6s + 9t), 6/(3r + 6s + 9t), 9/(3r + 6s + 9t)),
ft, ~ ), and a unit vector in the direction ofv = 4 i + 12j + 6 k
+; 4+3& (4i + 12j + 6k) = ~ i .+ ~ j + ~ k, so
4
l 1 1)
2 6 3) 1 + 2 + 3 - 23
Duh (1,1, 1) = '<'7h(
v
1, 1, 1) · U = ( 6•3•2
· ( 7•7•7
- 21
7
14 -42'
19. f(x, y) =
VxY => 'il f(x , y) = (Hxy)- 112 (y), Hxy)- 112(x)) = \2
/ yt=' ~)·so 'il!(2, 8) = ( 1, ~ ) .
yxy 2yxy
--
.
The unit vector in the direction of PQ = (5 - 2, 4 - 8) = (3, -4) is u = ( ~, - ~ ), so
.
.
1
(3
- 2
Duf(2,8)='il/(2,8)·
u= ( 1,4)·
5•-6" ) -5.
21. f(~,y) = 4yy'X => 'ilf(x,y) = ( 4y · ~x- 1 1 2 ,4v'x) = (2yjy'x,4.,fi).'
· 'il f(4 , 1) = {1, 8) is.the direction of maximum rat~ ofch~ge, and the maximum rate is·/'il /(4, 1)/ = v'1 + 64 = v'65.
=> 'il f(x, y) = (y cos(xy), x cos(xy)), 'il!(1 , 0) = (0, 1). Thus the maximum rate of change is
23. J(x, y) = sin(xy)
/'il f(1, 0)/ = 1 in the direction {0, 1).
25. f(x,y ,z) =
Jx2 + y2 + .z2
'il f(3, 6, -2) = (
=>
-7,w. ~· ~) = ( ~. ¥, -¥ ). Th~s the maximum rate ofchange is
27. (a) As in the proof of Theorem 15, Du f = /'il !I cos 0. Since the minimum value of cos(} is - 1 occurring when(} = 11', the
minimum value of Du f is - /'il f/ occurring when(} = 11', that is when u is in the opposite direction of 'il f.
(assuming \7f "f 0).
(b) f(x , y) = x 4 y - x 2 y 3
=> 'il f(x, y) = ( 4x3 y - 2xy3 , x 4 - 3x2 y 2 ), so f decreases fastest at the point (2, -3) in the
direction- 'il /(2 , - 3) = - {12, -92) = (-12, 92).
29. The directionoffastestchange is 'ilf(x,y) = (2x- 2) i + (2y- 4) j , so we need to find all points (x,y) where 'ilf(x,y) is
parallel to i + j
¢'>
(2x - 2) i + (2y - 4)j = k (i + j)
¢'>
k = 2x- 2 and k = 2y - 4. Then 2x- 2 = 2y- 4
y = x + 1, so the direction of fastest change is i + j at all points on the line y = x + 1.
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=>
216
0
CHAPTER 14
31. T =
PARTIAL DERIVATIVES
k
k
Jx2 + y2 +z2 and 120 = T(1, 2, 2) = -3 so k = 360.
- (1, - 1, 1)
(a
) uv'3 ,
2
DuT(1, 2, 2) = 'VT(1, 2, 2) · u = [.:._360( x 2 + y 2 + z2) -S/ (x, y, z)]
.
. {1,2,2)
(b) From (a), 'VT = -360(x
2
. u = - .12
3 (1, 2, 2). -4.. (1, -1, 1) = -~
v3
3v3
+ y 2 + z 2 ) - S/ 2 (x, y, z), and since (x, y, z) is the position vector of the point (x, y, z), the
vector - (x, y, z), and thus 'VT, always points toward the origin.
33. 'VV(x, Y •.z) = (lOx- 3y + yz, xz - 3x, xy), 'VV(3, 4, 5) = (38, 6, 12)
{a) Du V(3,4,5) = {38, 6, 12} · ~(1, 1, -1} =
'7a
(b) 'VV(3, 4, 5) = (38, 6 , 12), or equivalently, (19, 3, 6).
(c) IV'V(3,
4, 5)1 = v'382 + 6 + 12 = v'l624 = 2 v'406
2
2
-----t
-----t
35. A unit vector in the direction o(AB is i and a unit vector in the direction of AC isj. Thus D_..... f(1, 3) = f.,(1, 3) ~ 3 and
AB
D- f(l, 3) = / y(1, 3) = 26. Therefore \7 /(1 , 3) = (!.,(1, 3), /y(l , 3)) = (3, 26}, and by definition,
AC
-----t
DAD f (1 , 3) = V' f · u where u is a unit vector in the direction of AD, which is (fa, H). Therefore,
DAD f (1, 3) = (3, 26) . .( 153, H) = 3. fa + 26. H = 312;.
au;
b)=(o(au~+bv) ' 8(au+bv))
=\a au
bav au bav)= a (au
a
~ + {) , a ~- + ~
a , au;
~
+ b\av
~ 'a
'["7(
37. ()
a v au + v
y
uX
ux
X
uy
au , nu · au) -- nu
vu
uy
X
uy
ux
y
.= a 'Vu + b 'Vv
n - (
(d) '["7
vu -
39. f( x, y) = x
3
o(un) , 8(u")) -- ( nu
ax
oy
n- 1
n- 1
ox
n - 1 '["7
{)y
+ 5x 2 y + y 3 . "*
Duf(x,y) = 'Vf(x , y) · u = (3x 2 + lOxy, 5x2 + 3y2) · (~, ~) = ~x 2 + 6xy+ 4x2 + ¥v 2 = ~x2 + 6xy + ¥y2 . Then
D~.J(x,y) = Du (Duf(x ,y)] = \7 [Duf(x,y)] · u = (¥x + 6y, 6x + -\1Y) · (~, ~)
= l;s4x + Jfy + ¥x + ~y = 2is4x + ~ar,oY
and D~f(2 , 1) =
2iu4 (2)
+ 12866 (1) = 72754 .
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SECTION 14.6
DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
41. Let F(x, y, z) = 2(x - 2) 2 + (y - 1? + (z - 3) 2 . Then 2(x - 2)
2
0
217
+ (y - 1f + (z - 3) 2 = 10 is a level surface of F.
F:.(x, y,z) = 4(x - ·2)
=?
F,(3, 3, 5) = 4, F 11 (x,y,z) = 2(y- 1)
Fz(x,y, z) =2{z - 3)
=?
Fz{3,3, 5) = 4.
=?
F 11 (3,3,5) = 4, and
(a) Equation 19 gives an equation of the tangent plane at {3, 3, 5) as 4{x - 3) + 4{y - 3) + 4{z- 5) = 0
<=>
4x + 4y + 4z = 44 or equivalently x + y + z = 11.
. 20, the normaI I'me has symmetnc
. equat10ns
.
x- 3
y - 3
z- 5
. I l
(b) By EquatiOn
- - = - - = - - or eqUiva ent y
4
4
4
x - 3 = y - 3 = z - 5. Corresponding parametric equations are x = 3 + t, y =; 3 + t, z = 5 + t.
43. Let F(x , y, z ) = xyz2 • Then xyz2· = 6 is a level surface ofF and 'V F(x, y, z ) = (yz2 , xz 2 , 2xyz).
(a) 'V F(3, 2, 1) ~ (2,3, 12) is a normal vector for the tangent plane at {3, 2, 1), so an equation of the tange nt plane
is 2(x - 3) + 3(y - 2) + 12(z - 1) = 0 or 2x + 3y + 12z = 24.
(b) The norma l line has direction (2, 3, 12), so parametric equations are x = 3 + 2t, y = 2 + 3t, z = 1 + 12t, and
.
.
x-3 . y-2
z- 1
= - - = l 2.
3
symmetriC equations are - 2
45. Let F (x , y, z) = x + y + z - e"'Yz. Then x + y + z = e"'11 :; is the level surface F(x, y , z ) = 0,
and 'V F(x, y, z) = (1 - yze"'Y:;, 1 - xze"'11"", 1 - x ye"'11 z).
(a) 'V F(O, 0, 1) = (1, 1, 1) is a normal vector for the tangent plane at {0, 0, 1), so an equation of the tangent plane
is 1(x- 0) + 1{y - 0) + 1(z - 1) = 0 or x + y
+ z = 1.
(b) The normal line has direction (1, 1, 1), so parametric equations are x = t, y = t, z = 1 + t, and symmetric equations are
x = y =z- 1.
47. F(x, y , z) = xy + yz + zx, 'V F (x, y, z) = (y + z , x + z, y + x), 'V F{1, 1, 1) = (2, 2, 2), so an equation of the tangent
plane is 2:r; + 2y +2z = 6 or
x+ y + z = 3, and the normal line is given by x - 1 = y - 1 = z - 1 or x = y = z. To graph
3 - xy
the surface we solve for z : z = - -- .
x+y
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218
CHAPTER 14 PARTIAL DERIVATIVES
D
49. f(x, y) = xy => '\1 f(x , y) = (y, x}, '\1 !(3, 2) = (2, 3). '\1 !(3, 2)
is perpendicular to the tangent line, so the tangent line has equation
=> (2, 3) · (x- 3, x- 2) = 0 =>
'\1](3, 2) · (x - 3, y- 2} = 0
2(x - 3) + 3(y - 2) = 0 or 2x + 3y = 12.
I 2xo 2yo 2zo J
.
.
51. '\1F(xo, yo, zo) = \ (i2 • 1)2• C2 . . Thus an equatiOn oflhe tangent plane at (x 0 , y0 , zo) 1s
() .
. .
. .
2xo
2yo
2zo
( x5 y~
z~ )
(i2 x + 1)2 Y + C2 z = 2 a 2 + b2 + c2 = 2 1 = 2 smce (xo, yo, zo) IS a pomt on the elhps01d. Hence
xo
· Yo
zo
.
.
2 x + b2 y + 2 z = 1 1s an equation o
a
c
(
53. '\1 F xo, yo, zo
fth
e tangent p1ane.
2xo 2yo - 1 J '
.
. 2xo 2yo 1 2x5 2y5 zo
) = I\ (i2
• 1)2, c ,so an equatiOn of the tangent plane 1s (i2 x -1: 1)2 y - cz = (i2 + 1)2 - c
2
2
2
2
2xo
2yo
z
(x
y )
zo
zo
x
y
.
.
or - 2 x + -b2 y =- + 2 -% + b~ - - . But - = -% + b~, so the equat1on can be written as
a
c
a
c
c
a
I
2xo
2yo
z + zo
- 2 x+ - 2 y = - - .
a
b
c
55. The hyperboloid x 2 - y 2 - z 2 = 1 is a level surface of F(x, y, z) = x 2 - y 2 - . z2 and '\1 F (x, y , z) = (2x, -2y, -2z) is a
nonnal vector to the surface and hence a nonnal vector for the tangent plane at (x, y, z). The tangent plane is parallel to the
plane z = x + y or x + y- z = 0 if and only if the corresponding normal vectors are parallel, so we need a point (xo, yo, zo)
on the hyperboloid wh~re (2xo, -2yo, - 2zo) = c (1, 1·, ....,-1) or equivalently (xo, - yo, -zo) = k (1, 1, -1) for some k ¥: 0.
Then we must have x o = k , yo = - k , zo = k and substituting into the equation of the hyperboloid gives
k2 -
( -k)
2
~ k2 = 1 {:}
- k 2 = 1, an impossibility. Thus there is no such point on the hyperboloid.
57. Let (xo, y 0 , z 0 ) be a point on the cone [other than {0, 0, 0)]. The cone is a level surface of F(x, y, z) = x 2 + y 2 - z 2 and
'\1 F(x, y, z) = (2;t, 2y, - 2z), so '\1 F(xo, yo, zo) = (2xo, 2yo, -2zo) is a nomial vector to the cone at this point and an
equation of the tangent plane there is 2xo (x - xo) + 2yo (y - y0 ) - 2zo (z - zo) = 0 or xox + yoy - zoz = x~ + y5- z5.
But x& + v5 =
za so the tangent plane is given by xox + YOY - ZoZ = 0, a plane which always contains the origin.
59. Let F(x, y , z) = x 2 + y 2 - z. Then the paraboloid is the level surface F (x, y, z) = 0 and '\1 F (x , y ; z) = (2x,2y, -1), so
'\1 F(1, 1, 2) = '(2, 2,-1) is a nonnal vector to the surface. Thus the normal line at (1, 1, 2) is given by x = 1 + 2t,
y = 1 + 2t, z = 2- t. Substitution into the equation of the paraboloid z = x 2 + y~ gives 2- t = {1 + 2t)2 + {1 + 2t?
2 - t = 2 + 8t + Bf
{:} 8t2 + 9t = 0 {:}
t(Bt + 9) = 0. Thus the line intersects the paraboloid when t = 0,
corresponding to the given point (1, 1, 2), or when t = - ~,corresponding to the point (-%, -~, ¥).
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{:}
SECTION 14.6
DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
0
219
61. Let (xo, y0 , zo) be a point on the surface. Then an equation of the tangent plane at the point is
y
2..;Yo
X
-- + -- +
2y'Xo
Z
JXO+Jy0 + /Zo
2yZo
2
r.;:: =
-=-- + ..;Yo + ~zo .;c.
_]f_
y'Xo
v
=
r;.
. ·
·
. But JXo + ..;Yo + v'Zo = v c, so the equation IS
The x-, y-, and z-intercepts are foO,
-ICYo and .jCZO respectively. (The X- intercept is found by
setting y = z = 0 and solving the resulting equation for x, and they- and z-intercepts are found similarly.) So the sum of the
intercepts is .jC( JXo + ffo + v'Zo) = c, a constant.
63. If f(x, y, z ) = z - x 2 - y 2 and g(x, y, z) = 4x2 + y 2 + z 2 , then the tangent line is perpenoicular to both "il f and "ilg
at ( - 1, 1, 2). The vector v = "il f x "ilg will therefore be parallel to the tangent line.
We have "il f(x , y, z ) = ( - 2x, -2y, 1)
=?
"il f (- 1, 1, 2) = (2, - 2, 1), and "il g(x, y , z ) = (8x, 2y, 2z)
j
"Vg(- 1, 1, 2)= (-8, 2,4).Hencev="Vf x "Vg =
=?
k
2 - 2 1 = - 10 i-16 j-12k.
-8
2 4
Parametric equations are: x = -1 - lOt, y = 1 - 16t, z = 2 - 12t.
65. (a) The direction of the normal line ofF is given by "il F, and that of G by "VG. Assuming that
"il F -1= 0 -1= "VG, the two normal lines are perpendicular at P if "il F · "VG = 0 at P
(8Fj8x, 8F/8y,8F/8z) · (8G j 8x, 8G/ 8y, 8G/ 8z ) = 0 at P
#
-¢:?
F_..,G, + FvGv + FzGz = 0 at P.
(b) Here F = x 2 + y 2 - z 2 and G = x 2 + y 2 + z 2 - r 2 , so
"il F · "VG = (2x, 2y, - 2z) · (2x, 2y, 2z) = 4x2 + 4y2 - 4z 2 = 4F = 0, since the point (x , y, z ) lies on the graph of
F = 0. To see that this is true without using calculus, note that G = 0 is the equation of a sphere centered at the origin and
F = 0 is the equation of a right circular cone with vertex at the origin (which is generated by lines through the origin). At
any point of intersection, the sphere's normal line (which passes through the origin) lies on the cone, and thus is
perpendicular to tl1e cone's normal line. So the surfaces with equations F = 0 and G = 0 are everywhere orthogonal.
67. Let u = (a , b) and v = (c, d). Then we know that at the given point, Du f = "il f · u = afx + bfv and
Dv I = "il I · v = clx + dfv· But these are just two linear equations in the two unknowns fx and fv, and since u and v are
not parallel, we can solve the equations to find "V f = (f.,, f v) at the given point. In fact,
"Vf =
I d Du f - b Dv f , a Dv f - c D u f ) .
\
ad - be
ad - be
@) 2012 CcnJ:!ISC Lc:uning. All Rights Rosel"\·ed. May nol be SC4Med, cop k-d, or duplicated, or postod lo n publicly accessible wobsite, in wbolc or in port.
220
D
14.7
CHAPTER 14
PARTIAL DERIVATIVES
Maximum and Minimum Values
1. (a) First we compute D(1, 1) = f,x(1, 1) fvv(1, 1)- [f:cy(1, 1W = (4)(2) -
(1? = 7. Since D(1, 1) > 0 and
f o:x (1, 1) > 0, f has a local minimum at (1, 1) by the Second Derivatives Test.
2
(b) D(1, 1) = f.xx (1, 1) fuy(1, 1)- [fxu(1, 1W = (4)(2)- (3) = - 1. Since D(1, 1) < 0, f has a sadd le point.at (1, 1) by
'
the Second Derivatives Test.
3. In the figure, a point at approximately (1 , 1) is enclosed by level curves wh'ich are oval in shape and ind icate that as we move
away from the point in any direc~ion the values off are increasing. Hence we would expect a local minimum at or near (1, 1) .
The level curves near (0, 0) resemble hyperbolas, and as we move away from the origin, the values off increase in some
directions and decrease in others, so we would expect to find a saddle point there.
To verify our predictions, we have f(x, y) = 4 + x 3 + y 3 - 3xy
=> f x (x , y) = 3x 2 - 3y, f v (x, y) = 3y 2 - 3x. We
have critical points where these partial derivatives are equal to 0: 3x2 - 3y = 0, 3y
first equation into the second equation gives 3(x2 ) 2 - 3x = 0
2
-
2
3x = 0. Substituting y = x from the
=> 3x(x 3 - 1) = 0 => x = 0 or x = 1. T hen we have
two critical points, (0, 0) and (1, 1). The second partial derivatives are f xx (x, y) = 6x, fxu(x, y) = -3, and fvu(x, y) = 6y,
'
'
2
2
so D(x, y) = fxx(x , y) fuv(x, y)- [fxu(x, y)] = (6x)(6y)- (-3) = 36xy- 9. Then D(O, 0) = 36(0)(0) - 9 = - 9,
and D (1, 1) = 36(1)(1) - 9 = 27. Since D(O, 0) < 0, f has a saddle point at (0, 0) by the Second D erivatives Test. Since
D (1, 1) > 0 and fxx(1, 1) > 0, f has a local minimum at (1 , 1).
5. f(x, y) = x 2 + x y + y 2 + y
=> f x = 2x + y, fv = x + 2y + 1, fxx = 2, fx 11 = 1, fvv = 2. Then fx = 0 implies
y=-2x,andsubstitutionintofy=x + 2y + 1=0 gives x+2(~2x) +1=0
=> -3x=-1 => x = i·
Then y =-~and the only critical point is (t, -~).
D(x, y) = fxxfvv - (fru) 2 = (2)(2) - (1? = 3, and since_
D(~, - D = 3 > 0 and fxx(i, -~) = 2 > 0, !(~ , -~) = -~ is a local
minimum by the Second Derivatives Test.
7. f(x,y) = (x- y)(1- xy) = x - y - x 2 y+ xy 2
=> f x = 1 - 2xy +y2 , fv = - 1- x 2'+ 2xy, fx:n = -2y,
f xv = - 2x + 2y, fvv = 2x. Then fx = 0 implies 1 - 2xy + y 2 = 0 and / 11 = 0 implies -1 - x 2 + 2xy = 0. Adding the
two equations gives 1 + y 2 - 1- x 2 = 0
=>
y2 = x2
=>
y = ±x, but ify = -x then f x = 0 implies
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SECTION 14.7
1 '+ 2x 2 + x 2 = 0
=}
MAXIMUM AND MINIMUM VALUES
0
221
3x2 = -1 which has no real solution. l fy = x
then substitution into f, = 0 gives 1 - 2x 2 + x 2 = 0
=}
x
2
= 1 =}
x = ±1, so the critical points are (1, 1) and ( -1, -1). Now
D (1, 1) = ( - 2)(2) - 02 = - 4 < 0 and
D( - 1, - 1) = (2)( -2)- 0 2 = -4 < 0, so (1, 1) and ( -1, - 1) are
saddle points.
9. f(x, y) = y 3 + 3x y - 6x
2
2
-
6y 2 + 2
=}
f ,. = 6xy - 12x, ! 11 = 3y 2 + 3x2 - 12y, f x:t = 6y- 12, f xu = 6x,
fw = 6y - 12. Then f z = 0 implies 6x(y- 2) = 0, sox = 0 or y = 2. If x
3y2 - 12y = 0
3y(y- 4) = 0
=}
=}
= 0 then substitution into j y = 0 gives
y = 0 or y = 4, so we have critical points (0, 0) and (0, 4). If y = 2,
substitution into jy = 0 gives 12 + 3x2 - 24 = 0
=}
x2 = 4
=}
x = ±2, so we have critical points (±2, 2).
D(O, 0) = (- 12)( -12) - 02 = 144 > 0 and fu(O, 0) = - 12 < o•.so
f(O, 0) = 2 is a local maximum. D{O, 4) = (12){12) - 02 = 144 > 0
and f xx(O, 4) = 12 > 0, so f(O , 4) = -30 is a local minimum.
D( ±2, 2) = {0)(0)- {±12) 2 = -144 < 0, so (±2, 2) are saddle po ints.
11. f(x, y) = x 3 - 12xy + _8y
3
=}
y
fx = 3x 2 - 12y, f u = - 12x + 24y2 , fxx = 6x, fx 11 = - 12,/1111 = 48 y . Then fx = 0
implies x 2 ·= 4y and / 11 = 0 implies x = 2y 2 • Substituting the second equation into the first gives {2y 2 } 2 = 4y
4y 4 = 4y
=}
4y(y 3 - 1) = 0
=}
=}
y = 0 or y = 1. lfy = 0 then
x = 0 and if y = 1 then x = 2, so the critical points are (0, 0) and {2, 1).
D(O, 0) = {0){0) - ( - 12) 2 = - 144 < 0, so (0, 0) is a saddle point.·
D(2, 1) = (12)(48)- ( -12? = 432 > 0 and fxx(2 , 1) = 12 > 0 so
!(2, 1) = -8 is a local .minimum .
13. f(x,y)=e"'cosy
=}
f x=e"'cosy, / 11 =-e"'siny.
Now f a: = 0 implies cosy = 0 or y = f + mr for nan integer.
But sin ( f + mr) =1- 0, so there are no critical points.
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-4
222
D
CHAPTER 14
PARTIAL DERIVATIVES
l" = ( x 2 + y 2) e'~~ 2 - :r2 ( - 2x) + 2xe'~~2 -:r2 = 2xe'~~2 - :r 2 (1 - x. 2 - y 2 ),
fzx = 2xe'fl2 - o:2 ( - 2x) + (1- x 2 - y 2 ) ( 2x ( -2xe'~~2 -:r2) + 2e'fl2 - :r2) = 2e'~~2 -:r2 ((1- x 2 - y 2 )(1 - 2x2 ) - 2x2 ),
22
22
2
2
22
f xv = 2xe'11 - o: (- 2y) + 2x(2y)eY -:r (1 - x - y ) = ~4xyeY - x (x2 + y 2 ),
f 11 v = 2yeV2 - o:2 (2y) + (1 + x 2 + y 2 ) ( 2y ( 2yeV2 -:r2) + 2eY2 - :r 2 ) = 2e 712 - :r2 ((1 + x 2 + y 2 )( 1 + 2y 2 ) + 2y 2 ).
fv = 0 implies y = 0, and substituting into fx = 0 gives
2
2xe-"' (1 - x 2 ) = 0
~ x = o.or x = ± 1. Thus the critical points are
(0, 0) and (±1, 0). Now D(O, 0) = (2)(2) - 0 > 0 and f :r:x (O , 0) = 2 > 0,
so f(O, 0) = 0 is a local minimum. D(±l, 0) = (- 4e- 1 )(4e- 1 ) - 0 < 0
so (±1, 0) are saddle points.
17.f(x,y)=y2 - 2ycosx
~
j ; =2ysinx,fv=2y -2cosx,
fxx = 2y cos x, f ., 11 = 2 sin x, f v11 = 2. Then/:. = 0 implies y = 0 or
sinx = 0
~
x = 0, 1r, or 211" for -1 :-::; x :-::; 7. Substituting y = 0 into
f v = 0 gives cosx = 0
~
x = ~or 3; , substituting x = 0 or x = 21r
into f 71 = 0 gives y = 1, and substituting x = 1r into fv = 0 gives y = - 1.
Thus the critical points are (0, 1), ( ~. 0) , (1r, -1), ( 3; , 0), and (21r, 1).
D (~ , 0) = D
e
2"", 0) =
-4 < 0 so (~, 0) and
e;,0) are saddle points. D(O, 1) = D(1r, -1) = D(21r, 1) = 4 > 0 and
fa:x(O , 1) = fxx('Tr, - 1) = fxo:(27r, 1) = 2 > 0, so f(O, 1) = j(1r, -1) = f(27r, 1) = - 1 are local minima.
19. f(x, y) = x 2 + 4y2 - 4xy + 2 ~
fx = 2x- 4y, fv =By - 4x, f x:r: = 2, f, v = -4, / 1171 = 8. Then fx = 0
and fv = 0 each implies y = ~x, so all points of the form (xo, ~xo) are critical points and for each ofth~se we have
D(xo, ~xo) = (2)(8) - ( -4) 2 = 0. The Second Derivatives Test gives no information, but
f(x,y) = x 2 +4y2 - 4xy + 2 = (x- 2y} 2 + 2 ;:: 2 with equality if and only ify = ~x. Thus f(x 0 , ~xo) = 2 are all local
(and absolute) minima.
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SECTION 14.7
MAXIMUM AND MINIMUM VALUES
0
223
8·
7
8
6
4
2
6
z 5
-2.5
2.5
4
3
2
- 2.5
-2
- I
0
2
y
From the graphs, there appear to be local minimaofabout/(1, ±1) = f( - 1,±1) ~ 3 (and no local maxima or saddle
points). f, = 2x - 2x- 3 y- 2 , fv = 2y - 2x- 2 y- 3 , f xx = 2 + 6x- 4 y - 2 , fx11 = 4x- 3 y- 3 , / 1111 = 2 + 6x- 2 y-4 • Then
f, = 0 implies 2x4 y 2 - 2 = 0 or x 4 y 2 = 1 or y 2 = x - 4 • Note that neither x nor y can be zero. Now f v = 0 implies
2x2 y 4 - 2 = 0, and with y 2 = x- 4 this implies 2x- 6 - 2 = 0 or x 6 = 1. Thus x = ±1 and if x = 1, y = ± 1; if x = - 1, ·
y = ±1. So the critical points are (1, 1), (1, - 1),( - 1, 1) and ( - 1, -1). Now D(1, ± 1) = D( -1, ±1) = 64 - 16 > 0 and
fxx
> 0 always, so f(1, ± 1) = f (-1, ± 1) = 3 are local minima.
23. f(x,y) = sin x +siny+sin(x +y), 0 :S: x :S: 2rr, 0 :S: y :S: 2rr
y
From the graphs it appears that f has a local maximum at about (1, 1) with value approximately 2.6, a local minimum
at about (5, 5) with value approximately -2.6, and a saddle point at about (3, 3).
f :z: = cos x + cos(x + y), fv = co:;y + cos(x + y), f xx = - sin x- sin(x + y ), / 11 y = - siny- sin(x + y),
f xy
= - sin(x + 1J). Setting fx = 0 and / 11 = 0 and subtracting gives cos x - cos y = 0 or cos i = cos y. Thus x = y
or x = 2rr - y. l fx = y, f , = 0 becomes cosx + cos 2x = 0 or 2 cos2 x + cosx - 1 = 0, a quadratic in cosx. Thus
cos x = - 1 or ~ and x = rr, i , or r; , giving the critical points (rr, rr ), ( i, i) and ( 5; , 5; ) . Similarly if
x = 2rr - y , f :z: = 0 becomes (cos x) + 1 = 0 and the resulting critical point is (rr, rr). Now
D(x, y) = sin x sin y + sinx sin(x + y) +sin y sin(x + y) . So D(1r, 1r) = 0 and the Second Derivatives Test doesn't apply.
However, along the line y = x we have f(x , x) = 2sin x + sin2x = 2 sinx + 2sin x cos x = 2sin x (1 + cosx), and
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224
0
CHAPTER 14
PARTIAL DERIVATIVES
f(x , x) > 0 for 0 < x < 1r while f(x , x) < 0 for 1r < x < 21r. Thus every disk with center (1r, 1r) contains points where f is
positive as well as points where f is negat~ve, so the graph crosses its tangent plane (z = 0) there and (1r , 1r) is a saddle point.
D(~, ~) =
*> 0 and fxx (~,·?f) < 0 so/(~,~) = .tf! is a local maximum while D(
f xx ( 5; , 5; ) > 0, so f ( 63" , 6; ) =
5
6
3", ; ) =
~ > 0 and
-¥ is a local minimum.
4x(x2 - 2y) = 0, sox= 0 or x 2 = 2y. lfx = 0 then substitution into f 11 = 0 gives 4y 3 = -2
:::::}
•
1
y = - 1i2, so
( 0, - ~) is a critical point. Substituting x 2 = 2y into fv = 0 gives 4y3 - 8y + 2 = 0. Using a graph, solutions are
approximately y = - 1.526, 0.259, and 1.267. (Alternatively, we could have used a calculator or a CAS to find these roots.)
We have x 2 = 2y
=>
y = 0.259
x ~ ± 0.720 andy = ,1.267
=>
x = ±.J2Y, soy = -1.526 gives no real-valued solution for x, but
=>
x ~ ±1.592. Thus to three decimal places, the critical points are
( 0, -~) ~ (0, - 0.794), (±0.720, 0.259), and (±1.592, 1.267). Now since f xx = 12x2 - 8y, f ., 11 • = - 8x, f 1111 = 12y2 ,
and D = (12x 2 - 8y)(12y2 ) - 64x2 , we have D(O, -0.794) > 0, fx x(O, - 0.794) > 0, D(±0.720, 0.259) < 0,
D(±1.592, 1.267) > 0, and f xx (± 1.592, 1.267) > 0. Therefore /(0, _- 0.794) ~ - 1.191 and /(±1.592, 1.267) ~ - 1.310
are local minima, and (± 0.720, 0.259) are saddle points. There is no highest point on the graph, but the lowest points are
approximately (±1.592, 1.267, -1.310).
20
10
0
-6
y
27. f(x , y) = x 4 +y3 - 3x2 +y2 +x - 2y + 1
=> f x (x , y) = 4x 3 - 6x + 1 and f 11 (x , y) = 3y2 + 2y- 2. From the
graphs, we see thatto three decimal places, f., = 0 when x ~ -1.301, 0.170, or 1.131, and / 11 = 0 when y ~ -1.215 or
0.549. (Alternatively, we could have used a calculator or a CAS to find these roots. We could also use the quadratic formula to
find the solutions of / 11 = 0.) So, to three decimal places, f has critical points at ( - 1.301, - 1.215), ( - 1.301, 0.549),
(0.170, -1.215), (0.170, 0.549), (1.131, - 1.215), and (1.131, 0.549). Now since / zz = 12x2 - 6, fx 11.= 0, f 11 y = 6y + 2,
and D = (12x2 - 6)(6y + 2), we have D( -1.301, -1.215) < 0, D( - 1.301, 0.549) > 0, f xx( -1.301, 0.549) > 0,
D(0.170, - 1.215) > 0, f xx (0.170, - 1.215) < 0, D(0.170, 0.549) < 0, D(l.131, -1.215) < 0, D(l.131, 0.549) > 0, and
/ xz (l.131, 0.549) > 0. Therefore, to three decimal places, f( -1.301, 0.549) ~ -3.145 and /(1.131, 0.549) ~ - 0.701 are
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SECTION 14.7
MAXIMUM AND MINIMUM VALUES
0
225
local minima, !(0.170, - 1.215) ::::: 3.197 is a local maximum, and ( -1.301, -1.215), (0.170, 0.549), and (1.131 , -1.215)
are saddle points. There is no highest or lowest point on the graph.
3
z
29. Since f is a polynomial it is contint,Jous on D, so an absolute maximum and minimum exist. Here f x = 2x- 2, jy = 2y, and
setting fx = jy = 0 gives (1, 0) as the only critical point (which is inside D), where f (1, 0) = - 1. Along £ 1 : x = 0 and
f(O, y) = y 2 for -2 ~ y ::=:; 2, a quadratic function which attains its minimum at y = b, where f( O, 0) = 0, and its maximum
·aty = ±2, wheref(0,±2) = 4. Along £2: y = x- 2 forO ::=:; x ~ 2, and f(x,x- 2) = 2x 2 - 6x + 4 = 2 (x _ ~) 2 _ ~ .
a quadratic which attains its minimum at x =
£, where f (~, - ~) = - ~, and its maximum at x = 0, where f (0, - 2) = 4.
y
Along £3: y = 2 - x for 0 ~ x ::::; 2, and
f(x, 2- x) = 2x 2 - 6x + 4 = 2(x- ~) - ~.a quadratic which attains
(0,2)
its minimum at x =~~where f(~, ~) =-~,and its maximum at x = 0,
L,
2
(2,0)
X
where f (O, 2) = 4. Thus the absolute maximum off on D is f(O, ±2) = 4
and the absolute minimum is !(1, 0) = -1.
(0, - 2)
31. f x(x,y) = 2x +2xy, jy(x,y) = 2y +x2 , and setting f.,= f" = 0
gives (0, 0) as the only critical point in D , with .f(O, 0) = 4.
On £ 1: y = -1, j(x, - 1) = 5, a constant.
On £2: x = 1, /(1, y) = y 2 + 11 + 5, a quadratic in y which attai ns its
y
L,
(-1 , 1)
L•
(I, I)
L2
0
(-1,-1)
X
L,
(1, -1)
maximum at (1, 1), /(1 , 1) = 7 and its minimum at (1, - ~) , f(1 , - t) =.!f .
On £ 3: f(x , 1) = 2x 2 + 5 which attains its maximum at ( -1, 1) and (1, 1)
with f(±l , 1) = 7 and its minimum at (0, 1), f(O, 1) = 5.
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226
D
CHAPTER 14 PARTIAL DERIVATIVES
On .i4 : f( -1, y)
= y 2 + y + 5 with maximum at ( -1, 1 ), f( - 1, 1) = 7 and minimum at ( -1, -D. f( - 1, -~) = Jf.
Thus the absolute maximum is attained at both (±1, 1) with /(±1, 1) = 7 and the absolute minimum on Dis attained at
(0, 0) with f(O, 0) = 4.
33. f(x , y) = x 4 +y 4 - 4xy + 2 is a polynomial and hence continuous on D, so
it has an absolute maximum and minimum on D. fx(x, y) = 4x 3 - 4y and
Y
(0, 2)+ - - -L
- -'3'-------1
(3, 2) .
fv(x, y) = ·4y 3 - 4x; then f x = 0 implies y = x 3 , and substitution into
fv = 0
=>
x = y 3 gives x 9 - x = 0
=>
x(x8 - 1) = 0
=> x = 0
or x = ±1. Thus the critical points are -(0 , 0), (1, 1), and ( - 1, - 1), but only
(0,0)
(3,0)
X
(1, 1) with f(1, 1) = 0 is inside D . On £1: y = 0, f(x, 0) = x 4 + 2,
0 ~ x ~ 3, a polynomial in x which attains its maximum at x = 3, !(3, 0) = 83, and its minimum at x = 0, f (O, 0) = 2.
On £ 2 : x = 3, /(3, y) = y 4 - 12y + 83, 0 ~ y ~ 2, a polynomial in y which attains its m inimum at y = {/3,
/(3, {13) = 83- 9 .{/3 ~ 70.0, and its maximum at y = 0, ! (3, 0) = 83.
On L 3 : y = 2, f( x, 2) = x 4 - Bx + 18, 0 ~ x ~ 3, a polynomial in x which attains its minimum at x = q'2,
f( q'2, 2) = is- 6 ij2 ~ 10.4, and its maximum at x = 3, /(3, 2) = 75. On £ 4: x = 0, f(O, y) = y 4 + 2, 0 ~ y ~ 2, a
polynomial in y which attains its maximum at y = 2, f(O , 2) = 18, and its minimum at y = 0, f(O, 0) = 2. Thus the absolute
maximum off on Dis /(3, 0) = 83 and the absolute minimum is /(1, 1) = 0.
35. f,(x, y) = 6x 2 and fv(x, y) = 4y3 • And so f x = 0 and fv = 0 only occur when x = y = 0. Hence, the only critical point
+ y 2 = 1, y2 = 1 - x 2 so let
g(x) = f(x, y) =·2x 3 + (1- x 2 ) 2 = .x 4 + 2x 3 - 2x2 + 1,-1 ~ x ~ 1. Then g'(x) = 4x3 + 6x 2 - 4~ = 0 =>
.inside the disk is at x = y :::o 0 where /(0, 0) = 0. Now on the circle x
2
x = 0,
-2, or ~ - /(0,±1) = g(~) = 1, !(~, ±1 ) = g(~) = ~· and ,(-2,-3) is not in D. Checking the endpoints, we get
.f(-1, 0) = g( -1) = -2 and f(1 , 0) = g(1) = 2. Thus the absolute maximum and minimum off on Dare /(1, 0) = 2 and
f( - 1,0) = - 2.
A nother method: On the boundary x 2 + y 2 = 1 we can write x = cos (;I, y = sin 8, so f (cos 8, sin 8) = 2 cos3 8 + sin4 (;I,
0 ~ () ~ 27T.
37. f(x, y) = - (x 2 - 1) 2 - (x 2 y- x - 1) 2
=> f,(x, y) = - 2(x 2 - 1)(2x) - 2(x 2 y- x - 1)(2xy- 1) and
jy(X, y) = -2(x2 y - x - 1)x2 . Setting j y(x, y) = 0 gives either x = 0 or x 2y - x- 1 = 0.
There are no critical points for x = O,.since f:.,(O, y) = .:_2, so we set x 2y - x- 1 = 0
1
so f , ( x, x~ )
{:}
y= x~
X
1
[x =/= 0],
= ·-2(x2 - 1){2x)- 2(x 2 x ~ 1 - x - 1) (2x x ~ 1 - 1) = -4x(x 2 - 1). Therefore
f,(x, y) = fv(x, y)- = 0 at the points (1, 2) and ( -1, 0). To classifY these critical points, we calculate
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MAXIMUM AND MINIMUM VAL4ES
SECTION 14.7
0
227
f :u:(x, y) = - 12x 2 - 12x2 y 2 + 12xy + 4y + 2, f vv(x, y) = - 2x",
and f zv(x , y) = -8x3 y + 6x 2 + 4x. In order to use the Second Derivatives
Of-----1~
Test we calculate
- I
D(-1,0) = fxx(-1 , 0) / 7171 (-1,0) - [f:c 71 (- 1 , 0)]~ = 16 > 0,
-2
/ :r:c ( - 1, 0) = -10 < 0, D (1, 2) = 16 > 0, and f:r:c{ 1, 2) = -26 < 0, so
both ( -1, 0) and (1, 2) give local maxima.
X
39. Let d be the distance fr~m {2, 0, -3) to any point (x, y, z ) on the plane x + y + z = 1, sod= J(x - 2) 2 + y 2 + (z + 3)2
where z
= 1- x- y, and we minimize d2 ~ f(x, y) = (x - 2) 2 + y 2 + {4 - x- y) 2 • Then
j;,(x,y) = 2(x- 2) + 2{4- x - y)( -1) = 4x + 2y -12, f 71 (x,y) = 2y + 2(4 - x- y){-1) = 2x + 4y - 8. Solving
4x + 2y- 12 = 0 and 2x + 4y- 8 = 0 simultaneously gives x = ~. y = ~.so the only critical point is(~,~). An absolute
minimum exists (since there is a minimum di.s tance from the point to the plane) and it must occur at a critical poi nt, so the
shortest distance occurs for x = ~. y
= ~ for which d =
J(~ -
2)
2
+ (j) + (4 - ~ - ~) 2 = ji = ~2
41. Let d be the distance from the point {4, 2, 0) to any point {x, y, z) on the cone, sod = ,j(x - 4)2 + (y- 2)2 + z2 where
z 2 = x 2 + y 2 , and we minimize d 2 = (x - 4) 2 + (y - 2) 2 + x 2 + y 2 = f(x, y). Then
fz(x, y) = 2 (x - 4) + 2x = 4x - 8, f 71 (x , y) = 2 (y - 2) + 2y = 4y- 4, and the critical points occur when
f., = 0
=> x = 2, fu = 0 => y = 1. Thus the only critica l point is (2, 1). An absolute minimum exists (since there is a
minimum distance from the cone to the point) which must occur at a critical point, so the points on the cone closest
to ( 4, 2, 0) are (2, 1, ±v'5).
43. x + y + z = 100, so maximize f(x , y) = xy(l OO - x- y) .
f x = 100y- 2xy- y 2 , f 71 = 100x - x 2 - 2xy,
f, :r. = - 2y, f vu = -2x, fxv = 100- 2x- 2y. Then J,r; = 0 implies y = 0 or y = 100- 2x. Substituting y = 0 into
fv = 0 gives x = 0 or x = 100 and substi~ting y = 100 - 2x into fv = 0 gives 3x2 - 100x = 0 sox = 0 or 1 ~ .
1 0
1 0
Thus the critical points are (0, 0), (100, 0), (0, 100) and ( ~ , ~ ) .
D(O, 0) = D{100, 0) = D {O, 100) = -10,000 while D(1~ 0 , 1 ~ 0 ) = 10·~ 00 and f xx (1~0 , 1 ~0 ) = - 2 ~ < 0. Thus (0, 0),
(100, 0) and {0, 100) are saddle points whereas f( 1 ~ 0 , 1 ~0 ) is a local maximum . Thus the numbers are x = y = z =
45. Center the sphere at the origin so that its equation is x 2 + y 2
1 0
~ .
+ z 2 = 1·2 , and orient the inscribed rectangular box so that its
edges are parallel to the coordinate axes. Any vertex of the box satisfies x 2 + y 2 + z 2 = r 2 , so take (x, y, z) to be the vertex
in the fi rst octant. Then the box has length 2x, width 2y, and he ight 2z = 2 ,jr2 - x 2 - y 2 with volume given by
V(x , y) = (2x)(2y)(2 Jr 2 - x 2 - y 2 ) = 8xy Jr 2 - x 2 - y 2 forO < x < r, 0 < y < r. Then
V., = (Bxy) . ~(r2- x2- y2)-1 f2( -2x) + ,jr2 - x2 - y2. By=
2
Setting Vx = 0 gives y = 0 or 2x
8y(r2 - 2x 2 - y 2 ). ·and Vv = 8x(r 2 - x 2 - 2y 2 ) .
,jr2 _ x2 _ y2
. J r2 _ x2 _ y2
+ y 2 = r 2 , but y > 0 so only the latter solution applies. Similarly, V71 = 0 with x > 0
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228
0
CHAPTER 14
implies x
2
3x = r
2
PARTIAL DERIVATIVES
+ 2y2 = r 2 . Substituting, we have 2x 2 + y2 = x 2 + 2y2
2
::::>
x 2 = y2
::::>
y = x. Then x 2 + 2y2 = r 2
::::>
'*
x = .jr"i]3 = r I vf3 = y. Thus the oni,Y critical point is (r I vf3, r I ,/3 ). There must be a maximum
volume and here it must occur at a critical point, so the maximum volume occurs when x = y = r 1V3 and the maximum
.47. Maximize f(x, y) = x; (6- x- 2y), then the maximum volume is V = xyz .
f ., = i(6y ~ 2xy- y 2 ) = h(6- 2x- 2y) and fv = ~x (6 - x - 4y). Setting fx =:= 0 and j 11 = 0 gives the critical point
(2, 1} which geometrically must give a maximum. Thus the volume of the largest such box is V = (2)(1) ( ~) =
t·
49. Let the dimensions be x, y, and z; then 4x + 4y + 4z = c and the volume is
V = xyz = xv(ic- x - y) = ic:cy- x 2 y - xy 2 , x > 0, y > 0. Then V, = ~cy- 2xy- y 2 and Vy = ~ex- x 2 - 2xy,
so Vx = 0 = Vy when 2x + y = ic and x + 2y =~c. So lving, we get x =f-ie, y = 112 c and z = ic- x- y = f-ie. From
the geometrical nature of the ·problem, thi s critical point must give an absolute maximum. Thus the box is a cube with edge
length -f2c.
51. Let the dimensions be x, y and z, then minimize xy + 2(xz + yz) if xyz = 32,000 cm3 • Then
f(x,y) = xy + [64,000(x + y)fxy] = xy + 64,000(x- 1 + y- 1 ), fx = y- 64,000x- 2 , f v = x- 64,000y- 2 .
And fx = 0 implies y = 64,000/x2 ; substituting into / y = 0 implies x 3 = 64,000 or x = 40 and then y = 40. Now
D(x, y) = [(2)(64,000Wx - 3y- 3 - 1 > 0 for ( 40, 40) and 'f.,, (40, 40) > 0 so this is indeed a minimum. Thus the
dimensions of the box are x = y = 40 em, z = 20 em.
53. Let x, y, z be the dimensions of the rectangular box. Then the volume of the box is xyz and '
L =
J xz + y2 + z2 ::::> L 2 = x 2 + y 2 + z 2 ::::> z = J L 2 - x 2 - y 2 .
Substituting, we have volume V(x,y) = xy JL2- x 2 - y 2 (x,y > 0).
2
Vy = xJL2 -x2 -y2 -
2
2
2
2
2
2
2
JP - x2 - y2 . V., =Oimpliesy(L - x -y )=x y ::::> y(L -2x -y ) =0 ::::>
2x 2 + y2 = L 2 (since y > 0), and
xy
v;1 = o ·implies x(L2 - x 2 - y 2 ) = xy2 ::::> x( L2 - x 2 - 2y2 ) = 0 ::::>
x 2 + 2y 2 = L 2 (since x > 0). Substituting y 2 = L 2 - 2x2 into x 2 + 2y2 = L 2 gives x 2 + 2L2 - 4x 2 = L 2 ::::>
3x2 = L 2
::::>
x = L/vf3 (since x > 0) f!nd then y =
JL2- 2(L/vf3) = L/vf3.
2
So the on ly critical point is (Lf,/3, Llvf3 ) which, from th~ geometrical nature of the problem, must give an absolute
maximum. Thus th~ maximum volume is V(LI,/3, Ll,/3) = (LI,/3)
2
j L2- (Livf3)
2
2
-
(LI ,/3) = L 3 I (3 ,/3)
cubic units .
© 2012 Cen,SD.Gc Learning. All Rir)lts Rcsco·cd. May not be scanned. cop ied. or duplicaLCd, or poster.! to a publicly accessible website, in whole or in pnrt.
SECTION 14.8
.
.
n.
·
n
n
n
.~ Yi = m i~l Xi + i~l b = m
1
229
:L: [y,- (mxi +b)]2 . Then /m. = i=l
:L: - 2xi [Vi - (mxi + b)J = 0
i =l
n
n
n
·
t. = l
\= 1
•=1
n
implies :L: (x•Y•- mx; - bxi) = 0 or .:L: XiYi = m .:L: x~ + b .:L: Xi and fb =
i= l
0
n
n
55. Note that here the variables are m and b, and f{m, b) =
LAGRANGE MULTIPLIERS
:L: -2[yi - (mxi +b)] = 0 implies
1.=1
(ni~t Xi) + nb. Thus we have the two desired equations.
n
n
n
.
i =l
i= l
i= l
:L: 2x~, fbb = :L: 2 = 2n and fmb = :L: 2Xi. And fmm{m, b) > 0 always and
Now fmm =
equations do indeed minimize
f= d;.
i = 1
14.8
Lagrange Multipliers
1. At the extreme val';les off, the level curves off just touch the curve g(x, y) = 8 with a common tangent line. (See Figure I
and the accompanying discussion.)
We can observe several such occurrenc~ on the contour map, but the level curve
f(x , y) = c with the largest value of c which still intersects the curve g(x, y) = 8 is approximately c =59, and the smallest
value of c corresponding to a level curve which intersects g(x, y) = 8 appears to be c = 30. Thus we estimate the maximum
value off subj ect to the constraint g(x j y) = 8 to be about 59 and the minimum to be 30.
3. f(x , y) = x 2 + y 2 , g(x, y) = xy = 1, and \1 f = >.. \lg
From the last equation, x =I= 0 andy =I= 0, so 2x = >.y
y2 = x2
=>
=>
=>
(2x, 2y) = (>..y, >..x), so 2x = >..y, 2y = >..x, and xy = 1.
>.. = 2xjy. Substituting, we have 2y = (2xjy) x
=>
y = ±x. But xy = 1, so X= y = ± 1 and the possible points for the extreme values off are {1, 1) and
( -1, - 1). Here there is no maximum value, since the constraint xy = 1 allows x or y to become arbil:(arily large, and hence
f(x , y) = x 2 + y 2 can be made arbitrarily large. The mi~imum value is !{1 , 1) = f( - 1, -1) = 2.
5. f(x, y) = y 2 - x 2 , g(x, y) = tx 2
+y = 1, and \1 f = >.\lg => {-2x, 2y) = (t>..x, 2>..y) , so -2x = t>.x, 2y = 2>..y,
2
and ix2 + y 2 = 1. From the first equation we have x{ 4 + >..) = 0
=>
x = 0 or>. = - 4. If x = 0 then the third equation
gives y = ±1. If), = - 4 then the second equation gives 2y = -8y
=>
y = 0, and substituting into the third equation,
we have x = ±2. Thus the possible extreme values off occur at the points (0, ± 1) and (± 2, 0). Evaluating fat these points,
we see that the maximum value is f{O, ± 1) = 1 and the minimum is !{±2, 0) = -4.
7. f(x, y, z) = 2x + 2y + z, g(x, y, z) = x 2 + y 2 + z 2 = 9, and \1 f = >.. \1g
2>.y = 2, 2)..z = 1, and x 2 + y 2 + z 2 = 9. The first three equations imply x =
2
the fourth equation gives
2
(l) + ( l) + (
2
\ )
2
© 2012 Ccngogc Learning. All Rights Rcsc:rvcd. May not be =
= 9 =>
9
- 2
4)..
= 9
=>
(2, 2, 1) = (2>..x, 2>..y, 2>.z), so 2>.x = 2,
l• ·l·
y =
and z =
\ .
2
~ut substitution into
=> ).. = ±t, so f has possible extreme values at
ned. copi~; or duplicated, or posted to a publicly occ~ssiblc website, in whole or in p3r1.
230
0
CHAPTER 14
PARTIAL DERIVATIVES
the points (2, 2, 1) ~nd ( - 2, - 2, - 1). The maximum value off on x 2 + y 2 + z 2 = 9 is f(2, 2, 1) = 9, and the minimum is
J( -2, -2, - 1) = - 9.
9. f(x, y, z) = xyz, g(x, y, z) = x
2
+ 2y 2 '+ 3z2 = 6. V f = >.Vg
=:.
(yz, xz, x y) = .>. {2x, 4y, Gz). If any ofx, y, or z is
zero then x = y = z = 0 which contradicts x 2 + 2y2 + 3z2 = 6. Then.>.= (yz) /(,2x) = '(xz) / (4y) = (xy)/(6z ) or
x 2 = 2y 2 and z 2 = ~y 2 . Thus x 2 + 2y2 + 3z 2 = 6 implies 6y 2 = 6 or y = ±1. Then the possible points are
/j), (.,/2, ±1, -/j), (-.,/2, ± 1, /j), (- .,/2,±1, -/j). The maximum valu~ off on the ellipsoid is
-J:i, occurring when all coordinates are positive or exactly two are negative and the minimum is -7a occurring when 1 or 3 of
( .,/2, ±1,
the coordinates are negative.
11. f(x ,y,z) = x 2 +y2 +z2 , g(x,y, z) = x~ +y4 + z 4 = 1 =?
Vf = (2x,2y,2z)~ >.Vg = (4>.x 3 , 4>.y3 ,4>.z3 ).
Case 1: lf x f. 0, y f. 0 and z f. 0, then V f = >.Vg implies.>. = l /(2x2 ) = 1/(2y 2 ) = l /(2z 2 ) or x 2 = y2 = z 2 and
3x 4 = 1 or x = ± - 1- giving the points (±-L -L -L) (±-L _
VJ' VS' V3 ' · VJ'
V3
_!_
-L) (±-L -L _ _!_) (±-L _ _!_ __!_)
VS' V3 '
VS' VJ ' V3 '
VJ' W' V3
all with an !-value of v'3.
Case 2: If one of the variables equals zero and the other two are not zero, then the squares of the two nonzero coordinates are
equal with common value ~ and corresponding f value of .,/2.
Case 3: If exactly two of the variables are zero, then the third variable has value ± 1 with the corresponding f value ofl. Thus
on x 4 + y'1 + z4 = 1, the maximum value off is y'3 and the minimum value is 1.
13. f(x, y, z, t) =
X -1- y -1- Z -1-
t, g(x, y, z, t) = x 2 -1- y 2 -1- z 2 -1- e = 1 => (1, 1, 1, 1) = (2>.x, 2).y, 2).z , 2>.t), SO
). = 1/(2x) = 1/(2y) = 1/(2z) = 1/(2t) and x = y ~ z = t. But x 2 -1- y 2 -1- z 2 -1- t2 = 1, so the possible points are
(±~ , ±~, ±~, ±~). Thus the maximum value off is f(~, ~~ ~~ t) = 2 and the minimum value is
+ z = 1, h(x,y, z ) = y 2 + z 2 = 4 =:. Vf = (1,2,0), .>.Vg = (>.,>., >.)
and p.Vh = (0, 2p.y, 2p.z). Then 1 = .>., 2 = .>. + 2p.y and 0 = >. + 2~tz so p.y = ~ = - p.z or y = 1/ (2p.), z = -1/ (2p.).
15. f(x,y, z) = x + 2y, g(x,y,z) = x +y
Thus x + y + z = 1 implies x = 1 and y 2 + z 2 = 4 implies p. =±~.Then the possible points are (1, ±.J2, =FV2)
and the 'mfJ.Ximum value is f (1, .J2, -.J2) = 1 + 2 .J2 and the minimum value is f (1, - .,/2, .J2) = 1 - 2 .J2.
17. f(x, y, z ) = yz + xy, g(x, y , z ) = xy = 1, h(x, y, z) = y 2 + z 2 = 1
=:. V f = (y, x + z, y), >.Vg = (.>.y, >.x, 0),
p.Vh = (0, 2p.y, 2p.z). Then y = >.y implies>. = 1 [y f. 0 since g(x, y, z) = 1], x + z = .>.x + 2p.y andy= 2p.z. Thus
p. = z/(2y) = y/(2y) or y2 = z 2 , and so y 2 + z 2 = 1 implies y = ±72, z = ±72. Then xy = 1 implies x = ±.J2 and
© 2012 Ccng.agc Learning. All Rights Rescr\'cd. M:1y not be sc:mncd, copic:c.l, or duplicotcd, or posted to u publicly occcssiblc website, in wll(>Je or in po1rt.
SECTION 14.8
the possible points are ( ±-/2, ± 7:i ,
LAGRANGE MULTIPLIERS
0
231
72), (±V-2, ±~, - );) . Hence the maximum off subject to the constraints is
f(±J2,±~,±jz) = ~ andtheminimumisf(±J2,±~, 'f72) = ~·
Note: Since xy = 1 is one of the constraints we could have solved the problem by solving f (y, z) = yz + 1 subject to
y2
+ z2 = 1.
19. f( x, y) = x 2 + y 2 + 4x - 4y. For the interior of the region, we find the critical points: fx = 2x + 4, f u = 2y - 4, so the
only-critical point is ( - 2, 2) (which is inside the region) and f ( - 2, 2) = -8. For the boundary, we use Lagrange multipliers.
g(x, y) = x 2 + y 2 = 9, so "iJ f = >."ilg
(2x + 4, 2y - 4) = (2>.x, 2>.y). Thus 2x + 4 = 2>.x and 2y - 4 = 2>.y.
=;.
Adding the two equations gives 2x + 2y = 2>.x + 2>.y
x +y = 0
=;.
y = -x or>. - 1 = 0
x 2 + y 2 = 9 implies 2y = 9
2
=;.
y =
=>
=:.
x + y = >.(x + y)
=:.
(x + y) (>.- 1) = 0, so
>. = 1. But >. = 1 leads to a contradition in 2x + 4 = 2>.x, soy = - x and
±12. We have f ( ~· -72) = 9 + 12-/2 ~ 25.97 and
r( -12, ~) = 9- 12-/2 ~ - 7.97, so the maximum value off on the disk x ~ y :s: 9 is f ( 72· -12) = 9 + 12-/2
2
2
and the minimum is f( - 2, 2) = -8.
21. f(x , y) = e- xy . For the interior ofthe region, we find the critical points: f x = - ye-xu, / 11 = -xe-"'Y , so the only
critical point is (0, 0), and f(O, 0) = 1. For the boundary, we use Lagrange multipliers. g(x, y) = x 2 + 4y2 = 1
=;.
>. "iJ g = (2>.x, 8>.y), so setting "iJ f = >."iJ g we get -ye-"'11 = 2>.x and -xe-"'11 = 8>.y. The first of these gives
e- "' 11 = -2>.xjy, and then .the second gives - x( - 2>.xjy) = 8>.y
constraint x 2 + 4y2 = 1 gives X=
=> x 2 = 4y2 • Solving this last equation with.the
±72 and y=±~ . Now t(±~, 'f 2~) = e 1 ~ 1.284 and
1 4
f ( ±~ , ±;~) = e- 1/ 4 ~ 0.779. The former arc the maxima on the region and the latter are the min.ima.
23. (a) f(x , y) = x , g(x,y) = y 2 + x 4 - x 3 == 0
=;.
"ilf = (1, 0) = >.Vg = >.(4x3 - 3x2,2y). Then
1 = >.(4x3 - 3x 2 ) (1) and 0 = 2>.y (2). We have >. i= 0 from (1), so (2) gives y = 0. Then, from the constraint equation,
x 4 - x 3 =.0
=;.
x 3 (3; - 1) = 0
=;.
x = 0 or x = 1. But x = 0 contradicts (1), so the only possible extreme value
subject to the constraint is /(1, 0) = 1. (The question remains whether this is indeed the minimum of f.)
(b) The constraint is y 2 + x 4 - x 3 = 0
¢}
y 2 = x 3 - x 4 . The left side is non-negative, so we must have x 3 - x 4 :2: 0
which is true only for 0 :S: x :S: 1. Therefore the minimum possible value for f( x, y) =x is 0 which occurs for x = y = 0.
However,>."iJ g(O, 0) = >. (0- 0, 0) = (0 , 0) and "iJ f(O, 0) = (1, 0), so "iJ f(O, 0) # >. "iJ g(O, 0) for all values of>..
(c) Here "iJ g(O, 0) = 0 but the method of Lagrange multipliers requires that "iJg # 0 everywhere on the constraint curve.
© 20 12 Ccngagc Lcruning. All Rights Reserved. May not be scanned, copicc.l, or duplicated, or posted to a publicly accessible website. in whole or in p:m.
232
D
CHAPTER 14
PARTIAL DERIVATIVES
'* 'VP=(abLo:- K - o.,(1-a)bL'""JC'""), >..'Vg=(>..m, >..n).
1
25. P(L,K)=bLo.Kl -o., g(L,K)=mL+nK=p
1
Then ab(K/ L) -o. = >..m and (1- a)b(L/ K)"' =>..nand mL + nK = p, so ab(K/ L?-"'/m = (1 - a)b(L/ K) 0 /n or
1
naj[m(1- a)]= (L/ K)"'(L/ K) 1 -"'. or L = Kno/[m{1- a)]. Substituting into mL + nK = p gives K = (1- a)pjn
and L = apjm for the maximum production.
27. Let the sides of the rectangle be x andy. Then f(x, y) = xy, g(x, y) = 2x + 2y = p
'* 'V f(x, y) = (y, x)·,
>.. 'Vg = (2>.., 2>..). Then A= ~y = ~x i~plies x = y and the rectangle with maximum area is a square with side length iP·
29. The distance from {2,0, - 3) to a point (x,y,z) on the plane is d = .j(x- 2)2 + y 2 + (z + 3) 2 , so we seek to minimize
d = f(x, y, z) = (x- 2) 2 + y 2
+(z + 3) subject to the constraint that (x, y, z) lies on the plane x + y + z = 1, that is,
that g(x, y, z) = x + y + z = 1.
Then 'V f =A 'Vg
2
2
=*
(2(x- 2), 2y, 2(z + 3)) = (A, .A, .A), sox= (A+ 4)/2,
y = >./2, z = (>.. - 6)/2. Substituting into the constraint equation gives .A~
4
+
i
+ A;
6
'* 3A- 2 = 2 '*
= 1
>.. = ~ . sox = ~. y = ~.and z = -i· This must correspond to a minimum, so the shortest distance is
d=
J(~ - ~·
i + 3) 2 = ..fi = ~-
2) + ( ~ )2 + (-
31. Let f(x, y , z) = d 2 = (x- 4) 2 + (y- 2) 2 + z 2 • Then we want to minimize f subject to the constraint
'* (2 (x --; 4) ,2(y- 2) ,2z) = (2>..x,2>..y, -2-Xz), sox- 4 = J..x,
y - 2 = .Ay, and z = -J..z. From the last equation we have z + >..z = 0 '* z (1 +.A) = 0, so either z = 0 or>..= -1.
But from the constraint equation we have z = 0 '* x 2 + y 2 = 0 '* x = y = 0 which is not possible from the first
two equations. So ).. = - 1 and x - 4 = >..x '* x = 2, y - 2 = >..y '* y = 1, and x· + y
z = 0 '*
4 + 1 - z 2 = 0 '* z = ±v'S. This must correspond to a minimum, so the points on the cone closest to (4, 2, 0)
g (x,y,z) = x 2 +y2 - z 2 = 0. 'Vf = )..'\Jg
2
2
2
-
are (2, 1, ±v'S). '
=* 'Vf = (yz,xz, xy) = )..'\Jg =(>.., >..,>..). Then>.. = yz = xz = xy
33. f(x,y,z) = xyz,g(x,y,z) = x +y + z = 100
implies x = y = z =
1
~0 .
35. If the dimensions are 2x, 2y, and 2z, then maximize f(x, y, z) = (2x)(2y)(2z) = 8xyz subject to
'* (8yz , Bxz, 8xy) = .A (2x, 2y, 2z) =*
41JZ
4xz
4xy
. .
8yz = 2-Xx, Bxz = 2>.y, and 8xy = 2.Xz , so>.= - ·- = = .This gives x z = y z '* x = y (since z t= D)
X
Y
Z
g(x, y, z) := x 2 + y 2 + z 2 = r 2 (x > 0, y > 0, z > 0). Then 'V f = .A'V g
2
2
2
2
'* z = y so x = y = z '* x '= y = z, and substituting into the constraint
equation gives 3x = r
'* x = r/../3 = y = z. Thus the largest volume of such a box is
2
and xy = xz
2
2
2
2
,
2
2
2
2
. f ( ~, ~, ~) = 8 ( ~) (
7J) (~) = 3 ~ r
3
.
'
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SECTION 14.8
37. f(x ,y,z)
= xyz, g(x, y ,z) = x +2y + 3z = 6
LAGRANGE MULTIPLIERS
D
233
=> \lf = {yz,xz ,xy) = >..Vg = (>.. , 2>., 3>.).
Then >. = yz = ~xz = ~xy implies x = 2y, z = %Y· But 2y + 2y + 2y = 6 soy = 1, x = 2, z = ~ and the volume
is V = ~39. f( x, y, z) = xyz, g(x, y ,.z) = 4(x + y + z) = c
=>
\7 f = (yz, xz, x y), >.Vg = (4>., 4>., 4>.). Thus
4>. = yz = xz = xy or x = y = z = fie are the dimensions giving the maximum volume.
41 . If the dimensions of the box are given by x, y, and z, then we need to fi nd the maximum value of f(x, y, z ) = xyz
[x, y, z > 0) subject to the constraint L = ..jx2 + y 2 + z 2 or g(x, y, z ) = x 2 + y 2 + z 2 = £ 2 .
\7 f = )..\lg
yz
xz
(yz , xz, xy ) = >. (2x, 2y, 2z ), so yz = 2>.x => >. = - , xz = 2>.y => >. = - , and xy = 2.Xz
2X
Thus). = yz = xz .=>
~
~
x 2 = y 2 [since z =f. 0)
2Y ·
=>
=>
=> x = y and >.= y z = x y. => x = z [since y =f. 0].
2X
Substituting into the constraint equation gives x 2 + x 2 + x 2 = L 2
=>
x
2
2Z
= £ 2 / 3 · =>
x = L / V3 = y = z and the
3
maximum vo lume is (L/ V3) = £ 3 / (3 V3).
43. We need to fi nd the extreme values of f(x , y, z) = x 2 + y 2 + z 2 subject to the two constraints g(x, y , z) = x + y + 2z = 2
and h(x, y, z) = x 2 + y~ - z = 0. \lf = (~x, 2y, 2z}, >.\lg = (>. , >., 2>.) and ~\lh = (2J.Lx_, 2~y, -~). Thus we need
2x = ). + 2J.LX (1), 2y = >. + 2~y (2), 2z = 2>.- f.t (3), x + y + 2z = 2 (4), and x 2 + y 2 - z = 0 (5).
From (l) and (2), 2(x - y) = 2~(x - y), so if x =f. y, ~ = 1. Putting this in (3) give,s 2z = 2>.- 1 or >. = z +~. but p~1tting
~ = 1 into (l) says>. = 0. Hence z + ~ = 0 or z = -~. Then (4) and (5) become x + y- 3 = 0 and x 2 + y 2 + ~ ~ 0. The ·
last equation cannot be true, so this case gives no solution. So we must have x = y. Then (4) and (5) become 2x + 2z = 2 and
2x 2 - z = 0 which imply z = 1 - x and z = 2x2 . Thus 2x 2 = 1 - x or 2x2 + x - 1 = (2x - 1)(x + 1) = 0 so x= ~ or
x = - 1. The two points to check are (t, ~ . ~)and ( - 1, - 1, 2): /(~. ~. ~) = ~and f(-1, -1, 2) = 6. Thus(~,~. ~) is
the point on the ellipse nearest the origin and ( - 1, - 1, 2) is the one farthest from the origin.
45. f( x, y , z) = yex-::, g(x, y, z) = 9x2 + 4y 2 + 36z2 = 36, h(x, y , z) = xy + y z = 1.
\7 f = >.Vg + J-L\lh
=>
(yex-::, ex - z, - yex- z) = >.(18x, 8y, 72z) + i'(V, x + z, y), so yex-z = 18>.x + J.l.V, ex- z = 8>.y + ~(x + z ),
- yex- :: = 72>.z + ~y. 9x 2 + 4y 2 + 36z 2 .= 36, x y + yz = 1. Using a CAS to solve these 5 equations s imultaneously for x,
y, z , >., and ~ (in Maple, use the all valu es command), we get 4 real-yalued solutions:
X~ 0.222444,
y ~· -2.157012,
z ~ - 0.686049,
>. ~ - 0.200401,
~ ~ 2.108584
X~ - 1.951921,
y ~ - 0.545867,
z ~ 0.119973,
). ~ 0.003141,
~ ~ - 0.076238
X ~ 0.155142,
y ~ 0.904622,
z ~ 0.950293,
>. ~ - 0.012447,
f.' ~ 0.489938
X~ 1.138731,
y ~ 1.768057, .
z ~ -0.573138,
>. ~ 0.317141,
f.' ~ 1.862675
Subst ituting these values into f gives /(0.222444, -2.157012, - 0.686049) ~ -5.3506,
® 20 12 Ceng3gc (.earning.. All Righrs Re~r\'\,.-d. May not be scanned, copied, or duplicated, or posted to u publicly acct"Ssibh.: website, in whole or in 11.1r1 .
234
0
CHAPTER 14 PARTIAL DERIVATIVES
f( -1.951921, - 0.545867, 0.119973) ~ -0.0688, !(0.155142, 0.904622, 0.950293) ~ 0.4084,
!(1.138731, 1.768057, -0.573138) ~ 9.7938. Thus the maximum is approximately 9.7938, and the minimum is
approximately -5.3506.
47. (a) We wish to maximize j(x1, x2, ... , xn) = ytx1x2 · · · Xn subject to
g(x1,x2, .. . , Xn)
= X1 +x2 + ·· · +xn = c and X.;> 0.
\1 f = ( ~ (X!X2 · .. Xn).; -l (X2 .. · Xn) , ~(X1X2 .. · Xn)'~-l (X!X3 · .. Xn), ... , ~(XtX2 · .. Xn) -!,-l (xl · .. Xn-1) ~
and >. V g = (>., >., ... , >.),so we need to solve the system of equations
1/n 1/n
x1
1 ( XtX2 · • • Xn ).l.-1(
;;:
"
XtX3 • · • Xn) = A
x2
1/n 1 /n
X1
X2
1/n
,
= nAx
1
1/n
\
= nAX2
· · · Xn
· · · Xn
This implies n>.x1 = n>.x2 = · · · = n>.x... Note >. =I 0, otherwise we can't have all x.; > 0. Thus X1 = x2 -= · · · = Xn .
But X !
+ X2 + · ·· + Xn = c => nx1 = c
=>
X1
= ~n = X2 = X3 = · · · = Xn - Then the only point where f can
have an extreme value is (~ , ~, ... , ~). Since we can choose yalues for (Xt, x2, ... , Xn) that make f as close to
n
n
n
zero (but not equal) as we like, f has no minimum value. Thus the maximum value is
t(;,;, ....;) = \};-; ..... ;=;.
(b) From part (a), ~ is the maximum value of f. Thus j(x1; x2, . .. , x n )
n
,-,-,---,------~·--
x 1 + x2 + · ·· + Xn = c, so ytx1X2 · · · Xn :::;
X1
= ytx1x2 · · · Xn :::; ~But
n
+ X2 + · · •+ Xn . These two means are equal when f attams
. .
1ts
n
maximum value ~' but this can occur only at the point(;,*' . .. , * ) we found in part (a). So the means are equal only
when Xt
14
-
c
= X2 = X3 = · ·- · = Xn = -.
n
Review
CONCEPT CHECK
1. (a) A function f of two variables is a rule that assigns to each ordered pair (x , y) of real numbers in its domain a unique real
number denoted by f(x, y).
(b) One way to visualize a function of two variables is by graphing it, resulting in the surface z = f(x, y). Another method for
visualizing a function of two variables is a contour map. The contour map consists of level curves ofthe function which are
horizontal traces of the graph of the function projected onto the x y-plane. Also, we can use an arrow diagram such as
Figure I in Section 14.1.
© 2012 Cengage Le'ilming. All Rights Reserved. May nor be scnnncd. copied. or duplicntc:d, or postr..'tl ton publicly accessible wch~itc, in whole or in part.
CHAPTER 14 REVIEW
0
235
2. A function f of three variables is a rule that assigns to each ordered triple (x, y, z) in its domain a unique real number
f(x, y , z). We can visualize a function of three variables by examining its level surfaces f(x, y, z) = k, where k is a constant.
3.
f(x , y) = L means the values of f(x, y) approach the nwnber Las the point (x, y) approaches the point (a, b)
lim
(:z:,11)-+{a ,b)
along any path that is within the domain of f. We can show that a limit at a point does not exist by finding two different paths
approaching the point along which f( x, y) has different lim its.
4. (a) See Definition 14.2.4.
(b) Iff is continuous on JR2 , its graph will appear as a surface without holes or breaks.
5. (a) See (2) and (3) in Section 14.3.
(b) See "Interpretations of Partial Derivatives" on page 927 [ET 903].
(c) To find f ., regard 1J as a constant and diffe~entiate f(x, y) with respect to x. To find / y, regard x as a constant and
differentiate f( x, y) with respect toy.
6. See the statement of Clairaut's Theorem on page 931 [ET 907].
7. (a) See (2) in Section 14.4.
(b) See (19) and the preceding discussion in Section 14.6.
8. See (3) and (4) and the accompanying discussion in Section 14.4. We can interpret the linearization off at (a, b) geometrically
as the linear function whose graph is the tangent plane tothe graph off at (a, b) . Thus it is the linear function which best
approximates f near (a, b).
9. (a) See Definition 14.4.7.
(b) Use Theorem 14.4.8.
10. See (10) and the associated discussion in Section 14.4.
11. See (2) and (3) in Section 14.5.
12. See (7) and the preceding discussion in Section 14.5.
13. (a) See Definition 14.6 .2. We can interpret it as the rate of change off at ( Xo, yo) in the direction of u . Geometrically, if P is
the point (xo, yo , f(xo, Yo)) on the graph off and C is the curve of intersection of the graph off with the vertical plane
that passes through P in the direction u , the directional derivative off at (xo, Yo) in the direction ofu is the ·slope of the
tangent line to Cat P. (See Figure 5 in Section 14.6.)
(b) See Theorem 14.6.3.
14. (a) See (8) and ( 13) in Section 14.6.
(b) Du f(x , y) = "\7 f (x, y) · u or Du f(x, y, z) = "\7 f(x, y, z) · u
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236
0
CHAPTER 14 PARTIAL DERIVATIVES
(c) The gradient vector of a fu nction points in the direction of maximum rate of increase of the function. On a graph of the
function, the gradient points in the direction of steepest ascent.
15. (a) f has a local maximum at {a, b) if f(x, y) :5 f(a, b) when {x, y) is near {a, b).
(b) f has an absolute maximum at {a, b) if f(x, y) :5 f(a , b) tor all points (x, y) in the domain of f .
(c) f has a local minimum at (a, b) if f(x, y) ~ f (a, b) when (x, y) is near (a, b).
(d) f has an absolute minimum at (a, b) if f(x , y) ~ f(a, b) for aUpoints (x, y) in the domain of f.
(e) f has a saddle point at (a, b) if f(a, b) is a local maximum in one direction but a local minimum in another.
16. (a) By Theorem 14.7.2, iff has a local maximum at (a, b) and the first-order partial derivatives of j exist there, then
/ :r(a, b) = 0 and / 11 (a, b)= 0.
(b) A critical point off is a point (a, b) such that [.(a, b) = 0 and j 11 (a, b) = 0 or one of these partial derivatives does
not exist.
17. See (3) in Section 14.7.
18. (a) See Figure II and the accompanying discussion in Section 14.7.
(b) See Theorem 14.7.8.
(c) See the procedure outlined in (9) in Section 14.7.
19. See the discussion beginning on page 981 [ET 957]; see "Two Constraints" on page 985 [ET 961].
TRUE-FALSE QUIZ
1. True. f v(a, b) = lim f(a, b +h) - f(a , b) from Equation 14.3.3. Let h = ·y- b. Ash~ 0, y --+ b. Then by substituting,
h~o
h
,
we get f 11 (a, b) = lim
u~ b
f(a , y)-f(a,b)
b
.
y-
5. False. See Example 14.2.3.
7. Tru~. If f has a local minimum and f is differentiable at (a, b) then by Theorem 14.7.2, f x(a, b) = 0 and fu (a, b) = 0, so
V f(a,b) = (f,(a,b),fv(a,b)) = (0,0} = 0 .
9. False. V f(x , y) = (0, 1/ y).
11 . True. V f = (cosx, cosy), so IV / I = v'cos 2 x + cos2 y . But lcosBI :5 1, so IV!I :5 .J?.. Now
Du f(x, y) = V f · u = IVf l l.u l cos{:}, but u is a unit vector, so IDu f (x, y) l :5 .J2 · 1 · 1 = ./2.
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l
CHAPTER 14 REVIEW
0
237
EXERCISES
1. ln(x + y + 1) is defined only when x + y + 1 > 0
~
',
''
y > - x - 1,
so the domain off is {(x, y) I y > - x - 1}, all those points above the
y
''
''
'
line y = -x - 1.
3. z = f(x , y) = 1 - y 2 , a parabolic cylinder
5. The level curves are
J 4x2 + y2 = k or 4x 2 + y 2 = k 2,
k ::2::: 0, a fam ily of e ll ipses.
y
X
7.
y
9. f is a rational function, so it is continuous on its d omain.
2
Since f is defined at (1, 1), we use d irect substitution to
I
(©J (©J
(©J (©J
0
I
eva1uate
2
th I" .
e 1m1t:
lim
(z,v ) - (1 ,1) X 2
2x y
2(1)(1)
2
+ 2y 2 = 12 + 2 (1) 2 = -3 .
X
4
4
11 . (a) T.,(6 , 4) = lim T(B + h, ~ - T (B, ), so we can approximateT:c (6, 4) by considering h = ± 2 and
h- o
6 4
8 4
86 80
using the values g iven in the table: T x(6, 4) ~ T( , ) ; T ( , ) =
;
= 3,
. T.,(6, 4) ~ T( , ) ~ T(B , ) =
4 4
2
4
72
~280 = 4. Averaging these values, we estimate Tx(6, 4) to ~e approximately
3.'5 ° C / m . Similarly, Tv (6, 4} = lim T(B ,
4
h- 0
,.., (B
.Lv
+ h~ - T ( 6, 4) , which we can approximate with h = ±2:
) _ T(6, 6} - T(6 , 4) _ 75 - 80 _ _
'4 -
2
- - -2-
-
,.,., (
) _ T(6 , 2 ) - T(6, 4) _ 87- 80 _
A
. th
2 · 5 ' .L v 6 , 4 5 veragmg ese
_
- ~- - 3 ..
2
values, we estimate Tv(6, 4) to be approximately - 3.0° C / m.
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238
CHAPTER 14 PARTIAL DERIVATIVES
0
(b) Here u = (
12• )?.). so by Equation 14.6.9, Du T(6,4) = VT(6,4) · u = T, (6,4) ~ + Tv(6,4) ~·Using our
estimates from part (a), we have Du T(6, 4) l':;j (3.5) ~ + (-3.0) ~ = ~ l':;j 0.35. This means that as we move
through the point (6, 4) in the direction of u , the temperature increases at a rate of approximately 0.35°C/m.
.
Alternatively, we can use Definition 14.6.2: Du T( 6, 4) = lim
r(6+h~,4+h~)-T(6,4)
h-o
h
,
4
which ·we can estimate with h = ± 2 ,;2. Then Du T(6, 4) ~ T( 8 , 6 ) ~T( 6 , ) = 80 - ~O = 0,
2 2
2v2
D u T(6 , 4) ~
T(4, 2) - T(6,4)
74-80
.
(
)
3
0
I
tn
=~
= 3tn' Averagmg
these values, we have D u T 6, 4 ~ 27z ~ 1.1 C m.
-2v2
v2
- 2v2
_ !_ [T. (
)] - lim T,('x , y +h) - T:r. (x,y)
T. (
) _lim T., (6,4 +h) - T.,(6, 4) 1. h
()
C T.:V!/ ( X 1 y ) - a
x X1y h
, SO xy 6 1 4 h
W liC We Can
.
y
h-o
,
h-o
estimate with h = ±2. We have Tx (6, 4) ~ 3.5 from part (a), but we will also need values for T.,(6 , 6) and T,(6, 2). If we
use h = ±2 and the values given in the table, we have
T,(6 , 6 ) l':;j T(8, 6) - T(6 , 6) = 80 - 75 = 2.5 , T, (6 , 6) ~ T(4, 6)- T (6, 6) = 68 - 75 = 3 .5 .
2
2
-2
-2
Averaging these values, we estimate T.,(6, 6) ~ 3.0. Similarly,
2
Tx( 6 , 2 ) l':;j T(8, 2) ~ Tx(6, 2) = 90 ; 87 = l.S, T.,(6 , 2 ) ~ T(4, 2) ~ T(6, 2) = 74 ~ 87 = 6 .5 .
Averaging these values, we estimate T, (6, 2) ~ 4.0. Finally, we esti~ate T., 11 (6, 4):
~ T.,(6, 6)- T.,(6, 4) _ 3.0 - 3.5 _ _
T.:vy (6 ! 4) ~
2
-
2
-
0 •25 >
T. (
X)J
) ~ T.,(6, 2)- T.,(6, 4) _ 4.0 - 3.5 __
0 •25 •
61 4 ~
-2
-2
-
Averaging these values, we have Txy (6, 4) ~ -0.25.
13. f(x , y) = (5y3 + 2x2y) 8
=>
2 7
f .,= 8(5y3 + 2x 2·yf(4xy) = 32xy(5y 3 + 2x y) ,
2
2
2
3
2
7
fv = 8(5y3 + 2x2 yf (15y2 + 2x ) = (16x + 120y )(5y + 2x y)
17. S(u, v,w)
Sw =
= uarctan(vyfw)
S,. = a rctan(v y'W), Su = u ·
1
1
ufo
+ (vfo)2 (fo) = 1 +v2w '
u· 1 + (:JW) (v· ~w- 1 12 ) = 2Vw (~: v w)
19. f(x,y)
2
= 4x3 - x y2 =>
2
f x = 12x 2 - y 2 , f v = -2xy, f.,., = 24x, fv 11 = -2_:~:, fxv = fvx = -2y
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CHAPTER 14 REVIEW
21. f( x, y , z) = xky1z'"
0
239
f x = kxk-lylzm., ! 11 = lxky 1- 1 z m, fz = mxky 1z""'- 1 , f:"x = k(k- 1)xk- 2 y 1z"",
=>
f vv = l(l- 1)xky1- 2 zm, f zz = m (m -1)xky1z"'- 2 , f xv = f vcx: = klx k- 1y 1- 1z"', ! ex::= f ::;r; = kmx"- 1y 1zm- t,
f vz = f ::v = lmxk yl-1zm-1
23. z = x y
+ xe11 1"' =>
25. (a) zx = 6x + 2 => zx (l , -2) = 8 and z 11 = - 2y =>
zv(1, -2) = 4, so an equation of the tangent plane is
z - 1 = 8(x - 1) + 4(y + 2) or z = 8x + 4y + 1.
(b) A normal vector to the tangent plane (imd the surface) at (1, - 2, 1) is (8, 4, - 1). Then parametric equations for the normal
·
· d
.
.
x-1
y+ 2
z- 1
= - - = --=t·
8
4
Ime there are x = 1 + 8t, y = - 2 + 4t, z = 1 - t, an symmetnc equations are - -
27. (a) Let F(x , y ,z) = x 2 + 2y2 - 3z2 • Then F.,= 2x, Fy = 4y, Fz = - 6z, so F,.,(2, - 1, 1) = 4, F11 (2, - 1, 1) = - 4,
F: (2, - 1, 1) = - 6. From Equation 14.6.1 9, an equation of thetangent planeis 4(x - 2)- 4(y + 1) - 6(z - 1) = 0
or, equivalently, 2x - 2y - 3z = 3.
·
14620
· equatwns
·
fior the normaII'1 ~e are -X-(b) From Equatwns
. . , symmetnc
4
2
1
1
Y + = --=fl·
z= --=4
29. (a) Let F (x, y , z ) = x + 2y + 3z- sin(x yz ).. Then F,., = 1- yz cos(x yz ), Fv = 2 -
xz cos(xyz ), F: = 3 - x y cos(xyz) ,
so F:z: (2, -1, 0) = 1, Fy{2, - 1, 0) = 2,·F: (2, - 1, 0) = 5. From Equation 14.6. 19, an equation of the tangent plane is
1(x - 2) + 2(y + 1) + 5(z - 0) = 0 or x + 2y + 5z = 0.
1
1
'(b) From Equatwns
.
14 .6 .20, symmetriC
. equatiOns
.
<".
' are X- 2= -y +
.or the normaII me
- = -z or x- 2 = -y +- = -z .
.
1
2
5
2
5
Parametric equations are x = 2 + t, y = - 1 + 2t, z = St.
31. The hyperboloid is a level surface of the function F (x, y , z) = x 2 + 4y2 - z 2 , so a normal vector to the surface at (x 0 ,y0 , zo)
is \1 F( x 0 ,y 0 , zo) = (2xo, 8yo, - 2zo). A normal vector for the plane 2x + 2y + z = 5 is (2, 2, 1}. For the planes to be
parallel, we need the nonnal vectors to be parallel, so (2xo , 8yo, - 2zo} = k (2, 2, 1), or xo = k, yo = t k, and z 0 = - tk.
I
But x5 + 4y5 - z~ = 4
=>
k 2 + i k 2 - ik2 = 4
=>
k2 = 4
=>
k = ±2. So there are two such points:
(2, ~, - 1) and (-2, - t , I) .
33. f (x , y , z) = x
3
.jy 2 + z 2
yx3
zx 3
=> f x(x,'y, z ) = 3x 2 .jy2 + z 2 , f v(x, y, z) = ~· f , (x , Y.> z) = ~ ·
y2 +z2
so f(2 , 3, 4) = 8(5) = 40, f x(2 , 3, 4) = 3(4) J25 = 60, f v(2, 3, 4). = ~ = 2: , and /.:: (2, 3, 4) = ~ = ~ . Then the
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240
0
CHAPTER 14 PARTIAL DERIVATIVES
linear approximation off at (2, 3, 4) is
f(x ;y,z) ~ /(2,3,4) + J.,(2,3,4)(x ~ 2) + fv(2,3,4)(y - 3) + /z(2,3, 4)(z - 4)
= 40 +60(x- 2) + ¥CY - 3) + 352 (z- 4) = 60x+ 254 y+ 352 z - 120
Then (1.98) 3 J(3.01) 2 + (3.97) 2 = /(1.98,3.01, 3.97) ~ 60(1.98) + ¥(3.01) + ¥(3.97) -120 = 38.656.
du au dx au dy au dz
3
2 2
3
35. -d = ,---d +-;;--d +-;;- -d =2xy (1 +6p) + 3x y (per>+er>)+4z (pcosp+sinp)
p
p
uX
uy
.
p
uz
az
p
az ax
az ay
37. By the Cham Rule, as = ax as + ay as. When s = 1 and t = 2, X = g(l, 2) =;:: 3 andy = h(1, 2) = 6, so
&
. .
&
&~
&~
as = f,(3, 6)gs(1, 2) + fv (3, 6) h. (1, 2) = (7)( - 1) + (8)( -5) = - 47. Stmiiarly, Bt = ax 8t + ay at, so
az
at = f,(3, 6)gt(1, 2) + fv (3, 6) ht(1, 2) = (7)(4) + (8)(10) = 108.
az = 2x!'( X 2 - y 2) '
39. ax
az
ay = 1 - 2y!'( X 2 - y 2)
[w here !' = d( x2df- y2) ] . Then
az
az
2
2
I
2
2
Y Bx + x {)y = 2xyf'(x - y ) + x - 2xyf (x - y ) = x.
az
az
az - y
41 . -a = -a v+ -a 2
X
U
V X
2
a z
a
ax2 = Yax
and
(az).
2y az
a (az) 2y az
(a z
8 z - y)
- y (a z - y
a z )
au +xa8v + ~ax av =xaav+Y au ~ + 8v8u~ +~ 8v ~+auavY
- y
2
2
2
2
2
2
z
2y2 a 2 z
y 2 8 2z
= x 3 8v + Y &u2 - X2 &uav + x 4 av2
2y az
2 8
2
Thus
.
uv
smcey = xv = -
y
2
ory = uv.
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CHAPTER 14 REVIEW
45. j(x, y) = x 2 e- 11
u =
1
yf42+(-3)2
241
=> 'V f = (2xe- 11 , - x 2 e- 11 ), 'V f( -2, 0) = (-4, -4). The direction is given by (4, - 3), so
(4,-3} = { (4 ,-3) and Duf(-2,0} = 'Vf{-2,0} · u = (-4, -4) · t (4, -3) = t(- 16 + 12} = -~.
...
...
47. 'V f = (2x y,x2 + 1/(2JY)), I'Vf(2, 1)1 = 1(4, ~) I · Thus the maximum rate of change off at (2, 1} is
direction ( 4·,
D
o/ in the
! ).
49. First we draw a line passing through Homestea'd and the eye of the hurricane. We can approximate the directional derivative at
Homestead in the direction of the eye of the hurricane by the average rate of change of wind speed between the points where
this line intersects the contour lines closest to Homestead. In the direction of the eye of the hurricane, the wind speed changes
from 45 to 50 knots. We estimate the distance between these two points to be approximately 8 miles, so the rate of change of
wind speed in the direction given is approximately 50 8 45 =
51 . f(x,y)=x 2 - xy+y 2 + 9x -6y+10
=>
i = 0.625 knot/ mi.
f, = ·2x-y+9,
j 11 = - x + 2y- 6, f xx = 2 = / 1111 , f xv = - 1. Then f x = 0 and ! 11 = 0 imply
y = 1, x = -4. Thus the only critical point is ( - 4, 1} and fx:r:( -4, 1) > 0,
D(- 4, 1) = 3 > 0, so f( - 4, 1) = -11 is a local minimum.
53. f(x, y) = 3xy - x 2 y- xy 2
=> fx = 3y - 2xy - y2 , / 11 = 3x- x 2 - 2xy,
f ,, = - 2y, fv·u = - 2x, f xv = 3- 2x- 2y. Then f, = 0 implies
y(3- 2x- y) = 0 soy= 0 or y = 3 - 2x. Substituting into / 11 = 0 implies
x(3 - x) = 0 or 3x( - 1
+ x )' = 0. Hence the critical points are (0, 0), (3, 0},
(0, 3) and (1 , 1). D(O, 0} = D(3, 0} = D(O, 3) = ·- 9 < 0 so (0, 0), (3, 0), and
(0, 3) are saddle points. D(1, 1) = 3 > 0 and f xx (1, 1} = - 2 < 0, so
!(1, 1) = 1 is a local maximum.
55. First solve inside D. Here f ., = 4y2 - 2xy2 - y 3 , f 11
= 8xy - 2x2 y - 3xy2 •
Then f, =·o implies y = 0 or y = 4- 2x, but y = 0 isn't inside D. Substituting
y
(O, 6)
y = 4- 2x into / 11 = 0 implies x = 0, x = 2 or x = 1, but x = 0 isn't inside -D,
and when x = 2, y = 0 but (2, 0} isn't inside D. Thus the only critical point inside
D is (1, 2) and f (l , 2) = 4. Secondly we consider the boundary of D.
(6 ,0)
On L1: f(x, 0) = 0 and so f = 0 on L 1. On L2 : x = -y + 6 and
.t
f( - y + 6, y) = y 2 (6 - y)( - 2) = -2{6y2 - y 3 ) which has critical points
at y = 0 and y = 4. Then !{6, 0) = 0 while !(2, 4) = -64. On La: f(O , y) = 0, so f = 0 on £ 3. Thus on D the absolute
maximum off is !{1, 2} = 4 while the absolute minimum is /{2, 4) = - 64.
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242
0
CHAPTER 14 PARTIAL DERIVATIVES
57. f(x, y) = x 3 - 3x + y 4 - 2y2
2
z 0 l \"''<--1-l-1
-1
0
y
From the graphs, it appears that f has a local maximum f( -1 , 0} ~ 2, local minima !(1, ±1} ~ - 3, and saddle points at
(-1, ±1} and (1,0}.
To find the exact quantities, we calculate fx = 3x2 - 3 = 0
<=> x = ± 1 and fu = 4y3 - 4y = 0 <=>
y = 0, ±1, giving the critical points estimated above. Also fzx = 6x, fxu = 0, f uy = 12y 2 - 4,-so using the Second
Derivatives Test, D ( -1, 0} = 24 > 0 and f xx( -1, 0) = - 6 <.0 indicating a local maximum / ( -1, 0) = 2;
D (1, ± 1) = 48 > 0 and / u( 1, ±1} = 6 > 0 indicating local minima / (1, ± 1) = - 3; and D ( -1, ± 1) = - 48 and
D(1, 0) = - 24, indicating saddle points.
59. f(x,y) = x 2 y, g(x,y) = x 2 +y 2 = 1
y = >.. If x = 0 then x 2
+y
2
~
'ilf = (2xy,x2 ) = )..'iJg = (2>.x,2>.y). Then 2xy = 2)..x impliesx = Oor
= 1 gives y = ±1 and we have possible points (0, ±1) where f (0, ± 1) = 0. lfy =).. then
x 2 = 2)..y implies x 2 = 2y 2 and substitution into x 2 + y 2 = 1 gives 3y 2 = 1
~
y = ± 7:J and x =
corresponding possible points are ( ±jf, ±7:J). The absolute maximum is f ( ± jf,
61. f(x , y, z) = xyz, g(x,y,z) = ~2 + y 2 + z 2 = 3. 'ilf = )..'iJg
~
±jf. The
-Ja) = ~ while the absolute
(yz,xz,xy) = >.(2x, 2y,2z). lfany ofx, y, or z is
zero, then x = y = z = 0 which contradicts x 2 + y 2 + z 2 = 3. Then ).. = YZ ·=xz = xy ~ 2y 2 z = 2x2 z ~
2x
2y
2z
.
.
y 2 = x 2 , and similarly 2yz2 = 2x2 y ~ z 2 = x 2 • Substituting into the constraint equation gives x 2 + x 2 + x 2 = 3
~
I
x2 = 1 = y 2 = z 2 . Thus the possible points are (1, 1, ± 1), (1, -1, ± 1), ( - 1, 1, ±1), ( - 1, -1, ±1). The absolute maximum
is / (1, 1, 1) = /(1, -1, - 1) = f( -1 , 1, -1) = f( - 1, - 1, 1) = 1 and the absolute
minimum is /(1, 1, -1) = /(1, ....:.1, 1) = f( -1, 1, 1) = f( - 1, - 1, - 1) = - 1.
63. f(x ,y,z) = x 2 +y2 +z2 , .g(x,y,z) = xy 2 z 3 = 2
~ 'ilf = (2x, 2y,2z) = X'ilg = (>.y 2 z 3 , 2>.xyz 3 ,3)..xy2 z 2 ).
Since xy 2 z 3 = 2, x =I 0, y =I 0 and z =I 0, so 2x = )..y2 z 3 (I),
1 = )..xz3 (2),
2 = 3>.xy2 z (3~. Then (2) and (3) imply
2
2
~
= - 2 or y 2 = f z 2 soy = ±z q. Similarly (1) and (3) imply ~x3 = or 3x2 = z 2 sox = ± -jsz. But
xz
V3
y z
3 xy z
3 xy-z
?
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CHAPTER 14 REVIEW
D
243
xy2 z 3 = 2 so x and z must have the same sign, that is, x = ~ z. Thus g(x, y, z ) = 2 implies ~z Gz 2 )z 3 = 2 or
z = ±31 / 4 and the possible points are (±3- 1 / 4 , 3-1 / 4 y'2, ±3114 ), (±3- 1 / 4 , --,- 3.. :. 1 ; 4 ._/2, ±31 14 ) . However at each of these
points f takes on the same value, 2 J3. But (2, 1, 1) also satisfies g(x, y , z) = 2 and / (2, 1, 1) = 6 > 2 v'3. Thus. f has an
absolute minimum value of 2 J3 and no absolute maximum subject to the constraint x y 2 z 3 = 2. ·
. . . f( x, z ) = x 2 + - 2 + z 2 . T.I1en
Alternate solution: g ( x , y, z ) = xy2 z 3 = 2 .tmp I'tes y-., = - 2 3 , so mmtmtze
3
xz
fx = 2x -
2
X2 Z3
,
xz
4
24
6
f~ = - --;.
+ 2z, fxx = 2 + 3 3 , f ~~ =
+ 2 and fr~ = 2 4 • Now f x = 0 implies
XZ
X Z'
X Z5
X Z
2x3 z 3 - 2 = 0 or z = 1/x. Substituting into / 11 = 0 implies -6x 3 + 2x- 1 = 0 or x .= ~·so the two criti<;al points are
r
( ± ~, ± yt3).Then D ( ± ~ , ± V'3) = (2 + 4)( 2 + 234 ) - ( 7J
> 0 and f xa: ( ± ~, ± V'3) = 6 > 0, so each point
,- ~
, ± V'3).
4 .:fl,± :y'3), (±+
V3
V3
is a minimum. Finally, y 2 = _2_3 , so the four points closest to the origin are (± ~ ,
=
·
v3"
The area of the triangle is ~ ca sinO and the area of the rectangle is be. Thus,
65.
the area of the whole object is f(a , b, c)= ~ca sinO+ be. The perimeter of
the object is g( a, b, c) = 2a + 2b + e = P. To simplifY s in 0 in terms of a, b,
b
· and e notice that a 2 sin2
2
o+ ( ~e)
= a 2 => sino= ..!. v4a2 - c2.
2a
c
c
. '
Thus f(a, b, c)= 4 J4a 2 - c2 +be. (Instead of using 0, we could just have
used the Py1hagorean Theorem.) As a result, by Lagrange's method, we must find a, b, c, and A by solving "il f = A"il g which
gives the following equations: ca(4a2 - c2 ) - 1 12 = 2A (1), c = 2A (2),
*(
4a2 - c 2 ) 112 - i c2 (4a 2 - c2 ) - 1 12 + b = A
(3),and2a+2b +c = P (4). From(2),A= t candso(l)producesca(4a2 - c2 ) ....: 112 =c =>
(4a 2 - c2 ) 112 = a
2
4a2 - e2 = a2
(5),
=>
=> · c = v'3 a (5). Similarly, since {4a 2 - c 2) 112 = a and A= tc, (3) gives ~ - !:__ + b = ~ . so from
~ - 34a + b =
4
.;;a => -~ - v;a
2a + a(l + v'3) + v'3a=P
= -b =>
b
2
=% (1+ vf3) (6). Substituting (5) and (6) into (4) we get:
=> 3a + 2v'3a=P => a =
b = ·(2 J3- 3~ (1 + J3) p = 3 - 6 J3 p and
4a
p
= 2 J3 - 3 P and thus
3+ 2 J3
3
c= (2- V3)P.
© 2012 Ccngagc learni ng. Atl Ril:l>ts Rcset'·cd. May nol be SC3M<:d, copied. or dup UCill<'d. or posted 10 u pub licly •cccssiblc website, i n whole or in port.
D PROBLEMS PLUS
L
1. The areas of the smaller rectangles are A1 = xy, A2 = (L - x)y,
T
w
y
A 3 = (L- x)(W -y), A 4 = x(W - y). ForO :::;: x :::;: L,O :::;: y:::;: W, let
f(x, y) = Ai +A~+ A~ +A~
1
W-y
2
= x2 y 2 + (L - x?y 2 + (L- x) 2(W - v? + x (W- y)
2
X
L-x
= [x2 + (L- x?JIY2 + (W - y)2]
Then we need to find the maximum and minimum values of f(x , y). Here
f,(x, y) = [2x - 2(L- x)Jiy 2
+ (W - y) 2 ] = 0 => 4x '-- 2L = 0 or x = ~L, and
· j 11 (x , y) = [x 2 + (L - x) 2JI2y- 2(W- y)] = 0
=> 4y- 2W = 0 or y = W / 2. Also
f :tx = 4[y2 + (W - u?J, f v11 = 4(x2 + (L- x?J, and fxv = (4x- 2L)(4y- 2W) . Then.
D = 16(y2 + (W - y) 2 J!x 2 + (L- x) 2 ) - (4x- 2L) 2 (4y- 2W) 2 . Thus when x =~Landy= ~W, D > 0 and
f x:r: = 2W2 > 0. Thus a minimum off occurs at (~L, %W) and this ·minimum value is f(~L, ~W) = ~L 2 W 2 .
There are no other critical points, so the maximum must occur on the boundary. Now along the width ofthe rectangle let
g(y) = j(O, y) = f(L , y) = L 2 [y 2 + (W- Y? ]. 0 :::;: y:::;: W . Then g'(y) = L 2 [2y - 2(W- y)) = 0
~
y = ~W.
And g(%) = ~L2 W 2 . Checking the endpoints, we get g(O) = g(W) = L 2 W 2 • Along the length of the rectangle let
h (x) = f(x, 0) = f(x , W) = W 2 [x2 + (L- x?]. 0 :::;: x :::;: L. By symmetry h'(x) = 0 ~
x = ~Land
h( ~L) = %L2 W 2 • At the endpoints we have.h(O) = h(L) = L 2 W 2 • Therefore L 2 W 2 is the maximum value of f .
This maximum value off occurs when the "cutting" lines correspond to sides of the rectangle.
3. (a) The area of a trapezoid is ~ h(b1 + b2), where his the height (the distance between the two parallel sides) and b1 , ~ are
the Ieng!hs of the bases (the parallel sides). From the figure in the text, we see that h = x sinO, b1 = w - 2x, and
b2 = w - 2x + 2x cos 8. Therefore the cross-sectional area of the rain gutter is
~(x, 8) = ~xsin 0 [(w - 2x) + (w- 2x + 2xcos 8)) = (x sinO)(w - 2x + xcos8)
= wx sine - 2x2 sin B + x 2 sin B cos
e. 0 < X ::; ~w. 0 < B :::;: i
We look for the critical points of A : oAfax = w sin B - 4x sin B + 2x sin B cos B and
EJAfEJO = wxcos B- 2x2 cosB + x 2 (cos 2 8 - sin2 8), so {)A j ax= 0
cos B =
4
x- w = 2- w
2x
2x
(0 < 8 :::;: ~
~
sinO (w - 4x + 2x cosO) = 0
~
=> sin 8 > 0). If, in addition, oAf EJB = 0, then
0 = wxcosB - 2x2 cosO+ x 2 (2cos2 8 - 1)
=
~) - 2x2 ( 2 - ~) + x 2 [ 2 ( 2 - ~) - 1]
= 2wx -
~w 2 - 4x 2 + wx + x 2 [ 8- 4: + ~: - 1] = -wx + 3x2 = x(3x - w)
wx ( 2.-
2
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245
246
CHAPTER 14 PROBLEMS PLUS
0
Since x > 0, we must have x = ~w, in which case cosO=~. soB = f, sinO=~ . k = '{!w, b1 = ~w,ln = ~w.
and A= {!:w 2 • As in Example 14.7.6, we can argue from the physical nature of this problem that we have found a local
maximum of A. Now checking the boundary of A, let
g( B) = A(w /2, B) = ~ w 2 sin B - ~w 2 sin B + iw2 sin Bcos 6 = i w 2 sin 26, 0 < B ::; ~. Clearly g is maximized when
sin2B = 1 in which case A= iw2 . Also along the line B = ~. let h(x)
h'(x) = w- 4x = 0 *? x = i w, and h(~w) = w(~w) - 2(tw)
2
= A(x, ~) = wx- 2x2 , 0 < x < ~w . =>
= iw 2 . Since -kw2 < "If w 2 , we conclude that
the local maximum found earlier was an absolute maximum.
2
(b) If the metal were bent into a semi-circular gutter of radius r, we would have w
2
= 1rr and A= ~11'r 2 = ~1r(~) = ~7r.
2
2
../3-w-, 1t• would be better to bend the metal ·mto a gutter WI'th a sem1c1rcu
· · Iar cross-section.
.
>-
.
-w
S mce
211'
12
5. Letg(x ,y) = xf(~) · Theng,.(:,y) = !(~) +xf'(~) ( - : 2 )
= !(~ ) - ~ f'(~;) and
gy (x, y) = xf' ( ~) (;) = f' ( ~ ). Thus the tange?t plane at ( x 0 , y 0 , zo) on the surface has equation
[t (;~ ) - y x0 !' ( ~~ )] x + [r (~~)] y - z
1
0
= 0. But any plane whose equation is of the form ax+ by+ cz
=0
passes through the origin. Thus the origin is the ~ommon point of intersection.
y
7. Since we are minimizing the area of the ellipse, and the circle lies above the x -axis,
the ellipse will intersect the circle for only one value ofy. This y-value must
satisfY both the equation of the circle and the equation of the ellipse. Now
2
x
a2
2
2
+ Y2 = 1 =>
x 2 = a 2 (b2 -y2 ). Substitutingintotheequationofthe
b
b
2
circle gives ~2 (b
2
-
y ) +y
2
2
-
2y = 0
=>
2
In order for there to be orlly o ne solution to this quadratic equation, the discrirninant must be 0, so 4- 4a2 b
2
~ a = 0 =>
b2 - a 2 b2 + a 4 = 0. The area of the ellipse is A(a, b) = 1rab, and we minimize this function subject to the constraint ·
g(a, b) = b
2
-
a b + a = 0.
4
2 2
-rrb
2) (1),
2a(2a2 - b
.>. =
21rb
2
2 b(l7r~ a
2)
(2), b
= 47ra4
-:=:>
a
2
2
-
2 2
4
a b + a = 0 (3). Comparing (1) and (2) gives 2a( 2; b_ b2).
= ~ b. Substitute this into (3) to get b = ~
=> a =
jf.
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MULTIPLE INTEGRALS
15
D
15.1
Double Integrals over Rectangles
1. (a) The subrectangles are shown in the figure.
The surface is the graph of f(x , y) = xy and 6.A
3
v~
y
= 4, so we estimate
4
2
2
I: I: f(x;,y1 )6.A
i =l j = l
0
= j(2, 2) 6.A + !(2, 4) 6.A + f(4 , 2) 6.A + /(4,4) 6.A + !(6, 2) LU + !(6, 4) 6.A
2
4
6
X
= 4(4) + 8(4) + 8(4) + 16(4) + 12(4) + 24(4) = 288
3
2
,L: j (x.:, v1 ) 6.A = /(1, 1) 6.A + f(1, 3) 6.A + !(3, 1) 6.A + !(3, 3) 6.A + !(5, 1) LU + !(5, 3) 6.A
(b) V ~ ,L:
i=lj=l
= 1(4) + 3(4) + 3(4) + 9(4) + 5(4) + 15(4) = 144
3. (a) The subrectangles are shown in the figure. Since 6.A = 1 · ~ = ~ .we estimate
2
2
Jfn. xe-"' dA ~ I: I: f (x;1,Yi;) 6.A
y
11
i=l j = l
I
= /(1, ~) 6.A + /(1, 1) 6.A + /(2, ~) 6.A + /(2, 1) 6.A
I
2
= e- 1 1 2 (~ ) +e- 1 (~) +2e- 1 (~) + 2e- 2 ( ~) ~ 0.990
0
.
2
2
v
(b) ffn xe -"' 11 dA ~ I: I: f(x;, 1) 6.A
.
i=l j =l
1
1
I
2
2
X
5. (a) Each subrectangle and its midpoint are shown in the figure.
f
at each midpoint and estimate
y
4
3
.
.
.
.
0
2
X
y
= !(~, t) 6.A + !(~ , ~) 6.A + f(~, t) 6.A + ! (~. ~) 6.A
= ~e- l/8a) + ~e-318(~) + ~e-318(~) + ~e- 9/sa) ~ u 51
The area of each subrectangle is 6.A = 2, so we evaluate
2
1
.
.
.
.
2
2
.ffn.f(x , y)dA ~ I: I: f(x.: ,y1 ) 6.A
i = lj = l
= /(1, 2.5) 6.A + /(1, 3.5) AA
+ /(3, 2.5) AA + /(3, 3.5) AA
= - 2(2) + (-1)(2) + 2(2) + 3(2) = 4
0
2
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4
X
247
248
0
CHAPTER 15 MULTIPLE INTEGRALS
y
(b) The subrectangles are shown in the figure.
4
In each subrectangle, the sample point closest to the origin
is the lower left comer, and the area of each subrectangle is .6.A = ~.
3
2
Thus we estimate
4
ffnf(x ,y) dA ~ L::
4
L:: f(xii,Yii) .6.A
i= l j = l
= f (O, 2) .6.A
0
I
2
3
4
X
.
+ f(O , 2.5) .6.A + f(O , 3)..6.A + f(O, 3.5) .6.A
+ /(1, 2) .6.A + f(1, 2.5) .6.A + /(1, 3) .6.A + /(1, 3.5) .6.A
+ / (2, 2) .6.A + /(2, 2.5) .6.A + /(2, 3) .6.A + /(2, 3.5) .6.A
+ / (3, 2) .6.A + /(3, 2.5) .6.A + /(3, 3) .6.A + !(3, 3.5) .6.A
= -3(~) + <-5) G) + <- 6) (~) + <-4)(~) + <- 1) (~) + <- 2){ ~) + <-3)(~) + <-1)G)
+ 1G) + o(~) + (-1){~) + 1 (~) + 2(~) + 2G) ·;~- 1 (~) + 3G)
=- 8
7. The val~es off (x, y) = .j52 - x 2 - y 2 get smaller as we move farther from the origin, so on any of the subrectangles in the
problem, the function will have its largest value at the lower left comer of the subrectangle and its smallest value at the upper
right comer, and any otl).er value will lie between these two. So using these subrectangles we have U < V < L . (Note that this
is true no matter how R is divided into subrectangles.)
9. (a) With m = n = 2, we h~ve .6.A = 4. Using the contolir map to estimate the value off at the center of each subrectangle,
we have
2
2
ffn f(x, y) dA ~ L:; L:; J(xi, Yi) .6.A = .6.A[f(1 , 1) + /(1, 3) -t /(3, 1) + /(3, 3)] i'::j 4(27 + 4 + 14 + 17) = 248
i = lj=l
(b) /ave= Aln) ffn f(x, y) dA i'::j ft(248) = 15.5
11. z = 3 > 0; so we can interpret the integral as the volume of the solid S that lies below the plane z = 3 and above the
rectangle [- 2, 2] x [1, 6]. Sis a rectangular solid, thus ffn 3 dA = 4 · 5 · 3 = 60.
13. z
= f(x , y) = 4- 2y :2: 0 for 0 :::; y :5 1. Thus the integral represents the volume of that
part ofthe rectangular.solid [0, 1] x [0, 1] x (0, 4] which lies below the plane z = 4 - 2y.
So
ffn (4 - 2y) dA = (1)(1)(2) + H 1)(1)(2) = 3
X
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SECTION 15.2
ITERATED INTEGRALS
D
249
15. To calculate the estimates using a programmable calculator; we can use an algorithm
,.
similar to that of Exercise 4.1.9 [ET 5. 1.9). In Maple, we cari define the function
n
estimate
f( x, y) = v'1 + xe 11 (calling it f), load the student package, and then use the
' 1
1.141606
4
1.143191
middlesu m(middles um(f , x= 0 . . 1 ,m),
16
1.143535
y=O .. 1 , m);
64
1.143617
to get the estimate with n = m 2 squares of equal size. Mathematica has no special
256
1.143637
Riemann sum command, but we can define f and then use nested Sum commands to
1024
1.143642
command
calculate the estimates.
17. If we divideR into mn subrectangles, ffn k dA::::::
m
.f _t f (xi; . Yi;) ~A for any choice of sample points (xi;, Yi;).
<=13=1
n
But f(xi;. Yi;) = k always and L: L: ~A= area of R = (b- a) (d- c). Thus, no matter how we choose the sample
i=1j=l
m
n
""
n
points, L: L: f(xt;, Yii) ~A = k L · L: ~A= k(b - a)(d - c) and so
i=1j=l
i = lj=1
nt
ffnkdA=
15.2
lim
m
n.
n
L L: f(xi;.Yi;) ~A = rn,n--+oo
lim k L L ~A= lim k(b-a) (d -c)=k(b-a)(d- c).
i= l j=l
Tn,n---+oo
m ,n --+ooi = l j = l
Iterated Integrals
3. f 1
4
J02 (6x 2 y- 2x) dy dx = f 14 [3x 2 y 2 - 2xy]~=~ dx = f14 (12x2 - 4x) dx = [4x3 - 2x2 ] ~ = (256 - 32) - (4 - 2) = 222
5. J~ J0 y 3 e 2 "' d.yd:c = J0 e 2"' dx
2
4
7. J~ 3 J;
12
.r: y dy [as in ExampleS] = [te2 "']~ h
3
(
4
]:
= t(e4 - 1) (64 - O) = 32(e 4 · - 1)
(y+y 2 cosx) dxdy = J2 3 [xy+y 2 sin xJ::~ 12 dy
= J~3 (~y + y2) dy = [~y2 + h3]~3
= (9411' + 9 - (~; - 9)] = 18
9.
2
4(
4
/,2
(X
/,4
[
1 1·y ]
l
3 ) dx = [ ~x ln 2+~ ln lxlf
-+JL
dydx=
x
lnlvl+-·dx
=
x ln 2+---:
/,
2
y=l
.
2x
1
1
Y
? )
X
2
2 y=
1
1
X
= 8ln2 + ~ ln4 - ~ ln 2 = lf ln2 +3ln4 1 / 2 = ¥ ln 2
11.
f01 J01 v(u + v2) 4 dudv = J01 [iv(u + v 2 ) 5 ]~=~ dv = t j~ v ((1 + v2) 5 - (0 + v2) 5 ] dv
= t fo1 (v(1 + v2)5- vn] dv = Ht. i(l + v2)6 - J:\vl2]~
[substitute t = 1 + v 2 ==:> dt = 2v dv in the first term]
= to [(26 - 1) - (1 - 0)] = to (63- 1) = ¥o
1
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1
250
0
CHAPTER 15 MULTIPLE INTEGRALS
13. J~ f 01f r sin2 8 d8 dr = f~ r dr
J; sin2 8 d8 [as in Example 5] = J02 r dr J0 t (1 - cos 28) d8
1f
= [tr )~· ~ [0- ~ sin28)~ = (2 - 0) · ~ [(1r- ~ sin 27r) - (0- ~sinO)]
2
= 2. ~[(7r- 0)- (0 - 0)] = 7r
. 15. ffn sin(x - y) dA = Io1f
12
Iorr~ sin(x- y) dyd:r: = I; 12 [cos(x - y)]~~~l 2 dx = Iorrl 2 [cos(x - ~)- cosx] dx
2
= [sin(x-~)-si?x]
1f 12
0
.
= sinO-sin~-,- (sin(-~) - sinO]
=0-1-(-1-0) = 0
17.ffn X:~ 1 dA = 11
/_33
X::
1 dydx =
11
:r;2:
[t ln(x2 + 1)]: [h3] ~3
1 dx ;_: y2 dy =
= ~ (In 2 -In 1) . t(27 + 27) = 91n 2
19. Io1riG j~1rl 3 x sin(x + y) dy dx
= IorriG [-xcos(x + y)]~=~13 dx = J; 16 [x cosx- xcos(x +f)] dx
= x (sin x - sin (x + f) )~ 16 - J0 16 [sin x - sin ( x + f)] dx
[by integrating by parts separately for each term]
1f
=
i [~- 1)- (-cosx + cos(x + f)]~ 16 = - -&- [-~ + 0 - (-1 + ~)] = -q- 1 - f2
21. IInye-:z:il dA =I: I: ye-"'11 dxdy = 0 (-e-"' 11J ::~ dy = I:(-e-
J
3
2
il
+ 1) dy = [~e- 2 Y + y]~
= ~ e-6 + 3 - ( ~ + 0) = t~-6 + ~
23. z = f(x, y) = 4- x - 2y ~ 0 for 0 ~ x ~ 1 and 0 ~ y ~ 1. So the solid
is the region in the first octant which lies below the plane z = 4 - x - 2y
and above [0, 1] x {0, 1].
X
25. The solid lies under the plane 4x + 6y - 2z + 15 = 0 or z = 2x + 3y +
V = fin(2x + 3y + ;)'dA = I~ 1 I~ 1 (2x + 3y +
1
1/ so
Jf) dxdy = j~ 1 [x + 3xy + ¥xJ::~ 1 dy
2
= j~ 1 ((19 + 6y) - ( - 123 - 3y)) dy = I~ 1 (¥' + 9y) dy = [Yf y + ¥Y2 )~ 1 = 30- ( -21) = 51
27. V = I~ 2 f~ 1 (1 -
tx h
2
-
2
)
dx dy = 4
J; I (1 - ix
1
0
2
-
~y 2 ) dx dy
f2 [
1 3
1 2 ] :z; = 1 d
4 Jo
f2 ( 11
1 2) d
11
l 3] 2
83
166
= 4 Jo
X - fiX - iiY X :z:=O y =
i2- iiY
y = 4 [i2Y
- vY
0 = 4. 54= 27
29. Here we need the volume of the solid lying under the surface z = x sec2 y and above the rectangle R = (0, 2] ~ [0, 1r / 4] in
the xy-plane.
V =
I:
Iorr
14
2
x sec y dy dx =
I:
x dx Io"
14
sec2 y dy = [ ~ x 2 ] ~ [tan y] ~
14
= (2- O)(tan f - tan 0) = 2(1 - 0) = 2
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l
SECTION 15.3
DOUBLE INTEGRALS OVER GENERAL REGIONS
0
251
+ (y- 2? and above the plane z = 1 for -1 ~ x ~ 1, 0 ~ y ~ 4. The volume
of the solid is the difference in volumes between the solid that lies under z = 2 +x2 + (y- 2) 2 over the rectangle
31. The solid lies below the surface z = 2 + x
2
R = (-1, 1] x (0, 4) and the solid that lies under z = 1 over R .
V = J; J~ 1 (2 + x
2
+ (y :._ 2?J dxdy- f04 f~ 1 (1) dxdy = f04 [2x + 4x3 + x(y - 2?J:: ~ 1 dy- t 1 dx J;dy
= J; [(2 + ~ + (y - 2) 2 ) - ( -2- ~- (y- 2) 2 )] dy- [x]~ 1 (y]~
= f0 (lf + 2(y- 2) 2 ) dy- (1 - (-1)][4- OJ = ( 134 y + t(Y- 2?)~- (2)(4)
4
= [e; + ¥)- (o - 1nJ -8 = ¥-8 = ¥
33. In Maple, we can calculate the integral by defining the integrand as f
and then using the command int ( int ( f, x=O .. 1) , y=O . . 1) ; .
In Mathematics, we can use the command
Int egrate[f,{x,0,1}, {y,0,1} ]
We find that JJn x 5 y 3 e<IJY dA = 21e - 57 R:: 0.0839. We can use plot3d
(in Maple) or Plot 3D (in Mathematics) to graph the function.
35. R is the rectangle (-1, 1] x (0, 5]. Thus, A(R) = 2 · 5 = 10 and
5
1 f' r !(
) dA
1 r 5Jt 2 d d _ 1 rs [ 1 3 ]"' = 1 d _ 1 rs 2 d
1 (1 2] s
f,•"• = A(R)
Jn x,y
=wJo -lxy. x Y - wJo 3xv.,= - t Y -w Jo3Y Y= to 3Y o = e·
37.
!1
-xy
4 dA=
R1+X
1
11 11
-1
0
- xy 4 dydx =
1 +X
1
1
- x .4 dx
- 1 1
+X
11
0
·.
ydy [byEquatJOnS] butj(x) = - x- 4 is anqdd
1 +X
1
function so
1
f(x) dx = 0 by (6) in Section 4.5 [ET (7) in Section 5.5]. Thus/" {
xy 4 dA = 0. { y dy = 0.
ln 1 + x
- 1
lo
t and f01 f 01 f(x, y) dx dy = -t.
39. Let f(x, y) = ( x - ) 3 • Then a CAS gives J; f0 f(x, y) dydx =
1
x+y
To explain the seeming violation of Fubini's Theorem, note that f has an infinite discontinuity at (0, 0) and thus does not
satisfy the conditions ofFubini's Theorem. In fact, both iterated integrals involve improper integrals which diverge at their
lower limits of integration.
15.3 Double Integrals-over General Regions
2
3. J; J:2 (1 + 2y)dy dx = f0 [y + y J ~:: 2 dx =
1
.r; [x + x 2- x
2
-
(x2 ) 2 ) dx
= fo1(x - x4)dx = [tx2 - !xs)~ = t- i - 0+0 = fo
® 2012 Ccngagc LC<lllling. All Rights RJ:sc:m:d. May not be SC4lU1Cd, copied, or dupllcalcd, or posted to • publiCly occcssiblc website, in whole or in p:ll1.
252
7.
D
CHAPTER 15 ' MULTIPLE INTEGRALS
IIv y2 dA = I~t I~~-2 Y2 dx dy = I~t (xy2J ::~v-2 dy = I~l y2 [y - (- y- 2)] dy
1
= J-11 (2y3 + 2y2)dy = [ly4
+ 1y3J
= 12 + 13 - l2 + 13 = :!.3
2
3
- 1
9. ffv x dA =
·
f"
ro•ln"' x dy dx = ~" [xy]Y:'"oin "' dx = f" x sin x dx [ .
.lo J t
Jo
Jo
Y-
integrate by parts
wtth v.=x, rlv = sm x dx
]
= [- x cos x + sin x J~ = -1r cos 1r + sin 1r + 0 - sin 0 = 1r
y
11. (a) At the right we sketch an example of a region D that can be described as lying
betw~en the graphs of two continuous functions of X (a type I region) but not as
lying between graphs of two continuous functions ofy (a type IJ region). The
regions shown in Figures 6 and 8 in the text are additional examples.
0
(b) Now we sketch an example of a region D that can be described as lying between
X
y
the graphs of two continuous functions ofy but not as lying between graphs of two
continuous functions of x. The first region shown in Figure 7 is another example.
0
13.
X
As a type I region, D lies between the lower boundary y = 0 and the upper
y
(1, 1)
boundary y = x for 0 ::; x ::; 1, soD= {(x, y) I 0 $ x $ 1, 0 $ y $ x}. lfwe
describeD as a type ll region, D lies between the left boundary x = y and the
x =l
right boundary x = 1 for 0 ::; y ::; 1, soD = {( x, y) I 0 ::; y::; 1, y ::; x::; 1}.
X
Thus
3
IIv xdA =I; IC: xdydx = I 01 [xy] ~ =~ dx = I01 x 2 dx = !x ]~ = !(1 - 0) = ~or
.rf'v x dA = f 01
15.
I:
xdxdy =
I (tx J : : ~ dy = t J;(1 - y dy = t[y - h ] ~ = t((1- !) - 0] = ~ 1
0
y
2
2
)
3
The curves y = x - 2 or x = y + 2 and x = y 2 intersect when y + 2 = y 2
y 2 - y- 2 =0 ~ (y-2)(y+ 1)=0 ~
~
y = -1, y =2,so thepoints .of
inte rsection arc (1 , - 1) and (4, 2) . Tf we de~c ribe D as a type I regi on, the uppe r
boundary curve is y = ..fii but the lower boundary curve consists of two parts,
y =
Thus D = {(x, y) I 0::; X::; 1, -
-vx for 0 ::; x ::; 1 andy = x - 2 for 1 ::; x ::; 4.
vx::; y::; vx} u {(x, y) 11 ::; X::; 4, X- 2::; y::; vx} and
II0 y dA = Io1 I!5z y dy dx + f14 f.,~ y dy dx. If we describe D as a type II region, D is enclosed by the left boundary
x = y 2 and the right boundary x = y + 2 for -1 ::; y ::; 2, so D = {(x, y) I - 1 ::; y ::; 2, y 2 ::; x ::; y + 2 } and
.
.
.
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SECTION 15.3
DOUBLE INTEGRALS OVER GENERAL REGIONS
0
JJ0 y dA = f~ J:/ 2 y dx dy. In either case, the resulting iterated integrals are not difficult to evaluate but the region D is
1
more simply described as a type II region, giving one iterated integral rather than a sum of two, so we evaluate the latter
integral:
ffo ydA = f~tf:/2 ydxdy = f~t (xyJ ::~~2 dy = J~1 (y + 2- y2)ydy = f~t (y2 + 2y - ya) dy
= [jy3+y2- h4]~1 =
(! +4-4) - (-~ +1 - i) = ~
19.
0
21.
3
4 2
- y ] 1 = .§1
= [§.y
3
3 - 16- ~
3 + 1- ll
3
X
!2 !~ (2x -
y
- 2
-~
2
--2
/
- 2
y) dydx
12] !1=~
r.--;; dx
2
[ 2xy - 2Y
y =-y4- x
= }~ 2 [2x v'4 - x 2 :- H4 - x ) + 2x v'4- x2 + H4 - x 2)] dx
2
=J~24xv'4-x2dx= -H4-x2)3/2]2
= 0
-2
[Or, note that 4x v'4 - x2 is an odd function, so J~ 2 4x v'4- x2 dx = 0.]
2
y
23.
2
f1 J1-a: (
f1 [
2] y -l -x
)
V= 10
1 _., 1- x+2y dydx= 10 y-xy + y !1;: 1 _., dx
1
1
=
2
[ ((1- x ) -
2
x(1- x ) + (1 - x
2
?)
- ((1- x)- x(1- x) + (1- x) 2 ) ] dx
= J01 [(x 4 + x3 - 3x2 - x + 2)- (2x2 - 4x + 2)] dx
X
25.
0
3y xy dx dy = J21 [21 x 2 y ] .,"' == 17- 3y dy
V = J12J71
y
0
X
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253
254
27.
0
CHAPTER 15 MULTIPLE INTEGRALS
V = f 0 f~ - ~"' (6- 3x - 2y)dydx
2
y
(0,3)
r2(
= Jo 6y - 3xy - y
2] vv=
3- 1x
= o 2 dx
= J; (6(3 - ~x)- 3x(3 - ~x) - (3 - ~ x) 2] dx
= J; (~x2 - 9x + 9) dx = [~x 3 - ~x2 + 9x] ~ = 6 -0 = 6
29.
31.
y
V =
111~
0
{
11[2 ] Y= ~
y dy dx =
1L
0
1
1-
X
2
0
2
1
1
3 1
.
dx
y= O
1
= Jo - 2- dx = 2 [x - 3·-c Jo = 3
X
From the graph, it appears that the two curves intersect at x = 0 and
33.
at x ~ 1.213. T hus the desired integral is
x
d
fl. 213 [
x•l
X y u.'C - Jo
xy ] = "'- "' dX
JfD X dA ~~ Jorl .213J3xy = x4.
2
r L213(
= Jo
3X 2 - X 3 -
.J.
X
y
-
6) d X= [X 3 -
1
4X
4-
2
3
1
aX
G] O1.213
~ 0.713
35. The two bounding curves y = 1 - x 2 and y = x 2 - 1 intersect at (±1, 0) with 1 - x ~ x
2
2
-
1 on (- 1, 1]. Within this
region, the pla n~ z = 2x·+ 2y + 10 is above the plane z = 2 - x - y, so
2
2
•r1-_x (2x+2y + 10 ) dydx -,rl
V = _1 1 J,.
_ 1 Ji-~
., 2 _ ( 2-x-y) dydx
2 1
1
J
2
(
•
= J•l_1 J1-x
.,2_
2x +' 2y + 10- ( 2- x - y )) dy dx
1
1 [
3 2
Jy = l - :c2 dx
= j •l_1 f,l2-:c2
_ 1 (3x + 3y + 8) dy dx = f_ 1 3xy + 2Y + 8y y=z _
2 1
2
= J~ 1 [3x(l - x 2 ) + ~(1 - x 2 ) 2 + 8(1- x 2 ) - 3x(x2 - 1) - ~(x - 1)2 - 8(x2 - 1)] dx
1
4
3
2
= J-1 1 (-6x"3 - 16x2 + 6x + 16) dx = [-.:!x
- .l&x + 3x + 16x]
2
3
-1
= -~ - ~ + 3 + 16 + ~ 6
¥ - 3 + 16 =
64
3
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SECTION 15.3
DOUBLE INTEGRALS OVER GENERAL REGIONS
D
255
37. The solid lies below the plane z = 1 - x - y
y
or x + y + z = 1 and above the region
D={(x,y)IO~x~l,O~y~ 1 -x}
in the xy-plane. The solid is a tetrahedron.
X
,
X
39. The two bounding curves y = x 3 - x andy = x 2 + x intersect at the origin and at x = 2, with x 2 + x > x 3 - x on (0, 2).
· Using a CAS, we find that the volume is
V =
121x2 +x
o
Z
d d
y
'121:r:2 +:r:( y ,+
X=
X
0
x3- x
3 4
. XY
2) d
d
y -X=
xa - :z:
13,984,7:'!5,616
"
14,549,<>35
2
41 . The two surfaces intersect in the circle x 2 + y = 1, z = 0 and the region of integration is the disk D : x
Using a CAS, the volume is
'
43.
!In
(1- x
2
2
y ) dA =
-
D
y
2+ y ~ 1.
2
.
111~
(1 - x 2 - y 2 ) dydx = ~.
-Vl-x2
2
- 1
Because the region of integration is
D = {(x,y) I 0 ~ x ~ y, 0 ~ y ~ 1} = {(x , y) I x ~ y ~ 1, 0 ~ x ~ 1}
we have f 0 J~ f(x, y) dx dy = ffv f(x, y) dA = f0
1
1
J:
f(x, y) dy dx.
Because the region of integration is
45.
D = {(x,y) I 0 ~ y ~ cosx,O ~ x ~ 1rj2}
= {(x,y) I 0 ~ x ~ cos- 1 y, O ~ y ~ 1}
we have
foTr/2 focou f(x,y)dydx = ffv f(x , y)dA = fol f ocos-lv f( x,y)dx dy.
Because tlie reg_ion of integration is -
47.
D = {(x , y) I 0 ~ y ~' Inx, 1 ~ x ~ 2} = {(x ,y) J e 11 ~ x ~ 2, 0 ~ y ~ ln2}
In 2
we have
/, 211" f(x,y).dydx = ~~ f(x,y)dA = 11n 2[2 f(x,y) dxdy
x
l
49.
11
1
y
o
3
3u
0
ex 2 dxdy =
0
D
cV
131:r./3 ex dydx = 13[ex y ]u=x/3dx
2
2
o o
o
u=O
(3, 1)
X
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256
0
CHAPTER 15 MULTIPLE INTEGRALS
y
51.
53.
y
111</.2 cosx )1 + cos
1
y = sinx or
x= arcsiny
0
\
2 xdxdy
arcsmy
= f 01f
12
J;
1
""' cos x
~:os 2 x dy dx
J;12 COSX Vl + COS 2 X [y]~=~in:c dx
= f.o7r /2 COS X Vl + COS2 X sin X dx.
[Let u = cos x, du ,;, - sin dx, ]
=
X
X
dx = du/(- sinx)
0
~2 du = -Hl+u)
2 3/ 2]
=J10 -uvl+u
1
=Hv'B-1)=H2v'2- 1)
55. D = {(x, y) I 0 $ x $ 1, - x + 1 $ y $ 1} U {(x, y) I - 1 $ x $ 0, x + 1 $ y $ 1}
U{(x,y) I O$x$1, - 1 $y$x -1}U {(x,y) l -1 ::;x::; o, -1::;y::; -x- 1}, alltypel.
fir
x2 dA =
D
11,1 x dydx + 1011 x dydx + 11J•x-1 x dydx + 101-x-1x dydx
1
0
2
1
=4 {
1
{
f o l1 -"'
2
2
1-x
- 1
x 2 dydx
2
2
x+1
0
- 1
-1
-1
[by symmetry of the regions and because f(x, y) = x ~ 0]
2
2
2
e- 1 116 :::; e-<x +t?l $ e0 = 1 s ince et is art increasing fun~tion. We have A(Q) = ~1r (~) = fG, so by Property II,
2
2
2
2
e- 1116 A(Q) :::; J,h e-(x +112 ) dA:::; 1 · A(Q) ~ {Ge- 1116 :::; JJQ e-(x +112 ) dA:::; {G or we can say
.
2
2 2
0.1844 < JJQ e - <x +v l dA < 0.1964. (We have rounded the lower bound down and the upper bound up to preserve the
'
.
inequalities.)
59. The average value of a function f of two variables defined on a rectangle R was
defined in Section 15. I as !ave =
AlR) ffn f(x, y)dA. Extending this definition
y
(1,3)
to general regions D, we have fnvc = A(1D) ffv f(x, y)dA.
Here D = {(x, y) I 0$ x $ 1,0$ y $ 3x}, so A(D) = ~(1)(3) = ~and
f .... = A(1D) ffv f(x,y)dA = ~ fo1 fo3 x xydydx
= 1 fl [lxy2] v=3x dx = 1 f1gx3 dx = lx4]1 - 1
3 Jo
2
v=O
3 Jo
4
o -
4
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X
SECTION 15.3
DOUBLE INTEGRALS OVER GENERAL REGIONS
61 . Since m :S j(x, y) :S M, ffD mdA ~ ffD j(x, y) dA ~ ffD M dA by (8)
0
257
=>
mjf0 ldA ~ fj~ j(x, y)dA ~ lVI .ffD ldA by(?) => mA(D) ~ ffDJ(x,y)dA ~ MA(D) by (10).
y
63.
First we' can write JJ0 (x + 2) dA = JJD xdA + JJ0 2dA. But f(x, y ) =xis
an odd function with respect to x [that is, f( -x, y) = - f(x , y)] and Dis
symmetric with respect to x. Consequently, the volume above D and below the
-3
graph off is the same as the voiume below D and above the graph off, so
X
JJD x dA = 0. Also, .f.fD 2 dA = 2 ·A( D) = 2 · ~7r(3) 2 = 97r5ince Dis a half
disk of radius 3. Thus JJD(x + 2) dA = 0 + 97r = 97r.
65. We can write JJ0 (2x + 3y) dA = J J0 2x dA + JJD 3y dA. JJ0 2x dA represents the volume of the solid lying under the
plane z = 2x and above the rectangle D . This solid region is a triangular cylinder with length band whose cross-section is a
triangle with width a and height 2a. (See the first figure.)
(a,0,2a)
(O,b, 3b)
z= 3y
y
X
2
Thus its volume is~· a· 2a · b = a b. Similarly, JJ0 3ydA represents the volume of a triangular cylinder with length a,
triangular cross-section with width band height 3b, and volume
t ·b · 3b · a ~ ~ab • (Sec the second figure.) T~us
2
JJ0 (2x + 3y) dA = a2 b + ~ab2
67. ff0 (ax3 + In/+ va2 - x 2 ) dA = ff0 ax3 dA + JJ0 by3 dA + JJ'a Ja 2 - x 2 dA. Now ax 3 is odd with respect
to x and by 3 is odd witb respect to y , and the region of integration is symmetric with respect to both x and y,
3
so ffD ax 3 dA = ffD'by dA = 0.
JJ'v J a 2 - x2 dA represents the volume of the solid region under the
graph of z = Ja 2 - x 2 and above the rectangle p, namely a half circular
cylinder with radius a and length 2b (see the figure) whose volume is
t · 1rr2 h = ~7ra2 (2b) = 1ra2 b. Th~s
JJD (ax3 + by3 + Ja2 - x 2 ) dA = 0 + 0 + 1ra2 b = 1ra2 b.
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258
CHAPTER 15 MULTIPLE INTEGRALS
D
15.4 Double Integrals in Polar Coordinates
1. The 'region R is more easily described by polar coordinates: R = { (r, B) I 0 $ r::; 4, 0::; B::; 3; }.
Thus JJR f~x, y) dA = J:'~~' 12 f 0 f(r cos fJ, r sinO) r dr dB.
4
3. The region R is more easily described by rectangular coordinates: R = { (x, y) I -1 :=:; x :=:; 1, 0 :=:; y :=:; ~ x + ~}.
Thus JJRf(x,y) dA = J~ 1
5. The integral
J0("'+ 1)/ 2 f(x,y) dydx.
J:;t J r dr dB represents the area of the region .
2
1
y
0=31T
4
R = {(r,B) 11::; r::; 2, 7r/4 $ '0::; 37r/4}, the top quarter portion of a
ring (annulus):
t j r dr dO = ( J:;t d()) (f r dr)
J:;
2
1
2
1
X
= [B]3'11'/4 [!r2]2 = {3.,. _ .!!.) . 1 (4 _ 1) = 1!.. i! = 3'11'
'II' /4
2 -
1
4
4
2
2
2
4
7. The half disk D can be described in polar coordinates as D = { (r, B) I 0 ::; r ::; 5, 0 ::; B ::; 1r}. Then
ffv · x 2 y dA = J0"
J; (r cos B) (r sin B) r dr dB= (J cos Osin BdB) ( J~ r dr)
2
0.,.
4
2
= [-% cos3 OJ~ [ir 5 ]~ = -%(-1-1) · 625 = 12i 0
9. JJR sin(x
2
+ y2 ) dA = JJ:/ 2 J{ sin(r2 ) r dr dB= (!0.,.12 dO) (J13 r sin(r2 ) dr)
= [0]~12 [-~ cos(r 2 )]~
= (:~) [-~(cos9- cos 1)] = -;f(cos 1 - cos9)
_ ,2 _ u2 dA- J'll'/ 2 f2 -r2 d dB- J"'/2 dB r2
11 . f ·r
Jn e
-'11'/2 Jo e
r r - -'11'/2 Jo re-r2 dr
2
2
2
, = [B]"'/
[-le-r ) = 1r{-l)(e-4 - e0 ) = 1!.(1- e-4 )
-'11'/2
2
0
2
2
y
13. R is the region shown in the figure, and can be described
y=x
by R = {(r,fJ) I 0::; B ::; rr/4, 1$ r::; 2}. Thus
JJR ~ctan(y/x) dA = f0'~~' 14 f 1 arctan(tan B) r dr dO since yjx =tan 0.
2
Also, arctan( tan B) = B for 0 ::; B ::; 1r /4, so the integral becomes
. .lo
f":r/4 f20
-'"f'11' / 4"d(}
f21 r d r -- [!02)"'/4
[l2 r 2]21 -1
r dr uu
- Jo
u
2
o
.,-2. ~- _1_
32
2 -
a.t7r 2 ·
15. One loop is given by the region
D = {(r,O) 1-rr/6 S 0 $ rr/6, 0 S r $ cos30}, so the area is
11 !
dA =
D
7r
/61cos SO
-'11'/6
.
rdrdO =
0
~.,. /6 [ 1 2] r=coo 30 _
r
dO
-'11'/6 2
r=O
16
_! cos2 30 dfJ = 2 f"
}_71: 16 2
}0
= {"'
1[
16
_! ( 1 +cos 60 ) dO
2
2
6
1 sin 60]"'/ = ;;
= 2 0+G
0
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SECTION 15.4 DOUBLE INTEGRALS IN POLAR COORDINATES
17. In polar coordinates the circle (x- 1}2 + y 2 = 1
2 cos 0 = 1
x 2 + y 2 = 2x
is
r 2 = 2r cos 0
~
1'
259
= 2 cos 0,
= 1 is r = 1. The curves intersect in the first quadrant when
2
2
~
cos 0 = ~ · .~
and the circle x + y
¢}
0
0 = 7r / 3, so the portion of the region in the first quadrant is given by
D = {(r, 0} 11 ~ r ~ 2 cos 0, 0 ~ 0 ~ 7r/2}. By symmetry, the total area
O= Tr/3
is twice the area of D:
,/
r=2oos(J
2A(D} = 2ffvdA = 2Ja"/3 j l2coao rdrdO = 2fo"/3 ar2)~~~coao dO
= J;/ (4cos2 0- 1) d(} = J;/ [4 · ~( 1 +cos 20}- 1] dO
3
3
= J0" 13 (1 + 2 cos 20} dO= [0 + s~n20] ~;a = f + ~
21. The hyperboloid of two sheets - x 2 - y 2 + z2 = 1 intersects the plane z = 2 when -x2 - y 2 + 4 = 1 or x 2 + y 2 = 3. So tlie
solid region lies above the surface z = )1 + x 2 + y 2 and below the plane z = 2 for x 2 + y 2 :::; 3, and its volume is
II ( ) 1
V=
2-
+ x 2 + y 2 ) dA =
2
la " 1av"J ( 2 - ) 1 + r2) r dr dO
:r:2 +y2 ~3
= f~" dO f ov"J (2r- rv'f+'Ti) dr = [0] ~.,.. [r2 - t {1 + r 2 ?12 ] :
= 27r (3- i- 0 + 1) = ~7r
23. By symmetry,
2
V = 2
Ja 2 - x2 - y 2 dA = 2 1a "1aa )a 2 - r 2 rdrdO = 2 1az... dO
//
26. The cone z = Jx 2 + y 2 intersects the sphere x 2 + y 2 + z
V =
loar Ja2 - r2 dr
.
~+~~~
/1 (
)1- x
2
2
= 1 when x 2 + y 2 + ( Jx2 + y2 ) = 1 or x 2 + y 2 = ~· ~0
2- y 2- )x2+y2) dA = la27rlal/-/2 () 1 - r2 - r )rdrdO
.,2 + y2 ~ 1/ 2
= f~1r dO f 011V2 (r Jl-T2 - r 2 ) dr = [ 0]~7r [-~(1 - r 2 ) 3 12 - ~r3 ) :/-.12 = 271'( -~) ( ?z - 1) = f(2 - V2)
27. The given solid is the region inside the cylinder x 2 + y 2 = 4 between the surfaces z = J64 - 4x2 - 4y2
and z = - )64- 4x2 - 4y 2 • So
V =
x2
IIs [J
+ 112
64 - 4x2 - 4y2 - ( --:
JJ J
J 64 - 4x2 - 4y 2 ) ] dA =
4
2
+2
= 4J~.,. J~ ~ rdrdO = 4J:.,.. dO J: r v'16 - r 2 dr = 4 [ OJ~".
3 2
= 8Tr( - ~)(12 1
64 - 4x2 - 4y2 dA
+112 ~ 4
[-1(16- r 2 ) 312
J:
-162 ' 3 ) = 8
; .(64 - 24 v'3)
© 2012 Cengage Learning. All Rights Rc"S<m:d. May nol be seanned, copied, or duplicaled, or posl~ 10 a publicly accessible wcbsile, in whole or in part.
260
D
CHAPTER 15 MULTIPLE INTEGRALS
3
29.
/
{~ sin(x + y )dy dx = Jo{" Jor sin e) r dr dB
2
2
-3lo ·
= fo" dB J; r sin (r2 ) dr = [B)~ [-~cos (r2 )J~
= 1r (-~) (cos9 -1) = ~ (1 - cos9)
-3
0
3
X
y
31.
f0" 14 fov'2 (r cos B + r sin fJ) r dr dB == f 0" 14 (cos B + sin B) dB f 0v'2 1·2 dr
== [sin(}- cosB)~/
4
[i r 3 ]~
= (¥ - ¥- O+ 1] . i (2v'2- o) = ¥
33. D = {(r-, B) [ 0::; r ::; 1, 0 ::; 8::; 271' }, so
ffD e<x +1h dA = 1~ ...
2
fl
2
J; e<r
2 2
)
r dr dB =
J:" dB f rer dr = 271' J r er dr. Using a calculator, we estimate
4
1
0
4
1
0
,,.1 d
27rJo re
r ~ 4.5951.
35. The surface of the water in the pool is a circular disk D with radius 20ft. If we placeD on coord inate axes with the origin at
the center of D and define f(x, y) to be the depth of the water at (x, y), then the volume ofwater·in the pool is the volume of
the solid that lies above D = { ( x, y) I x 2 + y 2 ::; 400} and below the graph off (x, y). We can associate north with the
positive y-direction, so we are given that the depth is constant in the x -direction and the depth increases linearly in the
y-direction from f(O, -20) = 2 to f(O , 20) = 7. The trace in the yz-plane is a line segment from (0, - 20, 2) to (0, 20, 7).
The slope of this line is 20 ~(_?20 ) ~
t. ~o an equation of the line is z- 7 = t(y- 20)
=?
z = ty + ~· Since f(x, y) is
ty +~·Thus the volume is given by ffv f(x, y) dA, which is most conveniently evaluated
independent of x, f(x, y) =
using polar coordinates. Then D = { (r, fJ) I 0 ::; r ::; 20, 0 ::; B ::; 271'} and substituting x
= r cos B, y = r sin B~he integral
becomes
J;. .1~ tr sin + ~) r dr dB = 1~... [i4r sin B + ~r ]
20
2
3
(J
(
1
~ dB = g•· (10~ 0 s in O+ 900) dB
0
;::
0
= (- ~ cos B + 9008] ~" = 180071'
Thus the pool contains 18007r ~ 5655 ft3 of water.
37. As in Exercise 15.3.59, f,,.• = A(~J ffv f(x, y)dA. Here D = {(r, fJ) I a::; r::; b, 0::; ()::; 271'},
so A(D) = 1rb 2 -1ra2
1
!ave= A(D)
= 1r(b 2 - a 2 ) and
!1
_
1
-7r(b2 - a2)
1 lb
2
1
~
dA = 7r (b 2 1- a 2 )
D y x 2 + y2
"
0
. a
1 rdrdB = (b 1
o;
)
7r 2 - a 2
V r·2
12" ;·b
0
dB
a
dr
[e]2rr (]b
1
( )(b
)
2(b-a)
2
o , rn=11'(b2 - a2) 2 7r -a = (b+a)( b - a) = a+b
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SECTION 15.5
APPLICATIONS OF DOUBLE INTEGRALS
0
261
·Y
=
1rr/4 [
r
r
r cos 8 sinfJdrd8=
Jo
4
?r/4
2 3
4
] r=2
.!:.... cos8sin8
d()
o
J1
[sin
r = 1
4
2
4
151rr/ .
15
()] .,.;
15
sm8 cos9d(J = - - = = 4 0
4
2
0
16
•
.
2
2
41. (a) We integrate by parts with u = x and dv =·xe-x dx. Then du = dx and v = - ~e-x , so
f "" x 2 e- "'
Jo
2
dx = lim rt x 2 e-"' dx = lim ( - l xe- "'
t-+oo Jo
t-+oo
2
2
2
)t + Jort l e- "' dx)
2
2
0
= lim (_.!.te-t2) + .!. roo e-x 2dx = 0 + .!. roo e- "'2dx
2 Jo
2
t-+oo
= 41 f "" e_.,2 d X
2 Jo
[by !'Hospital's Rule]
2
[since e-"' is an even function]
-oo
[by Exerc ise 40(c)]
(b) Let u = ..JX. Then u 2 = x
dx = 2udu
.::?
.::?
2
f.0 .../Xe- "'dx
= lim I; .../Xe-"'dx = lim I0,/f.ue-" 2u du = 2I0 u 2 e- u du = 2 U.J7T)
·
t-oo
t-oo
2
00
00
[by part(a)] = ~ft.
#
15.5 Applications of Double Integrals
2
1. Q = IIv O"(x, y) dA = I~ I: (2x + 4y) dydx =I~ [2xy + 2y ] ~=~ dx
= I~ (lOx + 50- 4x- 8) dx = J~ (6x + 42) dx = [3x 2 + 42x]~ = 75 + 210 = 285 C
3. m
= IIv p(x, y) dA = I13 I14 ky 2 dydx = k I13 dx .{14 y 2 dy = . k [xJ~ [h 3 ]~ = k(2)(21) = 42k,
x = ~ IIv xp(x,y ) dA = 4 ~k I :
- = m
1 ffD yp(x , Y )dA -Y
1
42k
I14 kxy2 d ydx = 412 I13 xdx I 14 y 2 dy = f2 [tx2 ]~ [ky 3 ]~ = 4\(4)(21) = 2,
f 1a .f14kYa dy dx1 f 1sdx Jt
r4 ad _ 1 [ ]3 [1 4] 4 - 42
y y - 42 X 1 4Y 1 -
1 ( )( 211:;)
85
42 2
-4- = 28
Hence m = 42k, (x, Y) = (2, ~) .
r2f3
-x( x+y) dydx = Jo
r2 [xy+2Y
1 2]v=3-:z:
r2 [x ( 3- a x) + 1 ( 3 - x )2 - 1 x2] dx
5. m= Jo
.,12
v=x/ 2 dx =J~
2
2
8
=
.
2
3
J02 (-~x + ~) dx = [- ~(tx ) + ~x] ~ = 5,
r2J·3-x( x 2 + xy )dy dx = Jo
r2 (x 2y + 2xy
1 2]11=8:z: dx
r2 (92x - gX
9 3) d
1l=x/
= Jo
x = 92•
lvfv = Jo :r:/ 2
2
f2J3-v(
r2.[12xy2 + 3Y
1 3)11=3-x
d
r 2 (g - 2x
9 )dx = 9 .
lvfx = Jo
., 12 xy + y 2)dy dx = Jo
11=, 12 x = Jo
Mx) = (34' 23).
m '--;:;
- -)
(lvfy
Hencem = 6, (x_, y =
7. m = J~ 1 J~ -"' kydydx = k f~ 1 [~y2 ]~=~-"' dx = ~k f~ 1 (1 - x 2 ? dx = ~k f~ (1 - 2x2 + x 4 ) dx
2
2
1
= ~k [x- ix3 + %x5 ] ~ 1 = ~k (1 - i + -k + 1 - %+ %
) = 185 k ,
® 2012 Cengnge Learning. All Rights Reserved. Mny not tJe scanned, copied, or duplicated, or posted to a publicly acc•:ssiblc website, in whole or in pan.
262
D
CHAPTER 15 MULTIPLE INTEGRALS
Mv ·= .f~
1 ,{0 -:r kxydydx = kf~ 1 axy2 ]~=~-:r dx = ~kf~ 1 x(1 - x 2 ) 2 dx = ~ kf~ 1 (x- 2x 3 + x 5 )dx
1
2
2
2 - lx 4 + lx6 ] 1 - l k (1- l + l - l + l _ l) - 0
-- lk(lx
2
2
2
6
-1 - 2
2 . 2
6
2
2
6 '
2
1
1
3
2 3
] v=l-x dx = lk f
M x = J-1 1 Jo
r 1 -"'~ ky 2 dydx = kf-1
(1y
{ 1 (1'- 3x2 + 3x4 - x 6 ) dx
3
v=o
s -1 (1 - x ) dx = lk
3 . - 1
8 k , (-x , -)
_
Hencem -_ 15
y -_ (o, ·32k/105)
Bkf l ~
-
(o •74)·
9. Note thatsin(7Tx/ L) ~ 0 for 0 ::::; x::::; L.
m=
J:'
J;in(rrxfL) ydydx
= g t sin2 (7Tx/L) dx =ta x- 4~ sin(27rx/L) j~ = ~ L,
. d dx _ .! rL
· 2(11'XjL) dx
M.y -_ j 'L
o j·sin(7rx/L)
o
X Y Y
- 2 Jo X SID
integrate by parts with
[u
= x , dv = sin2 (7rx/ L) dx
]
r;, sin(27Tx/L)) ] ~ - ~ J [tx - 4~ sin(27Tx/ L)] dx
= t · x(tx -
0L
4
= l£2l [.!x2 + L2 cos(211'xj L)]L = .!£2 - l (.!£2 + L2 - L2) = l£2
4
2 4
41f2'
0
4
2
4
41f2' 41f2'
8
'
M:z: = J 0L J;in("'x f L ) y · y dy dx = f0L ~ sin3 (7Tx/ L ) dx = ~ f 0L [1 - .cos2 (7Tx/ L )] sin(7Tx/ L ) dx
[substituteu =cos(7Tx/L))
::}
du= -fsin(7Tx/L))
= t(- ~)[cos(7Tx/L) - 1cos3 (7Tx/L)]~ = - 3~ (- 1 + ~ -1 + i) = 9~L.
L _ _
Hencem = 4' (x,y) =
(L /8
2
L/ 4 '
11 . p(x, y) = k y = kr sin O, m = )~,.
4£/ (911'))
L /4
=
12
(L2 ' 16)
971' .
J; kr sinO dr dB= ik )~.,. sin Odtl = ik [- cosB]~ = ftk,
12
2
12
J0,.12 J; kr3 sinB cos BdrdB = i k f 0" 12 sinB cosB dB= ~k [- cos 28] ~ = lk,
1
12
12
M x = J0,.12 J0 kr 3 sin2 B dr dB = ik f 0"' sin2 BdB = lk [B + sin 28] ~ = fak .
12
Mv =
Hence (x, y) = (i, ~~) .
13.
p(x, y) = k jx2 +y2 = kr,
y
m = ffo p(x, y)dA = f 0"' f 1 kr · r dr dB
2
= k f0"' dB f 12 r 2 dr = k(11') [ ir3 ]~ = ~11'k,
= k [sinO]"' [ .! r 4]2 = k(O) (.!&) = O .
0
4
1
4
[this istobe expectedastheregionanddensity
function are symmetric about they-axis]
M:r = ffo yp(x, y)dA = fo,.t <r sin B)(kr) r dr dB = k
J0"' sinBdB f 12 r 3 dr
= k [-cosO] ~ (tr4 ] ~ = k(1 + 1) ( ~5 ) = lfk.
H ence (-x, -y ) -_ (o' 15k/ 2) _ (o ~)
14 ,. .
7-rrk/ :J
-
'
@ 2012 Ccnpgc Learning. All Rights Reserved. May not be seaMed, co pied, or duplicated, or posted to • publicly accessible website, in whole or In pori.
SECTION 15.5
15. Placing the vertex opposite the hypotenuse at (0, 0), p(x, y) = k(x
m =
J; foa-"' k (x + y dy dx = k foa [ax
2
2
)
2
-
x
3
2
APPLICATIONS OF DOUBLE INTEGRALS
D
+ y 2 ). Then
+ 4(a - x)3 ] dx = k[~ax3 - tx4 - 112 (a - x) 4 ]~ = ~ ka4 •
By symmetry,
Hence (x, y) = (~a, ~a).
17. I:r: =
JJ0 y 2 p(x, y)dA = .{~ 1 J; -x y 2 · ky dy dx = k j~ 1 [h 4J~:~-x dx = tk f~ 1 (1 - :t2 ) 4 dx ·
2
2
1
1
9
7
5
3
- ix + !!x - :!x + x]
-- l4 kf-1
(x 8 -4x6 +6x4 - 4x2 + l)dx = l4 k [lx
= .M..k
9
7
5
3.
- 1
315 '
2
2
Iy = fio x 2 p(x, y) dA = }~ 1 j~-"' kx~ydydx = k I~ 1 [~x2 y 2 J ~:~-"' dx = ~k t 1 x2 (1- x 2 ? dx
It
1 3
= 21 k• - 1 <X 2 - 2X 4 +X6) d X= 21 k [ aX
2
s X
s + "'fX
1 1] 1 _
- 1 -
8 k
105 '
and Io = I, + Iv = 36~k + t~sk = 38t8sk.
19. As in Exercise 15, we place the vertex opposite the hypotenuse at (0, 0) and the equal sides along the positive axes.
Ix = I~' I;-xy2k(x2 +y2)dydx = k.fo" .fo"- "'(x2y2 +y4 )dydx = k.fo"[~x2y3 + h5J~:~-x dx
3 x 3 - 1a2x 4 + !l.ax 5 - lx6 ) - l (a- x) 6 ] a = - 1-ka6
= k .f'a
[lx 2 (a- x) 3 + l(a
- x) 5] dx = k [l3 (la
0 3
5
3
4
5
6
30
0
180
'
Iv = I; I; -x x 2k(x 2 +y2 ) dydx = k J0a.f; -"'(x4 +x2 y2)dydx = k .f0a[x4 y + ~x 2 y 3 J::~-x dx
5 - lx 0 + l (l a 3 x 3 - !l.a 2x 4 + !!ax 5 - lx0 ) ) a - - 1- ka 6
-- k j 0·a [x 4 (a - x) + l3 x 2 (a - x) 3) dx = k [lax
5
6.
3 3
4
5
6
0 - 180
'
21. I,.
= IIo y 2p(x, y)dA = Io" I~ py dx dy =pI: dx I~' y dy = p[ x ]~ [~y ]~ = pb(th3 ) = tpbh3 ,
2
3
2
2
2
3
3
2
Iv = IIo x p(x, y)dA = Io" I~ px dxdy =pI: x dx I~' dy = P[ tx ]~ (y)~ = ipb h,
and m = p (area of rectangle) = pbh since the lamina is homogeneous. Hence ¥
= 2
3
2
l
l
b3 h
pbh
b2
= ...JL = L _ = m
3
2
I:r:
tpbh
=- = -h3
m
pbh
=>
andy = -
23. In polar coordinates, the region is D = {(r, 8) I 0 ~ r ~ a, 0 ~ 8 ~ ~},so
! 11 = .{[0 x 2 pdA = f 0"" 12
J; p(r cos 8) r dr d/J = p I
2
0""
12
cos2 dB .f0o. r 3 dr
= p(t8 + t sin 28)~ 12 (tr 4 )~ =~Of) (ta4 ) = ftPa 4 rr,
and m = p · A(D) = p ·
.l...pa4rr
a2
trra since the lamina is homogeneous. Hence ¥ = y = ~
= -4 => x = y = ~2 .
pa
2
2
2
4
1r
® 2012 Ccngogc Lc:uning. All Rights Re.~rwd. May not be scanned. copie-d. or duplicated. or posted to a publicly accessible website, in whole or in port.
263
264
D
CHAPTER 15 MULTIPLE INTEGRALS
25. The right loop of the curve is given by D = {(r, 0) I 0 ~ r :::; cos 20, - r./4 ~ () ~ 71'/ 4}. Using a CAS, we
4
find m = f fo p(x,y) dA = ff0 (x + y ) dA = r:~; f oco"
2
1
· x= -
m
1
y =-
m
(-x, _y )
i'!r
/lr
2
20
r 2 rdrd9 = ::. Then
j "'/4 [ os20 (rcos9)r rdrd8= -64 ~Tr/4 1cos20 r cos9dr d8= 16384J2 and
o
64
xp(x,y)dA= 37r
D
64 ~rr/4 1cos28
64
yp(x, y) dA = . (rsin 9) r 2 rdrd(J = 3 7r -rr/4 o
37r
= ( 16384v'2
1039571' '
o)
2
4
o
-Tr/4
371' - Tr/4
j"'/4
1039571'
0
cos20
-Tr/4 1 o
r 4 sin 9dr d0 = 0, so
.
The moments of inertia are
_ .[JD y 2 p ( x, y )dA -'
"/4 4 Jo
rcos20(r sm
. (})2 r 2 r dr dB -- J"/4
j ·cos20 r s sm
. 2(J d r d(J571'
4
I :r - J -Tr/
- 7r/4 o
- 384
- 105,
2
ly = JJ0 x p(x, y) dA = J~~;
4 J;
28
0
"
4
20
2 2
(Hos0) r r dr dO = J~~~ J 0c"" r 5 cos2 0 drdO = :~ +
1~5 , and
571'
Io = I, + I y = 192
27. (a) f(x, y) is a j oint density function, so we know JfR2 j(x, y) dA = 1. Since f(x, Y) .= 0 outside the
rectangle [0, 1) x [0, 2], we can say
J JR2 f(x,y)dA = f~oo f~oo f(x,y)dy 'dx .= f0
1
J; Cx(1 + y ) dy dx
1
2 1
= C Jo
f x [y + ly ]v= dx = c' f 4xdx ~ C (2x ] = 2C
2
v=;O
Jo
o
1
Then 2C = 1
2
2
=> C = t.
J 1 J; tx(1 + y) dy dx
(b) P (X :::; 1, Y ~ 1) = f~oo f~oo f(x, y) dy dx = 0
·
1 [
1 2] Y = 1 d
r1 1 ( 3) dx
1 2] 1
3
= Jor l 2x
Y + 2Y v=O x = Jo 2x 2
= 43 [ 2x
o = 8 or 0 ·375
(c) P (X+ Y :::; 1) = P ((X , Y) ED) where Dis the triangular region shown in
Y
the figure. Thus
· :s; 1)= If 0 f(x , y)dA = f 01 f 012:
1
P(X+Y
2x(1+y)dydx
2 ] v=l-:~: dx
= Jo
t lx
[y + ly
2
2
=o
1
11
2 - 2x + ~) dx
= Jof 1 lx(lx
2
2
2
1
3
2] 1
= l41
(x3 - 4x 2 + 3x) dx = l4[4
L•I - 4L
+ 3L2()
0
3
D
1
0
X
= 4~ ~ 0. 1042
. 29. (a) f(x, y) ;;::: 0, so f is a joint density function if Jf'it 2 f(x, y) dA = 1. Here, f(x, y) = 0 outside the fi rst quadrant, so
JJR2 f(x,y)dA= f ooo fo":'0.1e-(o.ox+?·211) dydx = 0.1 J~oc f ooo e-0.5:re-0.2y dy dx = O.l fooo e-o.sx dx f ooo e-0.2y dy
= 0. 1 lim
r t e- 0·5 "' dx
L--+oo Jo
lim rt e- 0·2 Y dy = O.l. lim (- 2e- 0·5 "'] t lim [- se-0·211 ] t
t-oo Jo
t.--+eo
0 t --+oo
0
= 0.1lim [- 2(e- 0 · 5 t - 1)] lim [- 5(e- 0·2 t - 1)] = (0.1). (-2)(0 - 1) · (-5)(0 - 1) = 1
t --+oo
t --+oo
Thus f(x, y) is a j oint density function.
.
'
© 2012 Ceng;~gc Le:uning. All Rights Rcsc:n·cd. lvfuy not be ocanncd, copied, or duplicated. or JXh1cd 10 a publicly acccssiblc website, in whole or in pan.
SECTION 1505
APPLICATIONS OF DOUBLE INTEGRALS
0
265
(b) (i) No restriction is placed on X, so
P(Y ~ 1) = f~oo I 100 f (x,y) dy dx =
Iooo Jt 0°1e-(OoSx+0. 2 y) dydx
= o ."1 roo e-0 · 5"' dx f 00 e- 0·211 dy = oo1 tim
Jo
rt e - o.s., dx t-oo
lim It e- 0 · 211 dv
1
t-+oo J o
1
= 001 lim [-2e- 0 · 5"']t lim [-5e- 0·211 ]t = 001 lim [- 2(e- 0·5 ' - 1)] lim [- 5(e-0 · 2 ' - e- 0 · 2 )]
t-oo
0 t-oo
1
t - oo
t-+oo
(0.1) 0 ( -2)(0- 1) 0 ( - 5)(0 - e- 0·2 ) = e- 0 · 2 ~ 008187
2 4
(ii) P(X $2, Y $ 4) = I~oo I~oo f (x,y) dyd.x = I 0 J0 Oo1e-<0·5 "'+0·2 Y) dydx
= 001 1~2 e- o.sx dx 1~4 e- 0 ·2 ll dy = 0.1 [-2e- 0 · 5 "'] ~ [.- 5e- 0 · 211 ]~
= (0.1) · (-2)(e- 1 -1) · (- 5)(e- 0·8 -1)
= (e- 1 - 1)(e- 0 · 8 - 1) = 1 + e- J.B - e- o.s - e- 1 ~ 0.3481
(c) The expected value of X is given by
Mt =
IIR2 x f(x, y) dA = 1~00 1~00 :z; [oo1e-(o.sx,+o. 2 v)J dydx
= 0.1 ];00 xe-O.ux dx ]; 00 e- 002!1 dy = 001 lim ];t xe- 005"' dx lim
0
t~oo o
0
rt e- 0 ·2 Y dy
t - oo ./o
To evaluate the first inteirat, we integrate by parts with u. = x and dv = e- 0·5 "' dx (or we can use Formula 96
in the Table of integrals): I xe- 0 · 5 "' dx = - 2xe- 0·5 " ' -
J - 2e- 0·5 "' do'l = -2xe- 0·5 "' - 4e- 0 ·5 "' = - 2(x + 2)e- 0 ·5"'0
Thus
JJ. 1 = 0°1 lim [-2(x + 2)e- 0·5 "'] '· lim [- 5e- 0·2 ll] t
0 t -+oo
t -+oo
= 001 t lim
( -2) [(t +
-...oo
= 0.1 (-2)(lim t +.
2)e- 0 · 5 t 2
t -+oo eO.at
0
2] lim ( - 5) [e- 0·2 t - 1]
t-+oo
- 2)(- 5)(- 1) = 2
[by !'Hospital's Rule]
The expected value ofY is g iven by
J1.2 =
Ifa2 y f(x , y) dA = Io00 fa"" v [oo1e-(0.5+0.2 y)]dy dx
= 001 ];0 e-0 · 5 "' dx .{;0 ye- 0 · 211 dy = 001 t-oo.
lim {;t e-o.sx dx lim rt ye- 0 · 2 Y dy
o
t-oo _J o
00
00
To evaluate the second integral, we integrate by parts with u = y and dv = e - 0·2 !1 dy (or again we can use Formula 96 in
the Table oflntegrals) which gives I ye- 0 ·211 dy = - 5ye- 0 · 2ll +
.r se-
0 211
dy =
·
-5(y + 5)e- 0 ·2 U 0 Then
JJ. 2 = 0.1 lim [- 2e- 0 · 0·"] t lim [ - 5(y + 5)e- 0 · 2 Y] '
0 t -+oo
t-oo
0
= Oollim [- 2(e-o. st _ 1)] lim (- 5[(t + 5)e- 0 · 2 t - 5])
t-+ oo
t -oo
=Oo1 (- 2)(- 1)o (-5)
t+5 - 5) = 5
lim - .,t
t- oo e 0 ....
0
(
[Py )'Hospital 's Rule]
310 (a) The random variables X and Y are normally distributed with JJ.1 = 45, JJ. 2 = 20, 0'1 = 005, and a 2 = Ool.
The individual density functions for X and Y, then, are !J (x) =
h (y) =
0.5
~ e- <:r-<11>)21°·5 and
27r
~ c- <Y- 20) / 0·02 0 Since X and Yare independent, the joint density function is the product
2
001
211'
.
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266
D
CHAPTER 15 MULTIPLE INTEGRALS
f(x, y) = /I(x)/2(y) =
2
e-(x-45) /0.s
1
0.5 ..,l2ir
Then P(40 <X< 50 20 < y
-
-
•
-
2
e-(y-20) /0.02 =
1
0.1 ..,f2ir
lQe- 2(x-45) 2 -so(v-2~)2.
"
.
<
25) = rso r2s f(x y) dy dx = !.!!. fso f25 e - 2(x-45)2-so(y-2o)2 dy dx
J 4o J2o .
'
.,. . 40 J 2o
·
Using a CAS or calculator to evaluate the integral, we get P( 40 $ X ~ 50, 20 $ Y $ 25) ~ 0.500.
(b) P(4(X- 45) 2 + lOO(Y- 20? $ 2) =· ffv ~ e- 2 <"'- 45 ) -so(v- 20 ) dA, where Dis the region enclosed by the ellipse
2
4(x- 45) 2 + 100(y- 20) 2 = 2. Solving for y gives y = 20
2
±to J2- 4(x- 45)2, the upper and lower halves of the
ellipse, and these two halves meet where y = 20 [since the ellipse is c•~ntered at (45, 20)]
~
4(x- 45) 2 = 2 ~
x = 4? ±' ~. Thus
!l
2
lQe-2(x - 45) -so(y- 20)
D
2
l 45+1/v'2 .
dA = lQ .
"
"
45-1/.,/2
2D+tn/2- 4(x-45)2
120-ftp,/2-
2
2
e-2(:.;-45) -so(y-20) dydi.
4(x-45)2
Using a CAS or calculator to evaluate the integral, we get P( 4(X - 45? + 100(Y - 20)2 $ 2) ~ 0.632.
33. (a) If f(P, A) is the probability that an individua l at A will be infected by <m individual at P, and k dA is the number of
infected individuals in an element of area dA, then f(P, A)k dAis the number of infectio ns that should result from
exposure of the individual at A to infected people in the element of area dA. Integration over D gives the number of
infections of the person at A due to all the infected people in D. In rectangular coordinates (with the origin at the city's
center), the exposure of a person at A is
JL
E =
kf(P,A)dA = k
JL fo
[20 - d(P, A)] dA = k
jL(1- -foJ(x- xo)2 + (y- YoF] dA
(b) If A = (0, 0), then
jL[
E = k
1-
=k
1
20
r = 20cos0
Jx2 + y2JdA
271' 1 10 (1' i r) rdrd() = 27T'k[~r - 1 r 3 ] 10
1
0
2
0
= 27T'k(50-
0
sn
'
=
60
0
2 0
~ 1T'k ~ 209k
For A at the edge of the city, it is convenient to use a polar coordinate system centered at A. Then the polar equation for
the circular boundary of the city becomes 1· = 20 cos() instead of r = 10, and the distance from A to a point P in the city
is again r (see the figure). So
= k J::~~ 2 (200 cos2 8 - 4~ 0 cos 3 8) dO = 200k J::~~ 2 [ ~ + ~ cos 20 - ~ (1 - sin2 0) cos OJ dO
= 200k [ ~ 8 +
t sin 2() - ~ sin 0 + ~ · ~ sin 0] =~ 12 = 200k [:f + 0 - ~ + ~ + :f + 0 - ~ + ~)
3
2
= 200k(i - ~) ~ 136k
Therefore the risk of infection is much lower ·at the edge of the city than in the middle, so it is better to live at the edge.
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SECTION 15.6 SURFACE AREA
0
15.6 Surface Area
1. Here z = f(x , y) = 2 + 3x + 4y and Dis the rectangle [0 , 5) x [I, 4), so by Formula 2 the area of the surface is
A(S) = ffD J[f,(x,y)] 2 + [!11 (x,y)]2 + I dA= ffo -../32 +42 +IdA= ../26ffDdA
= ../26 A( D) = ../26 (5)(3) = I5 ../26
3. z = f(x, y) = 6- 3x - 2y which intersects the xy-plane in the line 3x + 2y = 6, 'soD is the triangular region given by
{(x,y) I0:5 x :5 2,0:5 y :53- ~x}. Thus
A(S) = JJD J( -3) 2 + (-2) 2 +IdA= .JI4 ffo dA = JI4 A( D ) = JI4 (~ · 2 · 3) = 3 .JI4
5. y 2 + z 2 = 9
=* z = ~· fx = 0, fv = -y(9 - y2 ) - 112 =*
A(S) = h412
Jo2 + [- y(9- y2) -1/2)2 + 1 dy dx = h4 h 2 J9 ~2y2 + I dy dx
3
1
=t {
dydx = 3 t [sin- IL] = dx = 3 [(sin- 1 (~))x]~ = I 2sin- 1(~)
lo lo J9 - y 2
lo
· 3 v=O
2
11
2
7. z = f(x, y) = y 2 - x 2 with I ::; x 2 + y 2 ::; 4. Then
A(S) = Jf D J 1 + 4x2 + 4y2 dA = .{0 " f 1 Vf+"4r2 r dr dO = J:.,. dO J1 r -../I + 4r2 dr
2
2
= (0]~" u2(I + 4r2) 312
9. z
2
J: = {f(17vT7 - 5v'5)
= f(x, y) = xy with x 2 + y 2 :5 1, so /ex = y, f v =X =*
t"
2
2 + 1) 312] r=l dO
A(S)
= JJD Jy 2 + x2 + 1dA = 0 Jro -../r2 +I r dr dO= J;0 " [l(1·
.
3
r=O
=
A(S) =
!L
1
I:'lr H2-../2 - I) dO= 2I(2-../2 - 1)
x2 +y2
a2 - x 2 - y 2+ IdA
"/2 1 acos6 /
r2
- - + 1rdrd0
- 1r/2 o
a2 - r2
'lr f2
acoso
ar
= /_
drdO
2
- .r/2 o
· Va - r 2
= /_
1
= /_"/2 [- a Ja2...:. r2 ] r=acose dO
r=O
- 1rj2
1r/2
-a( J
-1r/2
= /_
a2 -
a2 cos 2 0- a) dO = 2a
2 1 -rr/2
0
(I -
Vl - cos 0) dO
2
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267
-I
268
0
CHAPTER 15 MULTIPLE INTEGRALS
2
2
2
2
2
2
13. z = f(x, y) =e-x - v , f.,= -2xe-x - v , fv = -2ye-x - v . Then
A(S)=
II
If
v( - 2xe- xLv2 )2+(- 2ye- x2 -v2 )2+ 1dA =
x2+y2_54
.j4(x2-t-y2)e- 2(x 2 +v 2 l-t- ldA.
x2+ 11 25-1
Converting to polar coordinates we have
A(S) =
J:" .{2 J 4r2e-' 2 + 1 r dr d(J = ;;11: dl1 .{ r V 4T 2er
0
= 27!'
.r; r V 4r e2
15. (a) The midpoints of the four squares are (i 1
2
0
2
2 r 2 + 1 dr ~ 13.9783
2 2
'' +
1 dr
using a calculator.
i ), (i %) , ( %, ~)~and (i ~). Here f(x y) = x + y so the Midpoint Rule
2
2
1
1
1
,
gives
A(S) = JJD .j[fx(X 1 y)F + [fv(x, y)J2 + 1 dA = jjD .j(2x) 2 + (2y) 2 + 1 dA
~ ~ ( J[2(~)J 2 + (2(iW + 1 + V[2(~)J 2 + [2(t~] 2 + 1
+
= ~(
V[2(~)] 2 + [2(i)] 2 + 1 + V[2(%W + [2(~)] 2 + 1)
.ft + 2 JJ + /¥) ~ 1.8279
(b) A CAS estimates the integral to be A(S) = J[D .)1+ (2x ) 2 + (2y )2 dA =
.1; J .)1 + 4x + 4y dy dx ~ 1.8616.
1
0
2
2
This agrees with the Midpoint estimate only in the first decimal place.
17. z = 1 + 2x + 3y -t--4y , so
2
·A(S) =
11
1+
(~~r + (~~r dA =
1 .f
11
4
4
v 1 +4+(3 + 8y)2dydx =
1
V 14+48y+64y2dydx.
\
Using a CAS, we have
J I~ .)14 + 48y + 64y dy dx = ¥ .Jl4 + ~ ln(ll J5 + 3 .Jl4J5) - t~ ln(3 J5 + v'i4 J5)
4
1
2
45 'P4
orV J.'% + -15 ln 11 J5 + 3 v'70
8
3
16
19: f(x, y) = 1 + x 2 y 2
J5 +. v'70 .
=> f, = 2xy 2 , fv = 2x 2 y. We use a CAS (with precision reduced to five significant digits, to speed
up the calculation) to estimate the integral
A(S) =
1!1~ Vfi
- 1
- vh-x.2
+ F~ + 1 dy dx =
.
'
1!1~ ..j4x2 y 4 + 4x4 y 2 + 1 dy dx, and find that A(S) ~ 3.3213.
- 1
-~
·
21. Here z = f(x , y) = ax+ fnJ + c, f.,(x 1 y) =a, / 11 (x, y) = b, so
I
A(S) = JJD ,Ja2 + bz + 1 dA = ,ja2 + b2 + 1JJD dA = ,Ja2 + b2 + 1 A(D).
23. If we project the surface onto the xz-plane, then the surface lies "above" the,' disk x 2 + z 2 ~ 25 in the x z-plane.
We have y = f(x, z) = x 2 + z 2 and, adapting Formula 2, the area of the surface is
A(S) =
J{
,;2 +z 2 .$25
.j(f,. (x ,z )]2+[fz (x,z)]2+1dA =
JJ
,J4x2 +4z 2 +1dA
,2 +=2 .$25
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SECTION 15.7 TRIPLE INTEGRALS
Converting to polar coordinates x = r cos fJ; z = r sin fJ we have
15.7 Triple Integrals
r xy~ 2 dV = Jo
ri Jo
r3 _ xyz 2 d y d z dx = Jo
ri Jo
r3 [ 1 xy2 z 2] v=v=2 d d ·
ri r3 3 2 d z d x
1. If,JB
2
1
1 z x = Jo Jo 2 x z
J2
= Jor1 (lxz3]
==3 dx = fl TI xdx = 27 x2 ] 1 = TI
2
==O
Jo 2
4
o
4
2
ru- "(2
· x - .y)cJ:J: dydz=Jor2 J;r·
3. J02 J;" Jo
=
r2= Jor tn x x e - y dy dxdz =
5. !12Jo
J; J
2
[
x=y- z
r2 Jo
r= 2 [(y- z) 2 - (y-z)y ] dydz
x 2 - xy] x=O
dydz =Jo
2
(z
0"
2
2
-
yz) dydz = J~ [yz
2
h 2 z] ~=~ dz = J02 (z4 - %z5 ) dz
2
-
!21 Jor2= [- x e -v] v=O
v=ln:r; d d = !2 r2z ( x z
xe - ln x + xe0) dxd z
1 Jo
Jt J:= (-i + x) dxdz = g [-x + %x2 J::~= dz
2
2
2
3
16
· = f 1 (- 2z + 2z ) dz = [- z + ~ z ] ~ = - 4 + 3 + 1 - ~ = ~
=
7. J0.,.
12
J~ J0"' cos(x+y+z)dzdxdy = j~rr/
2
g [sin(x+y +z)J::~ dxdy
= f0.,.12 J~ (sin(2x + y) - sin(x + y)J dx dy
= j~,./
2
[- %cos(2x + y) + cos(x + y)J::~ dy
= f 0,.12 [- ~ cos 3y + cos 2y + ~ cos y - cos y] dy
1. . 3
1 . 2
1 .
] .,. /2
1
1
1
= [ -6 SID y + 2 S Ln y - 2 sm y 0
= G. - 2 = -S
9.
JJJE ydV = J; fa"' J;::: ydzdydx = J; J; [yz]~==~~ dydx = f 03 f0"' 2y2 dydx
- r3 (1y3] y = x dx - r3 1x3 dx - lx4] 3 - 81 = l l
- Jo 3
- Jo 3
G
o- 6
2
11=o
11 1=~
+
4
11. / " ( (
} } E
- 2- zX
+Z
2 dV =
= f
1
4
1
4
71
0
X
1 1 .!
4
dx dz dy =
[z ·
1
Z
4
Z
71
tan - l ~] x = z dz dy
Z
x =O
f 11 (tan- (1)- tan- (o)] dzdy = f I:(~- o) dz' dy = ~ f 14 [zJ::: dy
4
1
1
4
1
= ~ f 14 (4-y)dv= ~ [4v - h 2 ]~ · = ~ (16-8 - 4+~) = 9;
13. Here E = {(x,y,z) I 0 :::; x:::; 1, 0 :::; y:::; -fo, O :::; z :::; 1 +x +y}, so
JJJE 6xydV = f01 fov'X J;+.,+u 6xydzdydx = J~ fov'X [6xyzJ ::~+"'+11 dydx
= f~ fov'X 6xy( 1 + x + y) dy dx = / 0 (3xy + 3x y2
1
2
2
+2x~ )~=~ dx
3
= t0 (3x2 + 3x3 + 2x5f2) ·dx = [x3 + ~x·l
+ :!x7f
2] 1 = 65
4
7
0
28
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0
269
270
D
CHAPTER 15
MULTIPLE INTEGRALS
HereT = {(x, y,z) I 0 $ x $ 1, 0 $ y $ 1 - x, O $ z $ 1- x - y}, so
15.
Jffr x2 dV ::::: fot J;-a: fot -a:-y x2 dz dydx = fot fot-x x 2(1- x- y) dy dx
= J; fot -a:(x2- ~3 - x2y ) dy dx = fot [x 2y.- xay - tX2y2J::~-a: dx
= J; [x2(1 - x)- x3 (1 - x) - t x2(1- x?] dx
4
2
3
5
4
3 1
- x + lx
) dx = [...!.
= j '01 (lx
2
2
10 x - l4 x . + l6 x ) 0
1
-- w
- 41 + s1 = oo1
The projection of E on the yz-plane is the disk y 2 + z2 $ 1. Using polar
17.
coordinates y = r cos 8 and z = r sin 8, we get
f.ffe x dV = ffv [J4~2 +4:2 x d.x] dA = ~ ffv [42 - (4y 2 + 4z2 ) 2 ) dA
= 8 J;""J:(1- r 4 ) 1· drd8 = 8 J~ " dB J0\ r - r 5 ) dr
2
X
_ 8(211") [lr2
_ lr6] I _ 16rr
2
6
0- 3
-
19. The plane 2x + y + z = 4 intersects'the xy-plane when
2x + y + 0
=4 =>
4
y = 4 - 2x, so
E = {(x, y, z) I 0 $ x $ 2, 0 $ y $ 4- 2x, 0 $ z $ 4 - 2x - y} and
V = fo2 J~l -2a: fo4-2x-v dz dy dx = fo2 fo4- 2:r. (4 - 2x - y) dy dx
= Jor2 [4y - 2x y - 21 Y2] vy ==4.-O 2x dx
= J; [4(4 - 2x)- 2x(4 - 2x) - t(4 - 2x? ] dx
=J; (2x
2
-
8x + 8) dx =
X
y
[~x 3 - 4x2 + 8x]~ ='¥
21. The plane y + z = 1 intersects the xy-plane in the line y = 1, so
E = {(x, y, z) I. - 1 $ x $ 1, x 2 $ y $ 1, 0 $ z $ 1 - y} and
V = JJJ ~ dV =
.C1 J:2f01- 11 dz dy dx = f~ 1 f:.2 (1 - y) dy dx
1 [y- 21 Y2] v=1
d
}'1 (1
'.!
1 ") dX
= ! -1
y=x2 X = - 1 2 - X + 2 X
_
-
[1
1
2X - . 3 X
3
+ lOX·
1
5] 1
1
1
1
l
1
1 _
R
- 1 = 2 - 3 + 10 + 2 - J + lO - 15
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(- 1, 1, 0)
SECTION 15.7 TRIPLE INTEGRALS
D
271
23. (a) The wedge can be described as the region
D = { (x,y,z) I y2 + z 2 ::; 1, 0 ::; x::; 1, 0 ::; y ::; x}
= { (x, y , z) I 0 ::; x ::; 1, 0 ::; y ::; x, 0 ::; z ::; } 1 - .Y2 }
So the integral expressing the volume of the wedge is
JffodV=fo1 fo"'J/1 -v2 dz dydx.
.
rl f"' n /l.- vz dz dy dx = 4" - 3·
l
(b) A CA S gives
Jo Jo Jo
(Or use Formulas 30 and 87 from the Table of Integrals.)
25. Here f(x, y, z) = cos(xyz) and~ V = ~ ·
t · ~ = ~,so the Midpoint Rule gives
1 1) + J(l4 •4•
1 43) + !(1
3• 4l) + f(l4 • 43• 43)
= s1 [!(14 • 4•4
4•4
+ !(~ ,
t. t) + !(~ , t. ~) + !(~, ~, t) + ! (~, ~. ~) ]
= ~ [cos -l4 +cos ~ +cos 6~ +cos f4: +cos 6~ +cos i4 +cos f4: + cos~~ ) ~ 0.985
27. E = {(x , y, z)
I 0 ::; x::; 1, 0::; z::; 1- x, 0 ::; y::; 2- 2z },
the solid bounde~ by the three coordinate planes and the planes
z = 1 - x, y = 2 - 2z.
X
y
29.
4
X
I
---~"--x' + 4z = 4
1
.t
- I
If D 1 , D2 , D3 are th~ projections of Eon the xy-, yz-, and xz-planes, then
D 1 = {(x, y) I - 2::; X::; 2, 0 ::; y::; 4 - x 2 } = {(x, y) I 0 ::; y::; 4, - v4 - y::; X::; .J4=Y }
I
.
D 2 = {(y,z) 1o ::; v ::; 4, - ~v4- y::; z::; ~v4- v} = {(y,z) 1-1 ::; z::; 1, o::; v ::; 4 - 4z 2 }
DJ = { (x,z) I x 2 +4z 2 ::; 4}
[continued]
(!) 2012 Ceog111:o lcamio~. All Rights Reserved. Moy 001 be scanned. copied. or duplicolcd, or poslcd lo o publicly occcssiblc wcb:iilc. io "'i1o lc or io JXU'I.
272
0
CHAPTER 15
MULTIPLE INTEGRALS
Therefore
E = {(x , y , z) 1 -2 -< x < 2'0< y< 4 ·_ x2 , - 12 yl4 - :z;2 - y <
- z <
- 1
2 yl4 - :z;Z - y }
= { (x , y,z) I 0:::; y:::; 4, - y'4""='1j:::; x:::; y'4""='1j, - h/4 - x 2 - y:::; z ~ ~yf4- x 2 - y}
= {(x,y,z) I -1 $ z 5: 1, 0 5: y 5:4 - 4z 2 , - yf4 - y- 4z2 :::; x:::; yf4 - y - 4z2 }
= { (x, y, z) I 0 5: y 5: 4, - ty'4""='1j $ z $ ty'4""='1j, - j4- y- 4z 2 :::; x:::; yf4 - y- 4z2 }
= {(x, y, z) I - 2 5: x 5: 2, - ~ J 4 - x 2 :::; z :::; t J 4 - x~, 0 :::; y :::; 4 - :z: 2 - 4z2 }
= { (x, y, z) I -1 5: z 5: 1, - .J4 - 4z 2 :::; x:::; .J4- 4z 2 , 0:::; y:::; 4- x
2
-
4z 2 }
Then
Iffsf (x,y,z) dV = !_ 2 ; 0
2
r4-x2 JJ4-x2-yf2
I
f4 - 4z
= I - tJo
2
2
(
)
(
)
-~/ / x,y,z
JV4-y-4z2
r4 f-14-ii JJ4-:z:2-y j 2
(
)
dzdydx=:Jo --14-ii -~ 12 ! x,y,z dzdxdy
,
f4 f-/4-!i/2
-.J4 _ _ 4 z 2 f x, y, z dx dy dz = Jo --14-ii/2
11
J V4-y-4z 2
- J 4 _ 11 _ 4 : 2 f ( x, y, z) dx dz dy
2
2
r4-x2 -4:2' f(x
. ,y,z) dydzdx= fl_ J J44:;2 r4-o: -4:
= f _22 J~/2
r;--:; Jo
~Jo
f ( x,y,z) dydxdz
1
-y4 - :z:2/2
-y4-4z2
y
31 .
(- 2,4,0;
JC
X
(2,4,0)
4
y
-2
2
JC
If D1, Dz, and Da are the projections of Eon the xy-, yz-, and xz-planes, then
D1 = {(x, y) I -2 $ x 5: 2, :z:2 ~ y 5: 4} = {(x, y) I 0 :::; y :::; 4, -y'Y $ x:::;
vY },
D2 = {(y,z) I 0 :::; y $ ,4, 0 :::; z 5: 2- ~y} = {(y,'z) I 0:::; z 5: 2,0 ~ y:::; 4 - 2z}, and
Ds = { (x, z) I - 2 $ x :::; 2, 0 5: z ~ 2 - ~~ 2 } = {(x, z) I 0 :::; i ~ 2, - .J4- 2z :::;. x $ v'4 - 2z}
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SECTION 15.7 TRIPLE INTEGRALS
E = { (x, y, z) I -2 ~ X~ 2, x 2 ~ y ~ 4, o' ~ z ~ 2-
Therefore
= { (x,y,z) I 0 ~ y ~ 4,
0
273
h}
-Jv ~ x ~ V"Y. 0 ~ z ~ 2 - tv}
= { (x,y,z)I O ~y~4,0 ~z~ 2-ty, -vv~ x~ JY}
= { (x, y, z) I 0 ~ z ~ 2, 0 ~ y ~ 4- 2z, -
VY ~ x ~ vY}
= {(x,y,z) l-2 ~ x ~ 2, 0 ~ z ~ 2 - tx2 , x 2 ~ y ~ 4 - 2z}
= {(x, y, z) I 0 ~ z ~ 2, - v'4 - 2z ~ x ~ y'4- 2z, x 2 ~ y ~ 4 - 2z }
1JJ~ f(x, y, z ) dV = I~ 2 I:2 I~- 1112 f(x,y,z) dzdydx = I 0 I~ I 02 -v 12 f (x, y, z) dz dxdy
4
Then
4
2
= 1~ J~-v/ 1~f(x, y,z) dxdz dy =I~ Io4 - 2 " J~f(x,y,z)dxdydz
2 f 2 - %2 / 2 !4- 2:: J(
r2 Jv'4=2z f4 - 2z (
.
= J-2
Jo
z2
x , y,z) d y d z d·X= Jo
- "'4 - 2z z2
I x ,y , z ) dydxdz
33.
y
The diagrams show the projections
of Eon the xy-, yz -, and xz-planes.
Therefore
-~------+--+
0
I X
1
Io
I{z 1~l - v f (x, y, z ) dz dydx =I: I/ I:-rJ f(x, y, z) dzdx dy = f Iol-,zIt f(x, y, z) dx dy dz .
1
0
rt rt- u1'tJ0
= JoJo
2
J( x,y,z ) d x d zdy= Jo
rt Jo
rt-../XJ1..;x z f (x,y,z)dydzdx
2
- z) Jl -z (
= Jflo./ ·(l
..;x f x,y,z) dydxdz
0
y
35.
z
X
I: I:Icf f(x,y, ~)dzdxdy = IIIEf(x , y ,z)dV where E = {(x,y,z) I 0 ~ z ~ y, y ~ x ~ 1, 0 ~ y ~ 1}.
If D 1, D2, and D3 are the projections of Eon the xy-, yz- and xz-planes then
Dt = [(x , y) I 0 ~ y ~ 1,, y ~ x ~ 1} = {(x , y) I 0 ~ x ~ 1, 0 ~ y ~ x},
D2 = [(y,z) I 0 ~ y ~ 1, 0 ~ z ~ y} = {(y,z) I 0 ~ z ~ 1, z ~ y ~ 1}, an~
D3 = [(x,z) I 0 ~ x ~ 1, 0 ~ z ~ x} = { (x, z ) I 0 ~ z ~ 1, z ~ x ~ 1}.
[continued]
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274
CHAPTER 15
0
MULTIPLE INTEGRALS
Thus we also have
E = {(x, y, z) I 0 ::::; x::::; 1, 0 ::::; y ::::; X, 0 ::::; z::::; y} = {(x, y , z) I 0 ::::; y::::; 1, 0::::; z.::::; y, y ::::; x ::::; 1}
= {(x,y,z) I 0 ::::; z::::; 1, z::::; y::::; 1, y::::; x ::::; 1} = {(x,y,z) I 0 ::::; x $ 1,0::::; z ::::; x , z::::; y ::::; x}
= {(x, y, z) I 0::::; z::::; 1, z::::; x::::; 1, z::::; y::::; x}.
Then
J; fv f~ f(x,y , z) dzdx_dy = f 0 J; f~ f(x,y ,z) dz dydz = f0 g
1
1
1
{=]; J;
I:
1:
f(x,y, z ) dxdz dy
f(x,y,z) dxdyd~ = J; J; f:'" f(x,y , z) dydzdx
"~--
= f 0 f~ [;"' f(x, y, z) dy dx dz
1
1
37. The region Cis the solid bounded by a circular cylinder of radius 2 with axis the z -axis for -2 ::::; z $ 2. We can write
fff0 (4 + 5x2 yz2) dV = fffc 4dV + fffc 5x2yz 2 dV, but f(x , y,z) = 5x2yz 2 is an odd function with
. respect toy. Since Cis symmetrical about the xz-plane, we have fffc 5x 2yz 2 dV = 0. Thus
39. m = JJJE p(x, y,z) dV = f 0 fofi fol+x+v 2dz dydx = J: J0fi 2(1 + x + y) dydx.
1
= Jorl (2y + 2xy + y2] y=D
y=fi dx = Jo
rl (2 v'x + 2x3/ 2 + x) dx = [ix3/ 2 + ix5/ 2 + lx2]
3
s
2
1
= so
79
0
'
My: = fffE xp(x, y, z ) dV = f 0 fofi J~+o:+v 2x dz dy dx = J; fofi 2x(1 + x + y) dydx
1
, = r1 [2 xy + 2x2y + xy2]v=fi dx = r1( 2xs12 + 2xsf 2 + x2) dx = [ixo/ 2 + ix7/ 2 + J.x3]
Jo
Jo
v=D
.
s
3
1
1
0
= 179
105
Mx~ = JJJE yp(x,y,z)dV = f 01 fofi J;+x+v 2ydzdydx = J01 f 0fi2y(1 +x+y)dydx
1
= Jor1 [y2 + xy2 + 1y3]
v=fi
r1 (x + x2 + 1x3/ 2) dx = [lx2 + lx3 + ~x:>/2) = 11
3
v=O dx = Jo
3
2
3
15
10
0
. = fffE zp (x , y, z ) dV = Jor l Jor fi Jor1+:r+v 2z dz d y .dx = J·l0 J·./i.
[z 2] ==D
~=l+:r+v d.y dx = Jo
r1 Jo
r./i. (1 + x + y )2 dy dx
lvlxv
0
=
J01 J/"(1 + 2x + 2y + 2xy + x 2 + y2) dydx = J; [y + 2xy + y 2 + xy2 + x 2 y .+ ty 3 J::f dx
1
r1 ( 'x + 1x3/2 + x + x2 + x5/2) dx _ ['l x3/2 + llx5/ 2 + l x2 + lx3 + 'lx7/ 2) _ 571
-_ Jo
v"'
3
3
1s
2 ·
3
7
- 2 10
0
.
79
.
(-
_ _)
Thus the mass IS 30 and the Center Of mass IS X, Y, Z
M:r= Mxv )
33 571)
= (My=
--:;;:;:--, --:;;:;:--,
m = (358
553 ' 79 > 553 ·
Mv: = J; f 0a f 0a [x3 + x(y 2 + z2 )] dxdydz = J; J; [ta4 + ~a2 (y 2 + z2 ) ] dydz
ru (.4a
1 s + 1a s + 2a
1 a Z 2) dZ = 4a
1 6 + 3a
1 6 = na
1 6
-_ Jo
= M :z:z = M xy by symmetry 0 fE and p(X, y, Z )
0
- - ) = (1 a, 7 a,' 7a.
)
H ence (x,y,z
12
12 12
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SECTION 15.7 TRIPLE INTEGRALS
43. I,. =
0
275
foL foL foL k(y2 + z2)dz.dydx = k foL foL (Ly2 + ~L3) dydx = k foL ~L4 dx = ~kLs.
By symmetry, l x = 111 =I.= ~k£5 •
2
2
45. 1::: = JJJE(x + y ) p(x, y, z) dV =
(b) (x,y, z) wherex =
[J;• k(x + y dz] dA =:z:2+y2:5a.2
JJ k(x + y )hdA
2
JJ
z2+y2:5a2
2
2
)
2
"* f~ 1 !:2f01- 11 x Jx2 + y 2dzdyd$, y = "* f J:z f 01-v y Jx 2 + y dz dy dx, and
2
1
~ = ~ t 1 J:2 J; - v z Jx 2 + y 2 dzdydx.
11
f Jo
r~ Jof"(1 + x + y + z ) d z d y dx = 332"" + 24
49. (a) m = Jo
1
(b) {X, y, z) = ( m- 1 J; fo~ f~ x(1 + x + y + z) dzdydx ,
m- 1 f 0 f0~ J~ y(1 + x + y + z) dzdydx,
1
m.
28
1
f 01 fo~j~ z(1 + x + y +z)dzdydx)
307r + 128
457r + 208 )
= ( 9n + 44 ' 45n + 220 ' 135n + 660
t {~ fo{"(x2 + y2)(1
. + x + y + z) dz dy dx = 68 240
+ 1571'
(c) 1::: = fo f o
51. (a) f(x, y, z) is a joint density function, so we know JJJR 3 f(x, y , z) dV = 1. Here we have
JJJR3 f(x,y ,z) dV = f~oo f~oo }:"00 f(x,y, z) dzdydx = f 0 J: J: Cxyzdzdydx
2
= C J: xdx f 0 ydy f 0 zdz = C[~x )~ [h )~ [~z )~ = 8C
2
Then we must have 8G = 1
2
2
2
2
=> C -- 18'
'
1 2] 1 [ 1 2] 1
1 ( 1 )3
1
= ii1 Jort x d X Jor1 11 dY Jorl z d z = ii1 [ 21 x2]1o [ 2Y
o 2z o = 8 2
= 1M
(c) P(X + Y + Z::; 1) = P((X, Y, Z) E E) where E is the solid regio·n ·in the first octant bounded by the coordinate planes
and the plane x + y + z = 1. The plane x + y + z = 1 meets the xy-plane in the line x + y = 1, so we have
P(X .+ Y + Z::; 1) = fff Ef(x, y, z) dV = j~1 j~ -x j 01-:z:-y kxyz dz dy dx
= k J; fot - :z: xy[~z2 J ;:~-:z:-y dydx = fs fo1 fol -z xy(l - x - y)2 dydx
= 1~ f 0 f 0 - "'[(x3 - 2x
1
1
2
+ x)y + (2x2 - 2x)y2 + x y 3 ] dydx
= 110 fo1 [(x3 - 2x2 + xHy2 + (2x2- 2x)h3 + x(fy'')J:: - :r: dx
1 f1(
1 ( 1 ) -- 192
Jo X - 4X 2 + 6X .3 -. 4X 4 + X 5) d X -- 192
30 -
1
57iii)
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276
0
CHAPTER 15 MULTIPLE INTEGRALS
1 L2 L2 L2
La
£ 3·222 = 8
55. (a) The triple integral wi ll attain its max imum when the integrand 1 - x 2 - 2y2 - 3z 2 is positive in the region E and negative
everywhere e lse. For if E contains some region F where the integrand is negative, the integral could be increased by
excluding F from E, and if E fails to contain some part G of the region where the integrand is positive, the integral could
be increased by including Gin E. So we require that x 2 + 2y 2 + 3z 2 $ 1. This describes the region bounded by the
e llipsoid x 2 + 2y2 + 3z2 = 1.
(b) The maximwn value of
JjJE (1 - x 2- 2y2 - 3z 2 ) dV occurs when E is the solid region bounded by the ellipsoid
x 2 + 2y2 + 3z 2 = 1. The projection of E on the xy-plane is the plaf1ar region bounded by the e llipse x
2
+ 2y2 = i, so
and
us ing a CAS.
15.8 Triple Integrals in Cylindrical Coordinates
From Equations I, x = r cos 9 = 4 cos
.1. (a)
i
= 4 · ~ = 2,
y '= rsinfJ = 4sin ~ = 4 · ~ = 2.J3, z = ~2, so the point is
3
Y ·
(2 , 2.J3, -2) in rectangular coordinates ..
I
- 2 •I
X
(b)
l (4, 3. -2)
(2.-¥. 1)
x = 2cos(-~) = 0, y = 2sin( - ~l) = - 2,
and z == 1, so the p o int is (0, - 2 , 1) in rectangular coordinates.
X
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SECTION 15.8 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES
0
277
3. (a) From Equations 2 we have r 2 = ( - 1) 2 + 12 = 2 so r = v'2; tan{;/= _: 1 = -1 and the point ( -1, 1) is in the second
quadrant of the xy-plane, so 8 =
s; + 2mr; z = 1. Thus, one set of cylindrical coordinates is ( v'2, 4'~~' 1).
3
,
(b) r 2 = ( -2) 2 + (2J3) 2 = i6 so r = 4; tanB = ~ = - v'3 and the point ( - 2, 2v'3) is in the second quadrant of the
xy-plane, so (} = 2;
+ 2mr; z = 3. Thus, one set of cylindrical coordinates is ( 4, 2; , 3).
5. Since (;I = f but r and z may vary, the surface is a vertical half-plane i~cluding the z -axis and intersecting the xy-plane in the
half-line y = x, x ~ 0.
1. z = 4- r 2 = 4 - (x2 + y 2 ) or 4 - x 2 - y 2 , SQ the surface is a circular paraboloid with vertex (0, 0, 4), axis the z-axis, and
opening downward.
9. (a) Substituting x 2 + y 2 = r 2 and x = r cos 8, the equation x
2
-
x + y 2 + z 2 = 1 becomes r 2 - r cos 8 + z 2 = 1 or
z 2 = 1 + r cos 8 - r 2 .
(b) Substituting x = r cosO andy = r sinO, the equation z = x 2 - y 2 becomes
z = (rcos8) 2 - (rsinB? = r 2 (cos2 8- sin 2 B) or z = r 2 cos 28. 1
0 ~ r ~ 2 and 0 ~ :z ~ 1 describe a solid circular cylinder with
11.
radius 2, axis the z -axis, and height 1, but - 11'/2 ~ 8 ~ 11'/2 restricts
the solid to the first and fourth quadrants of the xy-plane, so we have
a half-cylinder.
13. We can position the cylindrical shell vertically so that its axis coincides with the z-axis and its base lies in the xy-plane. If we
use centimeters as the unit of measurement, then cylindrical coordinates conveniently describe the shell as 6 ~ r ~ 7,
o ~ o ~ 211', o .~ z ~ 20.
15.
The region of integration is given in cylindrical coordinates by
E = {(r,B,z) 1- 11'/2 ~ 8 ~ 11'/2, 0 ~ r ~ 2, 0 ~ z ~ r 2 } . This
represents the solid region above quadrants I and IV of the xy-plane enclosed
by the circular cylinder r = 2, bounded above by the circular paraboloid
= x + y 2 ), and bounded below by the xy-plane (z = 0).
/ 2 j '2 rr
d d(}
I" / 2 f 2 [ ] z = r 2 d dB = I -71'/
'l ' / 2 2 Jof 2 r 3 drd(}
= - -rr/2 Jo rz z=O r
I -71'/
2 0 Jo rdz r
z = r 2 (z
2
2
T<
4 2
= I c-'lf/2
1f12 dB Jof 2 r 3 dr = toJr.- 1271'/2 (lr
]
4
o
= 11'(4- 0) = 411'
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278
D
CHAPTER 15 MULTIPLE INTEGRALS
17. In cylindrical coordinates, E is given by {(r, B, z} I 0 ~ B ~ 21r, 0 ~ r ~ 4, - 5 ~ z ~ 4}. So
19. The paraboloid z = 4 - x 2 - y 2 = 4 - r 2 intersects the xy-plane in the circle x 2 + y 2 =
4or r 2 = 4
=?
r = 2, so in
I
cylindrical co~rdinates, E is given by { (1·, B, z } 0 ~ (} ~ 1r/ 2, 0 ~ r ~ 2, 0 ~ z ~ 4- r 2 }. Thus ·
2
JJJE (x + y + z) dV = Io" 12 I; I04 (rcos{J + r sin(}+ z) r dzdr d(} = I0"12 I; [r2 (cos 0 + sin O) z + ~7·z2 ]~=~-r dr d{J
2
-r
=
Io"/2 J; [(4r2 - r 4 )(cos 0 + sin 9) + ~r(4- r· 2 ?] dr d{J
3 - l r 5 ) (cos(} +sin 0) - .1.. (4 - r 2 ) 3 ] r = Z d9
= .Jfo'1r / 2 [(1r
3
5
12
r=O
= I 0"
12
[~~(cosO + sin 0) + Jf] dO= [~(sin 0 - cos B)+ JfB]; 12
= lfi(1 - 0) + ¥. ~ - ~(0 - 1)- 0 = ~7r + \ 258
21 . In cylindrical coordinates, E is bounded by the cylinder,. = 1, the plane z = 0, and the cone z = 2r. So
E = {(r, 0, z) I 0 ~ 8 ~ 27r, 0 ~ r ~ 1, 0 ~ z ~ 2r} and
IIIE x 2 dV = J;'T( f 01 J;r r 2 cos 2 9 r dz dr dO = I02 J01 [r 3 cos2 0 z ]::~·· dr d9 = I;" Io1 2r4 cos2 8 dr dO
= I 02'T( [%r 5 cos2 B]~=~ dO= %I02" cos2 0 d(} = ~I;"~ (1 +cos 28) dO,;, t [B + t sin 28] ~1f = 2;
1f
23. In cylindrical coordinates, E is bounded below by the cone z = r and above by the sphere r 2 + z 2 = 2 or z = v'2 - r 2 . The
con~ and the sphere ·intersect when 2r2 = 2
r = 1, so E = {(r, 8, z) I 0 ~ 0 ~ 21r, 0 ~ r ~ 1, r ~ z ~ v'2 - r 2 }
=?
and the volume is
IIIE dV = J;" I01 .fr~ rdzdrdO =I;" I; [rzJ~::P drdO = I J (r~- r drd(}
1
2
2 3 2
3
= J0 dO Io (rv'2 - r 2 - r dr = 21r [-~(2- r ) 1 - ~r J:
2
0"
2
1f
1
0
2
)
)
= 27r (- ~) (1 + 1- 2312 ) = - j7r (2- 2V2) = ~7r (V2 - 1)
25. (a) The paraboloids intersect when x 2 + y 2 = 36 - 3x2 - 3y2
=?
x 2 + y 2 = 9, so the region of integration
is D = {(x,y) I x 2 + y 2 ::; 9}. Then, in cylindrical coordinates,
.E = {(r, B, z) I r 2 ~ z ~ 36 - 3r2 , 0 ~ r ~ 3, 0 ~ (} ~ 27r} and
V =
g"I; fr
36 3 2
r dz dr d(} =
2 -
r
J;"I; (36r - 4r3 ) dr dB = J;" [18r2 - r 4] ::~dB= J02 " 81 dO= 16271".
(b) For constant density K , m = J(V = 1627rJ( from part (a). Since the region is homogeneous and symmetric,
Nfyz
= Mxz = 0 and
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SECTION 15.8 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES
0
279
I: I:26 - 3r (zK) r dz dr d() = J( I~1f I: r [~ z2 ) ;:~~-3 ,. dr d()
=if I 02 " I: r((36- 3r2 ) 2 - r 4 ) dr d() = if I~-:r d8 I 03 (8r5 - 216r3 + 1296r) dr
M xv = I~1f
2
2
- K (27r) [.!!.r6 - 216 r4 + 1296r2] 3 = 11"!((2430) = 243011"J(
-2
6
4
0
~
11 z Mxz M:xv )
MThus (-x, __
y, z ) = ( , --, -- .
m
m
m
.
2430rrI<)
= (O , O, ~
= (0 , 0, 15).
2
27. The paraboloid z = 4x2 + 4y2 intersects the plane z = a when a = 4x2 + 4y or x 2 + y 2 = ia. So, in cylindrical
h!ci, 0 $ 8 $ 21r, 4r 2 $ z $a}. Thus
coordinates, E = { (r, 8, z) I 0 $ r $
12"1Va/2 1a
K r dz dr d8 = J(
m =
0
0
12"1Va/2(ar - 4r dr d8
3
4r2
- }(1211" [lar
-
2
0
2 -
0
r=,fi/2
d() --
r 4] r=O
T.(
.1·
12" ...!.. a
.
)
0
16
0
2 }(
l a 1r
2 d() --
8
Since the region is homogeneous and symmetric, lvfvz = Mxz = 0 and
Hence (x, y, z ) = (0, 0, ~a).
29. The region of integration is the region above the cone z = Jx 2 + y 2 , or z = r, and.below the plane z = 2. Also, we have
-2 :::; y $ 2 with - /4 - y 2 $ x $ )4 - y 2 which describes a circle of radius 2 in the xy-plane centered at (0, 0). Thus,
1 -~!
2
1~
2
..jz2+v2
- 2
xzdzdxdy=
r1f {
2
Jo Jo
1
2
2
2
(rcos8)zrdzd;d0 = { 1f {
1.
2
Jo Jo ,.
r
= J~1f I~ r (cos 8) [~ z ] ::~ dr d() = ~
2
2
r 2 (cos O)z dzdrd()
•
Io2 I~ r 2 (cos 0) (4- r 2 ) dr d8
Tf
2
4
3
5
= ~ I~" cos 8 d() I~ 4r - r ) dr = ~ [sin 8]~" [~r - ir ] ~ = 0
(
31. (a) The mountain comprises a solid conical region C. The work don~ in lifting a small volume of material !:. V with density
g(P) to a height h(P) above sea level is h(P)g(P) t:.V. Summing over the whole mountain-we get
W = IIIc h(P)g(P) dV.
(b) Here C is a solid right circular cone with radius R = 62,000 ft, height H = 12,400 ft,
and density g(P) = 200 Ibj ft3 at all points Pin C. We use cylindrical coordinates:
W = I~" IoH IoR(l- z/ H) Z • 200r dr dz d8 = 211" foH 200z ar2 J ~::(l -z/ H) dz
1fl ~
2
= 4007r
z
z
= 2001rR2 [2
= ¥-1rR H
2
2
2
-
( 1-
·~
r
dz = 2001rR
2
1fl
( z :-
2
2 2
~ + ~2 ) d7
3
2
z - ]H= 2001rR2 (H
2H
H)
+- -- + 3H
4H 2
2
3
4
2z
3
4
2
-
0
= ¥11"(62,000) (12,"400? ~ 3.1 x 10
2
19
!:._ = H -z = 1 _ .=_
R
H
ft- lb
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H
280
0
CHAPTER 15 MULTIPLE INTEGRALS
15.9 Triple Integrals in Spherical Coordinates
1. (a)
From Equat!ons 1, x = psin¢cos9 = 6 sin 2J cos i = 6 · ~ · ~ =
(6.¥.f)
If
1
y = p sin rP sin 9 = 6 sin ~ sin ~ == 6 · ~ ·
I I
I
I
'1r
I
I
I
7i / 6 :
z
I
I
I
!.
:1/- = ~, and
= pcos¢' = 6cos 2J = 6 · :1/- = 3VJ, so the point is(~ . ~ ,3../3) in
I
I
rectangular coordinates.
y
X
x = 3 sin 3_; cos ~ = 3 · ~ · 0 = 0,
(b)
y = 3 sin 3; sin ~ = 3 · ~ · 1 =
3
z= 3cos ;
¥, and
= 3 (-~) = ·~¥. so thepointis (o.¥ , - ¥ ) in
X
·rectangular coordinates.
.
.
z
0
3. (a) From Equations 1 and 2, p = .jx2 + y2 + z2 = ../02 + (-2)2 + 0 2 = 2, cos¢ =
= '2 = 0
p
cos 9 = p s:
rP = 2 sin~71'/ 2) = 0 => 9 = ;
3
(b) p = . v'1 + 1 + 2 = 2, cos¢ =
=>
¢'
71'
= 2'' and
3
[since y < 0). Thus spherical coordinates are ( 2, ; ,
-i).
z
- J2
371'
P
= - ::::} </> =
2
4 , and
X
- 1
cos 9 = - - = ::-:--:-::---:-:7
• psin ¢
2sin(311'/4)
- 1
1
2 (J2/2) -
J2
371'
4
-,..-::::--:- - - - => 9 = -
[since y > 0). Thus spherical coordinates
37!')
4 4 .
are ( 2, 371' ,
5. Since ¢ = ~, the surface is the top half of the right circular cone with vertex at the origin and axis the positive z -axis.
x 2 + (y - ~) 2 + z 2 = ~ . Therefore, the surface is a sphere of radius ~ centered at ( 0, ~, 0) .
9. (a) x = psin¢>cos8, y = psin ¢>sinO, and z = pcos ¢,so the equation z 2 = x 2
2
(p cos¢) = (p sin </> cos 8)
2
+ y 2 becomes
+ (p sin ¢> sin 8) 2 or p2 cos2 ¢J = p 2 sin2 ¢'. If p ;f 0, this becomes co~2 </> = sin2 rp. (p = 0
corresponds to the origin which is included in the surface.) There are many equivalent equations in spherical coordinates,
such as tan 2 ¢ = 1, 2 cos 2 ¢' = 1, cos 2¢ = 0, or even rP = ~ . ¢' = :~4,..
(b) x 2 + z 2 = 9
<*
(psin </>cos 8)
2
+ (pcos ¢') 2 = 9 <*
p2 sin2 ¢>cos2 9 + p2 cos2 ¢> = 9 or
p2 (sin2 ¢cos2 8 + cos2 ¢) = 9.
® 20 12 Ccngugc Lcamjng. All !Ugllls RcscrYcd. May not be scanned, copied, or duplit11ted, or posted to a publicly accc:ssible ""bsite, jn whole or in part.
SECTION 15.9 TRIPLE INTEGRALS IN SPHERICAL COORDINATES
0
281
11. 2 :S p :S 4 represents the solid region between and including the spheres of
0 :S ¢> :S f restricts the solid to that
radii 2 and 4, centered at the origin.
portion on or above the cone ¢> = f, and 0 :S 8 :S 1r further restricts the
solid to that portion on or to the right of the xz-plane.
13. p :::; 1 represents the solid sphere of radius I centered at the origin.
3
;
:S ¢> :S 1r restricts the solid to that portion on or below the cone ¢> = ~;.
15. z 2': Jx2 + y 2 because the solid lies above the cone. Squaring both sides ofthis inequality gives z 2 2': x 2 + y 2 ·
2z 2 2:_ x 2 + y 2 + z 2 = p
2
:::?
z
2
= p 2 cos2 ¢> 2': ~p
cos¢> 2': ~, or equivalently 0 :S ¢> :S
2
*.
:::?
:::?
cos ¢> 2': ~· The cone opens upward so that the inequality is
2
In spherical coordinates the sphere z = x +. y 2 + z 2 is p cos lp = p 2
2
:::?
p = cos¢. 0 :S p :S cos¢> because the solid lies below the sphere. The solid can therefore be described as the region in
spherical coordinates satisfYing 0 :S p :S cos ¢>, 0 :S ¢> :S
*.
The region of integration is given in spherical coordinates by
17.
E = {(p, 8, ¢>) I 0 :S p :S 3, 0 :S 8 :S 7r/ 2, 0 :S ¢> :S 1r/ 6}. This represents the solid
region in the fi rst octant bounded above by the sphere p = 3 and below by the cone
¢> = 1T/6.
J0" 16 J0" 12 .1; p2 sin ¢> dp dB d4> = ./~"16 s in 4> d¢> f0" 12 dB J: p2 dp
= [-cos¢>] ~
X
=
16
( 8]~ 12 [~Pa]~
(1- ~) G)(9) 9; (2- v'3)
=
19. The solid E is most conveniently described if we use cylindrical coordinates:
E = {(r,B,z) I 0
:s e :s i. O:s r :s 3,0 :s z :s 2}. Then
.1; ./~ f(r cos f)_ , rsinB, z) 1' dz d1·d8.
JJJE f(x , y, z) dV = f0" 12
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282
0
CHAPTER 15 MULTIPLE INTEGRALS
21. In spherical coordinates, B is represented by {(p, 0, 4>) I0 ~ p ~ 5, 0 ~ 0 ~ 27r, 0 ~ cf> ~ 1r}. Thus
2 2
JJJ8 (x 2 + y 2 + z ) dV = J; f 0 "' J:(p ) p sin cf>dpdO dl/> = Io"' sin 1/>dl/> J~"' dB I~ p dp
2
2 2 2
6
= [- coscf>]~ [0]~" [tp 7 ]~ = (2)(27r)(1s.~211)
= 312tJ07r ~ 140,249.7
23. In spherical coordinates, E is represented by {(p, 0, ¢) 12 ~ p ~ 3, 0 ~ 0 ~ 27r, 0 ~ cf> ~ 1f} and
IIIs(:c 2 + y 2 ) dV = ]~"'
g•r I (p s in2 4>) p sin cf>dpdO dcf> = Io"sin cf>dcf> I~"' dO I: p dp
3
2
2
2
4
3
I 0"'(1- cos2 1/>) sinl/>dl/> [ 0]~,. [ip5 )~ = [-coscf>+ ~ cos3 4>]~ (27r) : i(243 - 32)
= (1 - ~ + 1 - U (27r) Cil) = 16~:"
=
I
25. In spherical coordinates, E is represented by { (p, 0, 4>) 0 ~ p ~ 1, 0 ~ 8 ~ "i· 0 ~ 4> ~ ~}.Thus
2
JJJE xe"' + 112 +z dV = J0"/2 J0Tr12 I:(psin cf>cosO)eP p2 sin </l dpd(} dcf> = Io"'12 sin2 1/>d¢
2
2
.=
fo"12 H1 - cos 2¢>) d¢ IoTr12 cos(} d(} ( ~ p2eP
2
J: -I:
J;12 cos(} d(} j p3eP dp
1
0
2
2
peP dp)
2
[integrate by parts with u = p2 , dv = peP dp]
= a¢-
2
i sin 2¢]~/Z [sin(})~/2 ~p2 eP
[
2
-
~eP J: = (i- 0) (1- 0) (0 + ~) = i
27. The solid region is given byE= { (p, 0, cf>) I 0 ~ p ~ a, 0 ~ 0 ~ 21r, ~ ~ cf> ~ ~}and its volume is
V = ffis dV = I:/6 J~"' I; p2 sin cf>dpd(}drjJ = I:/: sincf>dl/> I~11' dO I; p dp
2
3
. = [-coscf>J;~: [0)~11' [~p )~ = ( -~ +
3
4) (27r) (ta3) = ' \- 7ra
1
3
29. (a) Since p = 4 cos 4> implies p2 = 4p cos 1/>, the equation is that of a sphere of radius 2 with center at (0, 0, 2). Thus
(b) By the symmetry of the problem Mv= = M x= = 0. Then
M, 11 = I:" I; 13 I 04 co• 4> p3 cos cf> sin cf> dp d¢ d(} = J~" I; 13 cos cf> sin r/> ( 64 cos4 rjJ) dcf> dO
--
2
f "' 64
Jo
2
[-16 cos() -•]
</>=71'13 dO -- Jo
f
.ll
'~' ¢=0
2 dO -- 21 7r
71'
Hence (x,y,z) = (0,0, 2.1).
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SECTION 15.9 TRIPLE INTEGRALS IN SPHERICAL COORDINATES
0
283
31. (a) By the symmetry of the region, My: = 0 and Mx: = 0. Assuming constant density K,
m = fffE K dV = K fffE dV = iJ< (from Example 4). Then ·
M .,y = Jjj~ z K dV = K
J;,. }~,. 14 j~o• ¢>(pcos ¢) p2 sin r/J dp drpd8 = K J;" J0,.14 sin ¢cos¢ [tP4] ==~os¢ d¢ d8
=if< J:w f0"' 14sin¢cos¢ (cos4 ifJ) .d(j)d(} = iK J:" d(} J;14 cos5 {j}sin (j)d¢
= iK
(8]~" [-i cos6 t/>] ~14 = iK(27r)(-i) [ c~r - 1] = -fiK (-~) = ;~ K
.. (- __) (My:
!vfu M.,u)
( 77rK/96 ) = (0,0, 12.
7)
m'm'm
= 0, 0, 1rf(jB
Thusthecentro1d1s x,y,z =
(b)' As in Exercise 23, x 2 + y 2 = p2 sin2 ¢ and
I: ~ JJJE (x2 + y2) J( dV = K J;" fo"/4 focos¢(p2 sin2 ifJ) p2 sin¢ dpd¢d8 = K J:.,.. fo"'/4sin3 t/> [kPsJ::~os ¢ d¢d8
iK J:" J; 14sin ¢cos"' ¢d¢ d8 =if< .r:w d(} j~w/ cos ¢ (1 - cos ¢) sin¢d¢
4
3
=
= tK [8]~"' [- t cos6 ¢ + ~-cos 8 ¢]~
8
2
8
2
14
(:ii) + 1(:ii) + 1 - 1] =
6
= 1J<(27r)
[-16
5
5
2
6
8
b!.](
5
(..ll.)
u wK
384 - 960
33. (a) The density function is p(x, y, z) = K, a constant, and by the symmetry of the problem M .,z = My: = 0. Then
M.,11 =
J:"' f J; Kp sin ¢ cos ¢ dpdrjJd8 = !1rKa J; sin¢ cos ¢ d¢ = t1rKa But the mass is K(volume of
0"
12
3
4
12
4
•
the hemisphere)= ~1rKa3 , so the centroid is (0, 0, ~a).
(b) Place the center of the base at (0, 0, 0); the density function is p(x, y , z) = K. By symmetry, the moments of inertia about
any two such diameters will be equal, so we just need to find I.,:
I; ' I ;(Kp sin ¢) p (sin ¢ sin
I., = J:"
2
2
2
2
2
(} + cos
2
¢) dpd¢d8
J
2
2
3
3
= iKa5 I 0 " [sin 8 (- cos¢+ t cos ¢) + (- t cos ¢)] ::~
= K I:.,.. 0" 12 (sin3 rjJ sin2 8 + sin r,b cos2 ¢) (ka5 ) d¢ d8
12
d(} = i Ka5 j~" [~ sin2 8 + i) dO
5 [1(18- l sin28) + 18] " = 1Ka5 [1(7r - 0) + 1(27r- 0)] = ..i.J(a5 7r
= lKa
5
3 2
4
3
0
5
3
3
15
2
35. In spherical coordinates z = ..jx2 + y 2 becomes cos¢ = s in ¢ or¢ = 'i· Then
V = J:" f 0" 14 f 0 p2 sin ¢ dpd¢d8 =
1
M.,11 = I:"'
.r:r. dO f0.,..14 sin t/>dr/J .{ p dp = 27r ( -4 + 1) (t) = %1r(2 - v'2),
1
0
2
J;14I: p3sin¢cos ¢ dpd¢ d8 = 27r[--! cos2¢] ~ {t) = t and by symmetry My: = M x: = 0.
Hence (x,y,z) =
14
(o,o,
(
8 2
~ v'2)).
37. In cylindrical coordinates the paraboloid is given by z = r 2 and the plane by z = 2r sili Band they intersect in the circle
r = 2 sin 8. Then JJJEz dV = g f 02 s in o.r:;•inO rzdz dr dO = 5ti'
[using a CAS].
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284
0
CHAPTER 15 MULTIPLE INTEGRALS
.Jx 2 + y 2 and below the sphere x 2 + y 2 + z = 2 in the first
39. The region E of integration is the region above the cone z =
2
octant. Because E is in the first octant we have 0 ~ B ~ %. The cone has equation 1/> = "i (as in Example 4), so 0 ~ 1/> ~ .;f,
and 0 ~ p ~ ../2. So Ute integral becomes
f0rr14 .fa"''12 fov"i (p sin¢ cos B) (p sin c/> sin t1) p 2 sin¢ dp dB d¢ ·
2
4
= J~-r/ sin3 1/>d¢ J 0rr/ sinB cost1dB
.J0v"i p4 dp = (Jorr/ 4. (1 - cos2 1/>) sin 1/> dl/>) [~ sin2 B]~12 [ip5 ] :
-- [13 cos3 '~'
A.. - cos A..] rr/4.. 1. l ( '2)
'~' o
2 5 V"'
5
4
5
[~
12 - ~
2 - ( la - 1) ] . M5 -~ fl15
-
-
41 . The.region of integration is the solid sphere x 2 + y 2 + (z - 2? ~ 4 or equivalently
2
p2 sin2 c/> + (p cos lj> - 2) = p 2 - 4pcos</> + 4 ~ 4
p ~ 4cos¢, so 0 ~ 8 ~ 211', 0 ~ c/> ~ %• and
=>
0:::; p ~ 4cos ¢.Also (x2 + y 2 + z2) 312 = (p2) 312 = p3 , so the integral becomes
J0" 1 2 J:" J04. cos<P (p3) p 2 sin ¢dpdB d¢ = J01112 J~ sin ¢ '[~lJ :~ cos,P dBd¢ = i J0" 1 2 f02 " sin c/> (4096 cos6 ¢) dOd¢
11
= i(4096)J0"
12
2
12
7
0
20 48
cos ¢sin ifJdifJ J 0 " dB = 3 (- ~cos ¢]~ [ B ]~"
= 20348 ( ~) (211') = 40i~rr
43. In ·cylindrical coordinates, the equation of the cylinder is r = 3, 0 ~ z ~ 10.
The hemisphere is the upper part of the sphere radius 3, center (0, 0, 10), equation
1·
2 + (z - 10) 2 = 32, z ~ 10. In Maple, we can use Ute coords=cylindrical option
in a regular plot3d command. In Mathematica, we can use Parametri c Plo t 3D.
45. If E is the solid enclosed by the surface p = 1 +
E = { (p, 8, ¢) I 0 ~ p ~ 1 +
t sin 60 sin 5¢, it can be described in spherical coordinates as
t sin 68 sin 5¢, 0 ~ 8 ~ 211', 0 ~ ¢ ~ ,1f}. Its volume is given by
V(E) = fJ{s dV = fo" fo2 rr J~l+(sin60sin5r/>)/u P2 sin cf>dpd(Jd¢ = 1~~" [using a CAS).
47. (a) From the diagram, z = r cot ¢ 0 to z = ~. r = 0
to r= asin ¢ (or use a 2 - r 2 = r 2 cot2 ¢ ). Thus
0
0
V = r 2" ra sin <Po r v'o.2 - r2 rdzdrdB
Jo Jo
J rcotr/> 0
z = rcot r/> 0
y
= 2:r [- (a2 - a2 sin2 c/Jo)3/2 - a3 sin3 ¢o cot¢o + a3]
2
;, ~7ra 3 [1 - (cos3 c/>0 + sin ¢ 0 cos 1/>0 )] = lrra3(1 - cos ¢ 0 )
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SECTION 15.1 0 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS
0
285
(b) The wedge in question is the shaded area rotated from B = 81 to B = 82.
Letting
V.i = volume of the region bounded by the sphere of radius P;
and the cone with angle ¢; (B = 81 to 82)
and letting V be the volume of the wedge, we have
y
V = (1122- 1l21)- (V12 - Vn)
· = 1(82 - B1)[pg(1- cos¢2 ) - p~(l - cos¢1 ) - p~(1- cos¢2 ) + p~(1- cos¢ 1 ) ]
= 1(82- 81) [(P~- p~)(1- cos ¢ 2 ) - (P~- pi) (1- cos ¢ 1)] = i(B2 - (Jt)[(p~- pi) (cos ¢ 1 - cos¢2 )]
Or: Show that V =
021P2 s in <l>2 1 rcot ¢ r dz dr dB.
1
1
p 1 s in 4> 1
01
r cot 4> 2
(c) By the Mean Value Theorem with f(p) = p3 there exists some p with p 1 ~ p ~ p2 such that
f(p 2 ) - f(p 1 ) = !' (p)(p 2 - p1 ) or p~ - p~ = 3Ji D.p. Similarly there exists¢ with ¢ 1 ~ ~ ~ ¢ 2
such that cos ¢ 2 - cos ¢ 1 ·= (-sin¢) D.¢. Substituting into the result from (b) gives
D. V = (p2 D.p)(B2- B1)(sin ¢) D.¢ = p2 sin¢> D.pD.¢ D.B.
15.10
Change of Variables in Multiple Integrals
1. x = 5u- v, y = u + 3v.
1) = l8xf8u 8x f 8v I =
TheJacobianis a(x
~( ,y)
v u,v
8y f 8u oyfav
5
1
-ll
1
3
=5(3) -(-1)(1) =16.
3. x = e-r sinB, y :;= er cos B.
8(x,y)
-- -
8(r, B) -
18xf8r 8xj8fJ I ~ -e-''sinB e-rcosBI
·
·
r .
= e-•·er sin2 B- e-re'· cos2 B = sin2 fJ- cos2 ()or- cos 2()
ayj8r ayjaB er cos() -e smfJ
.
.
5. x = ufv, y = vfw, z = wfu.
oxfou 8xfav 8x/8w
a(x,y,z) ayfau ayf8v 8yf8w
a(u,v,w)azjou 8z/8v 8z/8w
2
I
1/ v
- u /v 2
0
0
1/w
-v jw 2
- wfu2
0
1/u
2
= .!_11/w - vjw ! ( u)
0
-vjw ! 0 I 0
1/w I
v
0
1/u
- - V 2 .:..wfu 2
1/ u
+ -wfu2 0
=.!.
1
1
(...!_ - o) + ~ (o- ~)
+ o= -uvw
- - - - = o
u w
uvw
v uw
v2
2
1. The transformation maps the boundary of S to the boundary of the imageR, so we first look at side 8 1 in the uv-plane. S 1 is
described by v = 0, 0.::; u.::; 3, sox = 2u + 3v = 2u andy = u- v = u. Eliminating u, we have.x = 2y, 0 _::; x ~ 6. 8 2 is
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286
0
CHAPTER 15 MULTIPLE INTEGRALS
the line segment u = 3, 0:::; v :::; 2, sox= 6 + 3v andy = 3 - v. Then v = 3- y
x = 6 + 3(3- y) = 15- 3y,
=?
6 :::; x :::; 12. S3 is the line segment v = 2, 0 :::; u :::; 3, sox = 2u + 6 andy = u- 2, giving u = y + 2
6 $ X $ 12. Finally, S4 is the segment U = 0, 0 $ V $ 2, SO X = 3v and y = -V
:=}
X
=?
x = 2y + 10,
= -3y, 0 $ X $ 6. The image of
set S is the region R shown in the xy-plane, a parallelogram bounded by these four segments.
IJ
y
sl
(0, 2)
(3,2)
s.
s
sl
0
s,
(3, 0)
-
(6, 3)
T
(12,1)
0
II
X
(6,- 2)
9. sl is the line segment u = v, 0$ u ~ 1, soy= v = 'U and X = u 2 = y2 • Since 0 ~ u ~ 1, the image is the portion of the
2
2
parabola X = y ' 0 :::; y :::; 1. s2 is the segment v = 1, 0 :::; u ~ 1, thus y = v = 1 and X = u ' so 0 :::; X :::; 1. The image is
the l.ine segment y = 1, 0 $ X $ 1. 83 is the segment U = 0, 0 $ V $ 1, SO X = u 2 = 0 and y = V
:=}
0 $ y $ 1. The
image is the segment x = 0, 0 :::; y :::; 1. Thus, the image of S is the region R in the first quadrant bounded by the parabola
x = y 2 , they-axis, and the line y = 1.
y
T
u
0
X
11. R is a parallelogram enclosed by the parallel lines y = 2x- 1, y = 2x + 1 and the parallel lines y = 1- x , y
= 3 - x. The
first pair of equations can be written as y - 2x = -1, y - 2x = 1. If we let u = y - 2x then these lines are mapped to the
'
vertical lines u = - 1, u = 1 in the uv-plane. Similarly, the second pair of equations can be written as x + y = 1, x + y = 3,
and setting v = x + y maps these lines to the horizontal lines v = 1, v = 3 in the uv-plane. Boundary curves are mapped to
boundary curves under a !ransformation, so here the equations u = y- 2x, v = x + y define a transformation r - 1 that
maps R in the xy-plane to the square S enclosed by the lines u = -1, u = 1, v = 1, v = 3 in the uv-plane. To find the
transformation T that maps S to R we solve u = y - 2x, v = x + y for x, y: Subtracting the fi_rst equation from the second
gives v - u = 3x
=?
x = ~ (v - u) and adding twice the second equation to the first gives u + 2v = 3y
=?
y = Hu + 2v). Thus one possible transformation T (there are many) is given by x = Hv- u),y = ~('u + 2v).
3
v=3
s
u '= -1
u= 1
T
--+1
- I
0
v- 1
u
1
y=2t-1
'',f' :
•
X
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SECTION 15.10
CHANGE OF VARIABLES IN MULTIPLE INTEGRALS
0
287
13. R is a portion of an annular region (see the figure) that is easily described in polar coordinates as
R = {(r , 8) 11 ::; r ::; ../2, 0 ::; (} ::; 7T/ 2}. If we converted a double integral over R to polar coordinates the resulting region
of integration is a rectangle (in the rO-plane), so we can create a transformation T here by letting u play the role of rand v the
role of 0. Thus T is defined by x = u cos v , y = u sin v and T maps the rectangle S = {(u, v) I 1 ::; u ::; ../2, 0 ::; v. ::; 7T / 2}
in the uv-plane to R in the xy-plane.
v
y
E.
2
-
T
s
0
15 B(x, y) = 1
· 8(u, v)
2 1
1 2
1
II
X
= 3 and x - 3y = (2u + v) - 3(u + 2v ) = -u- 5v. To find the regionS in the uv-plane that
·
corresponds toR we first find the corresponding boundary under the given transform ation. The line through {0, 0) and (2, 1) is
y = ~x which is the image of u + 2v = ~(2u + v)
image of (2u + v) + (u + 2v) = 3
u + 2v = 2(2u + v)
=>
v = 0; the line through (2, 1) and (1, 2) is x + y = 3 wh ich is the
=> u + v = 1; the line through (0, 0) and (1, 2) is y = 2x which is the image of
=> u = 0. Thus S is the triangle 0 ::; v ::; 1 - u, 0 ~ u ::; 1 in the uv-plane and
jj~ (x - 3y) dA = j 0 j 0 -u ( -u - 5v) 131 dv du = -3 f 0 [uv + ~v 2 J::~-u du .
1
1
1
= - 3 j; (u - u
17 B(x, y) = 1
. 8( u, v)
2 0
0 3
2
+ ~{1 - u?) du = -3[~u2 - iu3 - ~(1- u?]~ = -3(~ - ~ + ~) = - 3
I= 6, x = 4u and the planar
ellipse 9x + 4y
.
2
2
ffn x 2 dA =
2
2
::; 36 is the image
~fthe disk u + v <- 1. Thus
2
2
JJ (4u2 )(6) dudv = f 02 "' f 01 (24r2 cos2 8)rdrd8 = 24J;.,.. cos 2 8d8 J01 r 3 dr
u2 +v2~ 1
4
= 24[~ x +~sin 2x]~"' ar ] ~ = 24(7r)(i) = 67!'
2
1
19. 8(x, y) = , /v - ufv ' =
o
8(u, v)
1
.!.,
xy = u, y =xis the image of the parabola v 2 = u, y == 3x is the image of the parabola
v
v 2 = 3u, and the hyperbolas x y = 1, xy =a are the images of the lines u = 1 and u = 3 respectively. Thus
jL dA = 11:
3
xy
21 . (a)
:ix, z~ =
y,
u,v,w
1 (In v3u - vfu)
3
u ( ; ) dv du =
u
a 0 0
0 b 0 = abc and since
ln
u= ~. v=
a
du = J: u In J3 du = 4ln J3 = 2 ln 3.
J!.b' w = ~the solid enclosed by the ell ipsoid is the image of the
c
0 0 c
JJJEdV =
JJJ
abcdudvdw = (abc)(volumeof tlie ball) = ~1rabc
u2+v2+w2 :S 1
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288
0
CHAPTER 15 MULTIPLE INTEGRALS
i
2
2
2
(b) If we approximate the surface. of the earth by the ellipsoid ;., +
+ ;
_= 1, then we can estimate
6 82
6 782
6 562
the volume of the earth by finding the volume of the solid E enclosed by the ellipsoid. From part (a), this is
fffE dV = ~11"(6378)(6378)(6356) ~ 1.083 X 1012 km3 •
(c) The moment ofintertia about the z -axis is I%=
JJJE(x2 + y 2) p(x, y, z) dV, where E is the solid enclosed by
I
xz yz zz
18(x y z)
2 + b2 + 2 = 1. As in part (a), we use the transformation x = au, y = bv, z = cw, so ( ' ' ) = abc and
a
.
c
8 u,v,w
I, =
JJJE (x 2+ y 2) kdV =
2 2+ b2 ·u2 )(abc) dudv dw
f.(.[
k(a u
u2 + v2+w2 ~ 1
2
= abck f 0" f 0 "
J01 ( a 2p2 s in2 ¢ cos2 () + b2 p2 sin2 ¢ sin2 8) p 2 sin ¢ dp d8 drjJ
1
1
= abck [a2 f0,.. f~,.. f0 (p2sin2 rP cos2 B) p2sin 1/> dp d() dlj> + b2 f0" f~,.. f0 (p2 sin2 rjJ sin2 B) p 2 sin rP dp dB dr!>]
= . a 3 bck 0" sin3 rjJ drjJ f0 " cos2 B dB
J; p4 dp + ab3 ck fo.,. sin3 rjJdrjJ f~" sin2 BdB J; p4 dp
= a3 bck [ ~ cos3 ¢-cos I/>] ~ [~B +
i sin2B]~" [iP ] ~ + ab ck [t cos rjJ- cosr/J]~ [~B- i sin 28] ~" [tP ]~
J
2
5
3
5
3
= a3 bck (t) (1r) ( t) + ab3 ck (1) (1r) (i) = rt7r (a 2 + b2) abck
8
((x, y)) =
8 u,v
23. Letting u = x - 2y and v = 3x - y, we have x = f.(2vu) and y = 0f(v- 3u). Then
0
215
~ -3/5
- 1 / 5 1/5
1 = .!.
5
and R is the image of the rectangle enclosed by the lines u = 0, u = 4, v = 1, and v = 8. Thus
25. Letting u = y- x, v = y + x, we nave y = !(u + v), x = Hv - u). Then
8(xy)
1.:...1/2 1/ 21
1
(u: v) =
i
= 2 and R is the
8
1 2 112
image of the trapezoidal region with vertices ( - 1, 1), ( -2, 2), {2, 2), and {1, 1). Thus
!h
R
112[ u]u=v
11" I 11
2
y -x
cos - dA-=
Y +X
'1
cos -u - - du dv = 2
2
-v
V
v sinV
1
u= -v
112
dv = -
2
.
27. Letu =x+y andv=-x+y.Thenu+v = 2y => v=Hu+v) andu - v=2x
1
-8(x
( 'y)
) = 11/2 -1/ 2! =.Now[u[= [x+y[< [x[+[y[<1
2
1/ 2
1/2
.
8 u, v
[v [ = [-x + y[ ~ [x [ + [y[ ~ 1 =>
2v sin{1) dv = ~sin 1
1
=> X=~(u-v).
=> - l<u<l,and
-
- 1 ~ v ~ 1. R is the image ofthe square
y
1
region with vertices {1, 1), (1, -1), ( - 1, -1), and (-1, 1).
So ffn e"+" dA = ~ f~ 1 f~ 1 e" dudv =
He"t v t = e 1 [
1
e-
1
X
.
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CHAPTER 15 REVIEW
0
289
15 Review
CONCEPT CHECK
m
n
,L f (xi;, Yi;) b.A, where b. A is the area of each subrectangle and (xi; , Yi;) is a
1. (a) A double Riemann sum off is ,L
i=lj = l
S3J!lple point in each subrectangle. If j(x, y) ;::: 0, this sum represents an approximation to the volume of the solid that lies
above the.rectangle R and be low the graph off.
(b) ffn. f(x, y) dA =
lim
f f== f(xi;. Yi;) b.A
m,n-+oo i = 1 j
1
(c) If f(x, y) ;::: 0, ffn f(x, y) dA represents the volume of the solid that lies above the rectangle Rand below the surface
z = f(x, y). Iff takes on both positive and negative values, ffn j(x, y) dA is the difference of the volume above R but
below the surface z = f(x , y) and the volume below R but above the surface z = f(x, y).
(d) We usually evaluate ffn. j(x, y) dA as an iterated integral according to Fubini's Theorem (see Theorem 15.2.4).
(e) The Midpoint Rule for Double Integrals says that we approximate the double integral
Riemann sum
JJR f(x, y) dA by the double
f f: f(xi, fi;) b.A where the sample points (:x,, fi;) are the centers of the subrectangles.
i=lj= l
(f) fnvc = A
~R) IIn f(x, y) dA where A (R) is the area of R.
2. (a) See ( I) and (2) and the accompanying discussion in Sectio n 15.3.
(b) See (3) and the accompanying discussion in Section 15.3.
(c) See (5) and the preceding discussion in Section 15.3.
(d) See (6) - (1 I) in Section 15.3 .
3. We may want to change from re~tangular to polar coord ina tes in a double integral if the region R of integration is more easi ly
described in polar coordinates. To accomplish this, we use ffn f(x, y) dA =
J: J: f (rcosB, rsin 9) r dr d(} where R is
given by 0 ::::; a ::::; r ::::; b, a ::::; B ::::; {3.
4. (a) m
= JJD p(x, y) dA
(b) M , = ffv yp(x, y) dA, M 11 = ffv xp(x, y) dA
(c) The center of mass is (x, y) where x == M" and y= lvfx .
m
m
(d) I , = ffv y p(x, y) dA I v = ffv x p(x, y) dA, I o = ffv (x
2
2
5. (a) P(a ::::; X::::; b, c:=; Y::::; d)=
1:J:
2
+ y 2 )p(x, y) dA
j(x, y)dydx
(b) f(x , y) ;::: 0 and JJ~~ f(x, y) dA = 1.
· (c) The expected value of X is JJ.
1= ffR2 xf(x, y) dA; the expected value ofY is p..z = JJT:/.2yf(x, y) dA.
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290
CHAPTER 15 MULTIPLE INTEGRALS
0
6. A(S) = JJ0 J [f.,(x, y))2 + [f11 (x, y))2 + 1 dA
(b) We usually evaluate f[JB f(x, y , z) dV as an iterated integral according to Fubini's Theorem for Triple Integrals
(see Theorem 15.7.4).
(c) See the paragraph following Example 15.7. 1.
(d) See (5) and (6) and the accompanying discussion in Section 15.7.
(e) See (10) and the accompanying discussion in Section 15.7.
(f) See (11) and the preceding discussion in Section 15.7.
8. (a) m =
JJJE p(x, y, z) dV
. (b) M z =
71
JJJE xp(x, y, z) dV, Mxz = JJJEyp(x, y, z) dV, Mxy = JJJEzp(x, y, z) dV.
· (- - - ) J Myz _
(c) The cent er of mass ts x, y, z w 1ere x = --, y
m
lvfx11
= -Mm-xz, an d _z = --.
m
9. (a) See Formula 15.8.4 and the accompanying discussion.
(b) See Formula 15.9.3 and the accompanying discussion.
(c) We may want to change from rectangular to cylindrical or spherical coordinates in a triple integral if the region E of
integration is more easily described in cylindrical or spherical coordinates or if the triple integral is easier to evaluate using
cylindrical or spherical coordinates.
10
I
I
a (x, y)
ax;au ax;av
ax ay ax By
· (a) a (u, v) = ay;au ayjav = au av - av au
(b) See (9) and the accompanying discussion in Section 15.10.
(c) See (13) and the accompanying discussion in Section 15.10.
TRUE-FALSE QUIZ
1': This is true by Fubini's T heorem.
3. True by Equation 15.2.5.
5. True. ~y Equation 15.2.5 we can write J~ f
this becomes
7. True:
1
0 f(x) f(y) dy dx = I~ f (x) dx I~ f(y) dy. But J~ f(y) dy = J~ f(x) dx so
I; f(x) dx .{ f(x) dx = [J; f(x) dxf.
1
0
JJ0 J 4 - x 2 - y 2 dA = the volume under the surface x 2 + y 2 + z 2 = 4 and above the xy-plane
= ~ (the volume ofthe sphere x 2 + y 2 + z 2 = 4) = ~ · t1r(2) 3 = 136 1r
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1
CHAPTER 15 REVIEW
0
9. The volume enclosed by the cone z = ...jx2 + y2 and the plane z = 2 is, in cylindrical coordinates,
V = J~" J~
J: r dz dr dO i= f
2
0"
J~
J: dz dr dO, so the assertion is false.
EXERCISES
1. As shown in the_contour map, we divideR into 9 equally sized subsquares, each with area b.A = 1. Then we approximate
ffn f(x, y) dA by a Riemann sum with m = n = 3 and the sample points the upper right comers of each square, so
3
ffnf(x,y)dA>:::J L:
3
L: f(x i, yj)b.A
i =lj=l
= b.A [!(1, 1) + /(1, 2) + /(1, 3) + !(2, 1) + !(2, 2) + f(2, 3) + !(3, 1) + !(3, 2) + !(3, 3)]
Using the contour lines to estimate the function values, we have
ffnf(x,y) dA >:::! 1(2.7 + 4.7 + 8.0 + 4.7 + 6.7 + 10.0 + 6.7 + 8.6 + 11.9] ~ 64.0
2
2
3. f 1 J~ (y + 2xeY) dx dy = J? [xy + x 2 e 11] ::~ dy = f 1 (2y + 4eY) dy = [y2 + 4eY] ~
= 4 + 4e 2 - 1 - 4e = 4e2 - 4e + 3
7.
J; J; J0~ ysinx .d zdy dx = f J; [(ysinx)zJ::F dydx·= f f y ...jl- y sinxdydx
0"
.
0"
1
0
2
= f 0" [- ~ (1- y2 ) 312. sin x)·u :l dx = f 0" ~ sin x dx = -~ cosx]~ = ~
y~
9. The regi?n R is more easily d~scribed by polar coordinates: R
= {(r, 0) I 2 s; r s; 4, 0 s; 8. s; 71"}. Thus
ffn f(x, y) dA = fo"f f (rcosO, r sinO) rdrdO.
4
2
. w hose area .IS g.tven
.
by Jo
fTr / 2 r•in 20 d dO .
The reg1on
Jo
r r
IS
11.
·{ ( r, 0) I 0 s; 0 s; %, 0 s; r s; sill 20}, which is th~ region contained in the
loop in the first quadrant of the four-leaved rose r =sin 28 .
./~
1
13.
J: cos(y dy dx J; g cos(y dx
2
)
=
2
)
dy
= f 01 cos(y2 ) [ x ] ::~ dy = J01 y cos(y2 ) dy
X
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291
l
292
CHAPTER 15 MULTIPLE INTEGRALS
0
!1.r +
y
17.
y
D
1
x2 dA =
111r..rx ,
o
= l2
0
o
.
1
11 + h )y-..rx
1
1 : x2 dy dx =
o 1
1+
X
2
x2 [
y:o
1
- - dx = [!·I In(1 + x )) o = !4 In 2
0 1 x2
2
X
fj ,0 ydA = Jfo2J,B-y~
ydxdy
112
19.
= fo2 y[x] ::~~v2 dy = J; y(8 ~ y2- y2) dy
3)dy = [4y
= f02 (8y - 2y
/L (xz +
y
21 .
y2)3/ 2 dA =
2
-
tY 4 ]~ = 8
17r/313 (r2)3/2r drd0
= 11r/3 dO 13r4dr = [0)~/3 [ir:;)~
5
1f 3
811f
=35=5
23. fffe xy dV
= .1: j~x J;+v xy dz dy dx = .{03 .fox xy [ z J::~+v dy dx =J; ./~"' xy(x + y) dy dx
= ]~3 J;(x 2 y + xy 2 ) dydx = .r; [tx2 y 2 + ~x1lJ;;:~ dx = J; (tx 4 + tx4 ) dx
6 / '3 4 d
[GX
1 5] 3
81
40 5
= 6,
0 X
X =
0 = 2 =. '
25. Iff v 2 z 2 d\l=f
E
1
- 1
2
J~
]'1 -v - z y2 z 2 dxdzdv=t- 1 ·rv'~-~~
y 2 z 2 (1-v2 - z 2 ) dz dv
-~ 0
-Vl -y2
2
2
= J;1r J; (r2 cos2 O)(r2 sin2 O)(l - 1·2 )rdrd() = J02 Tr Jd ;t sin2 2(J(r5 - r7 ) drdO
1 6
r = 1 d()
1 [0
. 49] 2,2Tr
"
= Jor2 rr S1 (1 - COS 40) [6r
- ii1 1' 8 ] r=O
= TIJ2
- 41 SID
0 = 192 = 96
3
= lls Jof " sin3 () dO = lll5 [- cos 0 + i-3 cos
OJ "o = !11
.
15
r4 ( 2
2) d d _
29 . V -_ Jro2 JJ
X + 4y
y X -
d . _ r2 ( 2
)d ._
u= l , X - Jo 3 X + 8 4 :r. - 176
10.2 [X 2 y + 34 y 3J u=4
31:
2
2
2
V = Jo
f j '11 r< - ul/ dzdxdy = rz rv (1-ly) dxdy
o .lo
.Jo .lo
2
y
X
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dx
CHAPTER 15 REVIEW
D
293
33. Using the wedge above the plane z = 0 and below the plane z = mx and noting that we have the same volume for .m < 0 as
for m > 0 (so use m > 0), we have
V = 2 foa/3
J/a2-9y2 mx.dxdy = 2 foal~ tm(a2 - 9y2) dy = m[a2y - 3y3)~/3 = m(ia3 - ia3) = ~ma3 .
35. (a) m = fo1 fo1 - v2 ydxdy = J~1 (y - y3) dy = ~- ~ = ~
2
rl rl- v
ri 2Y
1 (
1 (1
1
(b) Mv = Jo
Jo
xy dx dY = .Jo
1 - Y2)2 dY = -u
- 'y 2)3) o1 = 12'
c
2
c- -) = (_3 ,
r l Jo
r1 - v y 2 dx dy = r1 y 2 - y 4) d.y = 2 . H ence x, y
M, = Jo
10
15
1
s )
15 .
2
r1 r1-v
( 3
s) dy=u,
1
(c)lx=JoJo
y 3 dxdy= J·1
0 y - y
2
r1 rl - v
r1 1 (
_ -24
1 (1
1
ly = Jo
Jo
yx 2 d x d Y _- Jo
3Y 1 - Y2)3 dY- Y2)4] 1o -_ 24'
_ I
I _ 1 =2 _ .!.L_g _ 1
.lo - "'+ v- 8• Y - 1/ 4 - 3
=?
1
d= 2 _
Y - 73• an x -
= _
1/24 _
1/ 4 -
=
1
6
x = ?s·
=?
1
37. (a) The equation of the cone with the suggested orientation is (h - z) = ~ .jx 2 + y 2 , 0 $ z $ h. Then V = ~ 1ra2 h is the
volume of one frustum of a cone; by symmetry M 11:: = !vfx:: = 0; and
•= !! 11•-(11/a),jx2+y2
M;r;y
o
;r;2+y2 ~ 11 2
z dz dA =
12,.1a1(hfa)(a-r) rzdzdrd/J = 1r 1a.r h2 (a- r) ; dr
o·
ooo
·
2a
1a (a2r- 2ar2 + r 3) dr = -1rha2 (a-2 - -2a3 + -a4 ) = 1rh a
= -1rh
a2
12
2
2
4
4
2 2
4
0
Hence the centroid is (x,y,z) = (0,0, ~h).
(b) I , =
2"1"1(/•/.a)(a-r) r. 3 dz dr d() = 27r 1" -h (ar3 - r 4) dr = -27rh (a5- - -a5) = -7ra4-h
a
a
4
5
10
1 o o
0
0
39. Let D represent the given triangle; then D can be described as the area enclosed by the x - and y-axes and the line y = 2 - 2x,
or equivalently D = {(x , y) I 0 $ x $ 1; 0 $ y $ 2 - 2x }. We wantto find the surface area of the part of the graph of
z = x 2 + y that lies over D, so using Equation 15.6.3 we have
A(S)=
=
Jfv
1+
(~;y + (~~Y dA = Jfv v 1+(2x)2+(1)2dA=
11
1
2
-
2
;r;
)2 + 4x2dydx
.r; .../2 + 4x [y J~:~- ;r; dx = J;(2 - 2x) .../2 + 4x dx = · J 2 J2 + 4x2 dx- JC: 2x J2 + 4x2 dx
2
2
2
1
0
Using Formula 21 in the Table oflntegrals with a = J2. u. = 2x, and du = 2 dx, we have
J 2 J2 + 4x2 dx = x J2 + 4x 2 + ln(2x + J2 + 4x 2 ). lfwe substitute u = 2 + 4x2 in the second integral, then
du = Bx dx and J 2x ,J2 + 4x2 dx = ~ J .JU du = i · tu 312 = i (2 + 4x 2 ? 12 . Thus
.
A(S) = [xJ2 + 4x 2 + 1n(2x + J2 + 4x2) - t{2 + 4x2 ) 312
= J6 + ln (2 + JB) - ~(6) 3 1 2 - ln J2 +
= ln( J2 + v'3) +
1
Jo
4- = ln 2t# + {1
4- ~ 1.6176
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294
0
CHAPTER 15 MULTIPLE INTEGRALS
41.
f
rJ
3
2
9
- x
1o 1-J9-x2
(x3 + xy2 )dydx = {
3
{~ x(x +y )dydx
2
2
1o 1-~
= J::~~ 2 f~ (r cos O)(r ) r dr dO
2
= J::~~2 cos 0 dO
= (sino]
J; r dr
4
:::/2 (ir5]~ = 2 ·. i (243) =
4 6
:
= 97.2
43. From the graph, it appears that 1 - x2 = e"' at x ~ - 0.71 and at
x = 0, with 1 - x 2 > e"' on ( -0.71, 0). So the desired integral is
JJ"n y2dA ~ f~o.n J.1.,- x2 y2 dydx
= l3 Jo
[(1 - x2)3 - e3"'] dx
-0. 71
=-o.25
45. {a) f(x , y) is a joint density function, so we know that
JJR2f( x, y) dA = 1. Since f(x , y) = 0 outside the rectangle
(0, B) x (0, 2], we can say
JJR2 f (x,y) dA = f~oo f~oo f(x ,y) dydx = J: J: C(x + y) dydx
= C .r; [xy + h2J ~:~ dx = C J03(2x + 2) dx = C[x2+ 2x)~ = 15C
Then 15C = 1 :::?
C = ft.
(c) P(X + Y ~ 1) = P ((X, Y) E D) where Dis the triangular regiol} shown in
y
the figure. Thus
P(X +Y ~ 1) =
JJ0 f(x,y) dA = f 01J;-x -ft (x+y)dydx
1
=· ..1..
rl [
+ 12Y 2]u=
15 J o xy
u=O -"' dx
=
-
fs f; [x(1 - x) + ~ ( 1 - x?] dx
1
f1(1
30 Jo
.z
47.
-
0
I
X
.2)dX -- 30
1 [
1 3]1
1
X- 3X 0 = 45
X
( , zdzdydx=
)
J.l0 .r.1-=Jv"ii
- v"ii ! (x,y,z) d xdydz
J_l 1 j'l.,2 j 0'1- IJ fx,y
0
X
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CHAPTER 15 REVIEW
0
295
49. Since u = x- y and v = x + y, x = t(u + v) andy= ~(v- u).
8(x
Thus -'- y) =
8(u,v)
I-1/2
1/2 ·1/2
1/2 1= -12 and flRx+y
x-y dA = 1/ .o -u (1)
- du dv = - £4
-dv = -In 2
v 2
,
v
·
4
2
51 . Let U = y - X and V = y +X SO X = y :... U = (V - X) - U
I
-.
=:>
_ 2
X
2
= ~ (V - U) and y = V - ~ ( V - U) = ~ (V + u).
I~~::~~ = I ~:~ ~~~~I = 1-~ (~) - ~(~)I = -'~I = ~.· R is the image under this transformation .o f the square
J
with vertices (u, v) = (0, 0), (-2, 0), (0, 2), and ( -2, 2). So
This result could have been anticipated by symmetry, since the integrand is an odd function of y and R is symmetric about
the x-axis.
53. For each r such that Dr lies within the domain, A(Dr) = 71'1'2 , and by the Mean Value Theorem for Double Integrals there
exists (xr, Yr) in D,. such that f (xr, Yr) = ~
7TT
so lim ~
r-o+ 1fT
j'lv,.
{ f(x, y) dA. But lim (xr, Yr ) = (a, b),
r-o+
j'{ f (x, y) dA = lim f(xr, y,.) = f(a, b) by the continuity of f.
} Dr
r--+O+
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D PROBLEMS PLUS
1.
y
Let R ,=
5
u;= Ri, where
1
R , = { (x , y ) I x+ y ~ i + 2, x +y < i+3, 1 ~ x ~ 3, 2 ~ y ~ 5}.
4
5
3
5
ffn [x+y) dA = I: f fn .[x+y)dA= L: [x+ylffn . dA , since
2
i =1
x+y=5
0
2
3
i= l
1.
'
[x + y] = constant = ~+ 2 fo r (x, y) E R; . Therefore
Jfn[x + y) dA = E?=1 (i + 2) [A(R;)]
X
= 3A{R I) + 4A(R2) + 5A(R3) + 6A(&) + 7A(R s)
= 3(4) + 4(~) + 5{2) + 6(~) + 7 ( ~) = 30
rb
I
t [J.t cos(t-)dt dx
I
3. f ,,.• = b _ ala f (x )dx = I - 0 Jo
=
?
J; J:cos(t dt dx = ,{ J; cos(t dx dt
2
1
0
)
2
)
]
x=t
[changing the ordeqlf intcsrntion]
0
1
5. Since lxvl < 1, except at (1, 1), the formula for the sum of a geometric series gives -I --
- ~
.
f (xy)", so
=
n~
•.
"' 1 1 "'
00
=
00
=
L.., n+1 ' n+1
n=O
L.., (n+11)2
n= O
1
= 1!! + 211 + 32"1 + · · · = "L..,n'""= 1 ~1
1
--
7. (a) Since ixyz i < 1 except at (I , I , I), the formula for the sum of a geometric se;ies gives -
1- X 1JZ
111111
o
11lll1
E(xy z )n, so
n=O
11111·1
1 - d:r;dydz=
00
00
- _I:
(x yztdxdy dz = I:
xyz
o . o o n =O
n=O 0
o o 1
=
00
(xy z tdx dydz
o
0
00
= I: .f01 x" dx ] [ ,[01 y"' dy] [ Jf o1 z" dz ] = I: - 1-1 · - 1- 1 · - 1- 1
[
n=O
00
n =O n
.
1
1
1
1
.
n+
+
n+
1
00
= ...~o (n + 1)3 = 13 + 23 + 33 + . .. = n~1 n 3
(b) Since 1-xyzl < 1, except at (1, 1, I ), the formula for the sum of a geometric series gives
111111 +
1
o o o 1
xyz
dx dy dz =
111·11·1I: (
00
o
o o n=O
00
-xyz)" dx dy dz = 2:
n=O
1
1 +xyz
= f= (- x yz)" , so
n= O
11.1111(
o
o
o
-xyz)" dx dy dz
·
= n=o
f (- 1)". (1~1 x" dx] [J; y"dy] [J; z" dz] = n=O
f (-1) "n-+1-1- -n +1- 1 . -n +1- 1
00
00
(- 1)"
1
1
1
(-1)" - 1
= n~o (n + 1)3 = 13 - 23 + :P - . . . = n~o
n3
© 2012 Ccngagc: Lc:.1ming. All Rights Resc!rv.:d. May not be scanned, copied. or duplicated, or posted to a publicly oc~ssiblc website. in whole or in parl
[continued]
297
298
D
CHAPTER 15 PROBLEMS PLUS
To evaluate this sum, we first write out a few tenns: s = 1 - ; 3 + ; 3 a1
.
;
3
+ ;~ - ; 3 ~ 0.8998. Notice that
= 7\ < 0.003. By the Alternating Series Estimation Theorem from Section 11.5, we have .Is- s61~ a1 < 0.003.
This error of0.003 :-viii not affectlhe second decimal place, so we have s .~ 0.90.
.
9. (a)x=rcosO, y =rsiDO, z=z.
au.
8u ox au ay au az au
au. .
Then or= axar + ayar + ozor =ox cosO+ oysmOand
I ou
ou . (} 8u
(} d
S.lffil. 1ary
{)(} =-ax TSID + oy rcos an
82u
a 2 u '2 . 2
8 2 u. 2 2
8 2u 2 .
au
8u .
ao2 = 8x2 r SID 0 + 8y2 r cos (} - 2 oy ax r SID 0 cos (} - OX r cos (} - {}y r SID 0. So
82 u
lau
1 8 2u _ 8 2 u
a2 u
82 u .
a 2 v.
.
au cosO
8u sinO
!:i:2 + + 2 <>o2 + !i'2 = !i"2 cos 2 0 + !i'2 sm2 0 + 2 .<~.. <> cos 0 sm 0 + -;:;---- - + -;:;---- vr'
r
-ar
r
u
uz
ux
uy
82u . 2
vyuX
82u
2
uX
82v.
. (}
+ ux.
~sm O + ~cos 0 - 2~a sm
uy
uy X
cos
r
uy
(}
8ucos9
8usin9
82u
- ox - r - - oy - r - + 8z 2
82u
82 1~
82 u
=
- +-+
-{)z2
. 8x2
{)y2
(b) x = psinrf>cosO, y = psin ¢>sin0, z ·= pcos¢>. Then
au
au~
8u8v
&&
&.
& ..
au .
op = ox op + oy op + oz ap = OX sm¢> cosO+ {}y sm¢> s m9 + {)z cos rf>, and
. . 1 I 8u
8u
-~..
8u
-~.. . (} ou
S1m1ar
y Br/> = &xp.cos'l'cos 9 + oypc~'l'sm - azpsinrj>,and
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r
CHAPTER 15 PROBLEMS PLUS
au
au
.
A. •
And afJ = - ax psm.I' Slll
f)
D
299
. A.
()
h'l
+ au
ay pSID'I'COS , W I e
a2u
a2u 2 2
·
a2u
·2
2
=
-2 - a
a p sin if> cos() sin 0 + -a 2 p2 sin ¢ sin ()
afJ
y X
X
-2
2
. 2 A.
au psm'l'
. A. cos 0 - -a
au p sm
. 'l'
A. • 0
'l'cos2 0 --a
sm
+ aa yu2 p2 sm
X
y
Therefore
2
2
2
2 au cot¢> au
1 8 u
1
a u
-aap2u + - + -p2-a¢
- +- + ~--;:opap
p2 aq?
p2 sin2 if> aB2
= ::~ [(sin2 ¢> cos 2 B) + (cos2 ¢> cos2 B) + sin2 B)
+ ~~ [(sin2 ¢> s in 2 0) + (cos2 ¢> sin2 0) + cos2 OJ + ~~ [cos2 ¢> .+ sin2 ¢)
+ au [ 2 sin2 if> cos()+ cos2 ¢> c~ 0 - sin2 ¢> cos 0 - cos 0]
ax
psm¢>
+ au [ 2sin ¢> sinB + cos 2 ¢> s~n()- sin2 ¢> sin()- sinO ]
2
ay
psm¢>
But 2 sin2 ¢>cos 0 + cos2 if> cos 0 - sin2 ¢>cos 0 - cos 0 = (sin2 if>+ cos 2 if> - 1) cos 0 = 0 and simi larly the coefficient of
aujay is 0. Also sin2 ¢co?>2 0 + cos2 if>cos2 0 + sin2 e = cos 2 fJ (sin2 ¢ + cos2 ¢) + s in2 () = 1, and simi larly the
coefficient of
a2uj8y is 1. So Laplace's Equation in spherical coordinates is as stated.
2
11. Io"' I~ Io" f(t)dtdz dy = IIIE f(t)dV , where.
y
· E = {(t, z, y) I 0 ~ t ~ z, 0 ~ z ~ y, 0 ~ y ~ x}.
t =z
If we let. D be the projection of Eon the yt-plane then
D = {(y, t) I 0 S t S x, t S y S x}. And we see from the diagram
that E = { (t, z , y) I t ~ z ~ y, t ~ y ~ x, 0 ~ t ~ x }. So
}~"' I~ I; f(t) dt dz dy = Io"' I /' .ftv f (t ) dz dy dt = Io"' [ft"' (y- t) f(t) dy] dt
= J-;' [(h 2 - ty)f(t) J~:; dt = Io"' [~x - t x- ~t + t 2 ]f(t) dt
2
= Io"' [~x2 - tx + tt2 ] f(t) dt = Io"'
2
ax 2tx + t )f(t) dt
2
-
2
= ~ I;(x - t) 2 f(t) dt
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300
0
CHAPTER 15 PROBLEMS PLUS
13. The volume is V = Jff R dV where R is the solid region given. From Exercise 15.10.2 l(a), the transformation x =au,
= cw maps the unit ball u 2 + v 2 + w 2 :::; 1 to the solid ellipsoid
x2
y2
z2
i
o(x y z)
2 + -b + 2 :::; with o( ' ' ) = abc. The same transformation maps the
y = bv, z
a
2
c
u,v,w
X
plane u + v + w = 1 to -
a
. R.
+ -yb + -cz . = 1. T.hus tl1e regton
111 xyz-space
corresponds to the regionS in uvw-space consisting of the smaller piece of the
. unit ball cut offby the plane u + v + w = 1, a "cap of a sphere" (see the figure).
We will need to compute the volume of S, but first consider the general case
where a horizontal plane slices the upper portion of a sphere of radius r to produce
a cap of height h. We use spherical coordinates. From tile figure, a line through tile
origin at angle ¢ from tile z-axis intersects the pl_ane when cos¢ = (r - h)/a
=:>
a = (r· - h)/ cos¢, and the line passes through the outer rim of the cap when
a = r =:> cos¢ = (r.- h)/r =:> ¢ = cos- 1 ((r- h)/r). Thus the cap
is described by { (p, 8, ¢) I (r- h)/ cos¢:::; p:::; r, 0:::; (}:::; 21r, 0:::; ¢:::; cos- 1 ((r- h)/ r)} and its volume is
1
v = Jof27r .Jrcos
- ((r-h)/r) rr
2 sinA.d dd.d(}
o
J(r-h)/ cos 4> P
"~' P
'I'
= f27r
f COS - l ((r-h)/r)
Jo
Jo
3P
112" 1cos-
= 3
0
0
3 .I 0
((r-h)/r) [ 3
r sin¢ -
A._ 1 (
r cos 'I'
2
r
d¢ d(}
p = (•·-h)/cos¢
1
·
_ 1 f27r [ - 3
-
[l 3 sin¢] P=•·
_ h)3
cos
(r _ h?
]
¢ sin rjJ d¢ dO
3
cos
- 2 ,;.] ¢=cos~- ¢ =O
1
((r- h)/•·) d()
.
(r-h) - 2(r-h)3(rh)- +r3+ 2 r -h) 3] d()
- r-
r7f' [-r3 - r = 31 ./o
2
1
1(
(This volume can also be computed by treating the cap as a solid of revo lution and using the single variable disk method;
see Exercise 5.2.49 [ET 6.2.49].)
To determine the height h ofthe cap cut from tl1e unit ball by the plane
u + v + w = 1, note that the line u = ·u = w passes through the origin with
direction vector (1, 1, 1) which is perpendicular to the plane. Therefore this line
coincides with a radius of tile sphere that passes through the center of the cap and
h is measured along this line. The line intersects the plane at ( t, t, t) and the
sphere at (
plane u + u + w = I
'7a, '7a, '7a). (See the figure.)
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CHAPTER 15 PROBLEMS PLUS
The distance between these points is h =
V=
IlL
V -jg - t Y= v'3 ( -jg - t) =
3(
dV=
2
Ills 1:(~:~:~~ ~
= abc· 1rh (r - th) = abc· 1r ( 1 -
= abC7r ( ~ -
dV = abc
1-
Ills
73. Thus the volume of R is
dV = abcV(S)
7-J Y[1 - i _(1- 73)]
"7a) (j + aTa) = abC7r ( ~ - ~) ~ 0.482abc
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0
301
16
0
VECTOR CALCULUS
16.1 Vector Fields
1. F (x, y) = 0.3 i - 0.4j
All vectors in this field are identical, w ith length 0.5 and
parallel to (3, -:-4) .
y
3. F (x,y) = - ~ i + (y - x) j
T he length of the vector - ~ i + (y - x) j is
vt. +
(y - x) 2 . Vectors along the line y =X are
horizontal with length ~.
5 F(
) - y i +x j
.
x,y - Jx2 +y2
The length of the vector ~ is 1.
x2 + y2
7. F (x, y , z) = k
All vectors in this fie ld are parallel to the z -axis and have
length 1.
9. F(x,y,z) = x k
,,
At each point (x, y, z), F (x, y, z) is a vector of length lxl.
For x > 0, all point in the direction of the positive z -axis,
while for x < 0, a ll are in the direction of the negative
z -axis. In each plane x = k, all the vectors are identical.
X
]j
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303
304
0
CHAPTER 16
VECTOR CALCULUS
11. F(x, y) ·= {x, -y) corresponds to graph IV. In the first quadrant all the vectors have positive x-components and negative
y-components, in the second quadrant all vectors have negative x - and y-components, in the third quadrant all vectors have
negative x -components and positive y-components, and in the fourth quadrant all vectors have ~ositive x - and y-compdnents.
In addition, the vectors get shorter as we approach the origin.
13. F (x, y) = (y, y + 2) corresponds to graph I. As in Exercise 12, aJI vectors in quadrants I and II have positive x~components
while all vectors in quadrants III and IV have negative x-compo nents.Vectors along the line y = -2 are horizontal, and the
vectors are independent of x (vectors along horizontal lines are identical).
15. F (x, y, z) = i + 2j + 3 k corresponds to graph IV, since all vectors have identical length and direction.'
17. F (x, y, z)' =·xi+ y j + 3 k corresponds to graph III; the projection of each vector onto the xy-plane is x i + y j , which points
away from the o rigin, and the vectors point generally upward because their z-components are all 3.
19.
4.5'
.
1~I ' '
\
~
I I
I
- 4.5 1 l
I
I
..
I
'
I
I
' I
' I I
.
..
I
' .'
I
I
I
'
For F (x, y) = (0,
o'} we must have y 2 - 2xy = 0 and 3xy- 6x2 = 0.
The first equation holds if y = 0 or y = 2x, and the second holds if
4.5
I
I
I
I
x = 0 or y = 2x. So both equations hold [and thus F (x, y) = 0] along
t
the line y = 2x.
I ~
\
' \
\
The vector field seems to have very short vectors near the line y = 2x.
I
-4.5
21. f(x, y) = xe"'Y =>
'i7 f(x , y) = f:,(x, y) i + jy (x, y) j = (xe'"Y · y + e'"Y) i + (xe"'Y. x) j = (xy + l )e"'Y i + x 2 e'"!l j
23. 'ilf(x,y, z) =f:z:(x,y,z)i+fy(x,y,z) j +f::(x,y, z ) k =..)
x
i +. J
y
j +..)
z
~k
x 2 +y2 +z2
x 2 +y2 +z2
x 2 +y2 +z
25. f(x,y) = x 2 - y
=> 'il f(x,y) = 2xi - j .
The length ofV' f(x, y) is ..)4x2 + 1. When x =I= 0, the vectors point away
from they-axis in a slightly downward direction with length that increases
as the distance from the y -axis increases.
27.Wegraph\lf(x, y)=
2x
2
22
l +x+
y
i+
4y
jalongwith
2
2y2
1 +x+
a contour map off.
The graph shows that the gradient vectors arc perpendicular to the
level curves. Also, the gradient vectors point in the direction in
which f is increasing and are longer where the level curves are closer
together.
© 2012 Ccngogo Learning. All RigltlS Rc:scrvcd. May nol be scanned, copied, or duplicated. or posted 10 o publicly occc:ssible website, in whole or in pan.
I
SECTION 16.2
29. f ( x , y) = x 2 + y 2
LINE INTEGRALS
0
305
=> V f( x, y ) = 2x i + 2y j . Thus, each vector V f (x, y ) lias the same direction and twice the length of
the position vector of the point (x , y), so the vectors all point directly away from the origin and their lengths increase as we
move away from the origin. Hence, V f is graph III.
31 . f (x, y) = (x + y) 2
=> V f (x , y) = 2(x + y) i + 2(x + y) j. The x- andy-components of each vector are equal, so all
vectors are parallel to the line y = x . The vectors are 0 along the line y = --:x and their l~ngth increases as the distance from
this line increases. Thus, V f is graph II.
33. At t = 3 the particle is at (2, 1) so its velocity is V (2, 1) = (4, 3). After 0.01 units of time, the particle's change in
location should be approximately 0.01 V(2, 1) = 0.01 (4, 3) = (0.04, 0.0~) . so the particle should be approximately at the
point (2.04, 1.03) .
35. (a) We sketch the vector field F (x; y ) = x i - y j along with
several approximate flow lines. The flow lines appear to
be hyperbolas with shape similar to the graph of
y = ±11x, so we might guess that the flow lines have
equations y = C / x.
(b) If x = x(t) andy = y(t) are parametric equations of a Aow line, then the velocity vector of the flow line at the
point (x , y) is x' (t) i + y' (t) j . Since the velocity vectors coincide with the vectors in the vector field, we have
=> dx Nt = x, dy I dt = - y . To solve these differential equations, we kno~
x' (t) i + y' (t) j = x i - y j
dxldt = x
=> dx/ x = dt => In lxl = t +. C => x = ±~t + c = Aet for some cons~nt A, and
dy I dt = - y
=> dy / y = -dt => In Ivi = - t + I< => y = ±e- t + K = B e - t for some constant B. Therefore
x y = A et B~-t = AB = constant. If the flow line passes through (1, 1) then (1) (1) = constant= 1
=> x y = 1 =>
y = 1l x, x > 0.
16.2 Line Integrals
1. x = t 3 andy = t, 0 :::; t :::; 2, so by Formula 3
.Ia y ds ~ 1\
3
3
(
~~ Y+ ( ~~ Ydt =
1
2
3
2
t y'(3t )2 + (1)2 dt =
1\a
yf9t 4 + 1 dt
3 2 2
3 2
= ..l.
. 1 (9t4 + 1) / ) O = ..l.
(145 v'I45- 1)
:ill 3
G4 (145 1 - 1) or ..l.
54
3. Parametric equations for C are x = 4 cos t , y = 4 sin t, - ~ :::; t :::; ~ . Then
fc x y 4 ds = J:~~2 (4 cos t) (4sin t)4 y'( -4sin t )2 + (4cos t)2 dt = J:~~ 2 4 5 cost sin4 t yf16(sin2 t + cos2 t ) dt
2
2
= 4 5 Jrr/
(sin4 t cos t )(4) dt = (4) 6 [l5 sinGt) Tr/
= 2 ·540 = 1638.4
- -rr/ 2
- Tr/ 2
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308
CHAPTER 16
0
VECTOR CALCULUS
5. If we choose x as the parameter, parametric equations for Care x = x, y = v'xfor 1:::; x:::; 4 and
fc ( x2y3- v'x) dy = J;' [xz. (Vx)3 - v'x] 2 ~ dx = 4J14 (x3- 1) dx
x] 41-- !2 (64 - 4 - 14 + 1) -- ill
8
= 12 [14 x ·1 y
7.
(2, 1)
On cl: X = x, y = ~X => dy = ~ dx, 0 :::; X :::; 2.
On Cz: x = x, y = 3 - :v => dy = -dx, 2 :::; x:::; 3.
Then
f 0 (x + 2y) dx + x 2 dy = fc 1 (x + 2y} dx + x 2 dy +.J02 (x + 2y} dx + x 2 dy
=J: [x+2(~x)+x 2 (~)]dx+J: [x+2(3-x}+x2 (-1}]dx
2
= f 0 (2x + ~x ) dx +
2
J23 (6 - x- x 2 ) dx
9. x = 2sint, y = t , z = -2cos t, 0:::; t:::; 7T. Then by Formula 9,
!fltf + C¥tf dt
fc xyz ds = J07T (2 sin t)(t)( - 2 cos th/ (!!fd + {
= J; -4t sin t cost .J(2 cos t)2 + 11)2 + (2 sin t) 2 dt = J0 - 2tsin 2t .J4(cos 2 t + sin2 t) + 1dt
7T
= -2 y'5 j~'ff t sin2tdt = -2 y'5 [-~ t cos 2t
+'i sin2t] ~
integrate by parts with ]
[u
t , dv s in 2t dt
=
=
= -2¥'5(-~ - 0) = J57T
11 . Parametric equations for Care x = t, y = 2t, z = 3t, 0 :::; t :::; l. Then
15. Parametric equations for Care x = 1 + 3t, y = t, z = 2t, 0 :::; t :5. 1. Then
fc z2 dx + x 2 dy + y 2 dz = J;(2t? · 3 dt + (1 + 3t}~ dt + t 2 · 2 dt = J; (23t2 + 6t + 1) dt
= [¥tl + 3t2 + t]~ = ¥ + 3 + 1 = ¥
17. (a) Along the line x = - 3, the vectors ofF have positive y-components, so since the path goes upward, the integrand F ·Tis
always positive. Therefore f.e l F · dr = f.e 1 F · T ds is positive.
.
© 2012 Cengoge L<aminH. All Riglns Rese" ·ed. May not be scanned, copied, or duplicated, or pos1ed to n publicly accessible website, in whole or in part.
SECTION 16.2
LINE INTEGRALS
0
307
(b) All of the (nonzero) field vectors along the circle with radius 3 are pointed in the clockwise direction, that is, opposite the
direction to the path. So F · Tis negative, and therefore fc 2 F · dr = fc 2 F · T ds is negative.
19. r(t) = llt 4 i + t 3 j,, so F (r (t)) = ( llt4 )(t3 ) i + 3(t3 ) 2 j = llt 7 i + 3t6 j and r' (t) =· 44t3 i + 3t2 j . Then
fc F · dr = J; F(r(t)) · r '(t)dt = J;(ne · 44e + 3t6 · 3t2 )dt = J; (484t 10 + 9t8 )dt = [44t 11 +t 9 )~ = 45.
21 . fc F · dr =
J; (sin t\ cos(-t t
2
),
4
2
) · (3t , - 2t, 1) dt
= J;(3t2 sint3 - 2tcose + t 4 ) dt = [- cost3 - sint + tt 5 )~ =~-cos 1- sinl
2
fc ~ ·d~ = i
=
i
2
F (r (t )) · r '(t)dt =
2
2
2
[ e t- t -
i
2
2
2
2
2
[et-t et :rsm(e-t ) . ( - '2 te- t ) ] dt
2
2te - t sin ( e - t ) J dt ::::J 1.9633
25. x = t 2 , y = t 3 , z = t 4 so by Formula 9,
fc xsin(y + z) ds = J;(e) sin(t3 + t 4 )vf(2t)2 + (3t2)2 + {4t3 )2 dt
= f 0 t 2 sin(t3 + t 4 ) J4t2 + 9t 4 + 16t6 dt ::::: 15.0074
5
27. We graph F ( x, y) = (x - y) i + x y j and the curve C. We see that most of\he vectors starting on C poi~t in roughly the same
direction as C, so for these portions of C the tangential compon~nt F · T is positive. Although some vectors in the third
quadrant which start on C point in roughly the opposite direction, and hence give negative tangential components, it seems
reasonable that the effect of these portions of C is outweighed by the positive tangential components. Thus, we would expect
fc F · dr = fc F · T ds to be positive.
To verifY, we evaluate fc F · dr: The curve C can be represented by r (t) = 2 cost i + 2 sin t j , 0 $ t $
3
;,
so F (r(t)) = (2 cost- 2 sin t) i + 4 cos tsin t j and r ' (t) = - 2 sin t i + 2 cos t j . Then
fc F · dr = ./;"12 F (r (t)) · r ' (t) dt
=
J;"12 [-2 sin t(2 cost - 2 sin t) + 2 cos t(4cos t sin t )) dt
= 4J0 ,. 12 (sin2 t - s in tcost + 2sin tcos 2 t) dt
3
[using a CAS)
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308
0
CHAPTER 16
VECTOR CALCULUS
(b) r (O) = 0, F(r(O)) = (e- 1 , 0);
2.1
F(r(ll ) /
(l 272' , F(.r (72
1 ) _
r ( 72
-
2'
1 )
1 )) _
1 )·
- /\ e-1/2 ' 4,72
'
r(1) = (1, 1), F (r(1)) = (1, 1).
In order to generate the graph with Maple, we usc the 1 ine command in
the plot t ools package to define each of the vectors. For example,
0 ~--....:.....:....c!...__
_ _+i 2. 1
-0.2
vl : =line ( [0 , 0),. [exp(-1), 0 ] ):
generates the vector from the vector field at the point (0, 0) (but without an arrowhead) and gives it the name vl. To show
everything on the same screen, we use the display command. In Mathematica, we use List Plot (with the
Plot Jo ined - > True option) to generate the vectors, and then Show to show everything on the same screen.
31. x = e-t cos4t, y = e-t sin4t, z = e-t, 0 ~ t :::; 2rr .
Then:= e- 1 (-sin4t}(4) - e-tcos4t = - e- 1 (4sin4t+cos4t),
dy = e- 1 (cos4t)(4)- e-t sin4t = -e- t( - 4cos4t + sin4t), and dd.z = - e- L, so
dt
t
(~~ y ~; y ~: y
+ (
= v(-e- 1 )2((4sin 4t + cos4t)2 + ( - 4cos4t + sin4t)2 + 1]
+ (
= e-t J16(sin 24t + cos2 4t) + sin2 4_t + cos2 4t + 1 = 3 ..f2 e-t
J~ x 3 1/ z ds = J0211" (e-t cos4t) 3 (e-t sin 4t) 2 (e- 1) (3 ..f2 e-t) dt
Therefore
172·704 ..f2(1 - e- 14-.r)
-- Jo
r 2 -.r3.,J2e- 7tcos3 4t sin2 4tdt- 5,632,706
33. We use the parametrization x = 2 cost, y = 2 sin t, - ~ :::; t :::; 'i. Then
ds =
2
(',~n + (~~) dt = v( -2sin t) 2 + (2cost) 2 dt = 2dt, som = J~ kds = 2kJ::~~2 dt ~ 2k(7r),
2
Hence (x,·il) = · (~, 0).
35. (a) x = ..!._ { xp(x , y , z) ds, y = ..!._ ; · yp(x , y, z) ds, z = ..!._ { zp(x, y, z) ds where m = ./~ p(x , y, z) ds.
m}c
m
c
m }c
(b) m = fc kds = k J~-.r V4sin2 t + 4cos2 t + 9 dt = k vTIJ;"" dt = 2rrkJI3,
x=
z=
1
1 211"
iM 1 2-.r 2k v'i3 costdt = 0,
JI3
2kVlJ sintdt = 0, y =
27rk 13 0
27rk 13 0
2
M { -.r (k Vi3) (3t) dt = 23 (2rr2 ) = 3rr. Hence (x, y, z) = (0, 0, 37r).
1
27rk
13
/o
7r
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SECTION 16.2
LINE INTEGRALS
D
309
37. From Example 3, p(x, y) = k(1- y), x = cost, y = sint, and ds = dt, 0 :S t :S 1r ::::}
1:. = Icy2 p(x,y) ds = IoT< sin2 t [k(1- sint)] dt = ki0" (sin2 t - sin3 t) dt
= ~k I 0.,. (1- cos2t) dt- k Io"(1- cos2 t)sintdt
[
Let u = cost, du = - sin t dt ]
in the second integral
= k[~ + I1- 1 (1-u2 )du] =k (~ - ~)
lv = Icx 2 p(x, y)ds = ki0.,. cos2 t(1 - sint)dt = ~ I 0.,.(1 +cos2t)dt- kio" cos2 tsintdt
= k(~- i), using t?e same substitution as above. ·
39. W = I c F · dr = I;.,. (t- sint,3 - cost)· (1 - cost,sint) dt
=
I;.,. (t - t cost - sin t + sin t cos t + 3 sin t - sin t cos t) dt
=I;,. (t- tcost + 2sin t) dt = [~t 2 - (tsin t + cost) - 2 cos t)~,..
integrate by .parts ]
[ in the second term
41. r (t) = (2t, t, 1 - t), 0 :::; t :::; 1.
W = Ic F · dr = Io (2t- t 2 , t- (1 - t? ,1 - t- (2t) 2 ) · (2,1, - 1) dt
1
= Io (4t- 2t2 + t- 1 + 2t - t 2 - 1 + t + 4t2 ) dt =I; (t2 + 8t - 2) dt = [lt3 + 4t2 - 2t)~ = ~
1
43. (a) r (t) = at2 i + bt3 j
::::}
v (t) = r' (t) = 2at i + 3bt2 j
a(t) = v' (t) = 2a i + 6btj, and force is mass times
::::}
acceleration: F(t) = ma(t) = 2mai + 6mbtj.
(b) W = f c F · dr = J0 (2mai + 6mbtj ) · (2at i + 3bt2 j) dt = I 0 (4ma2 t + 18mb2 t 3 ) dt
1
1
= [2ma2 t 2 + ~mb2 t 4 )~ = 2ma 2 + ~mb2
45. Let F = 185 k. To parametrize the staircase, let x = 20 cost, y = 20 sin t, z = ~~ t = 1!- t, 0 :::; t :::; 61r ::::}
W = f c F · dr =I:.,. (0, 0, 185) · ( -20 sin t, 20 cost, ~) dt = (185) ~ J:.,.. dt = (185)(90) :::::: 1.67 x 104 ft-lb
47. (a) r (t) = (cost, sin t}, 0 :S t:::; 271", and let F =(a, b). Then
W = I c F · dr = I ;" (a, b) · (- sin t,cost) dt = I~'lf(-asint + bcost) dt = [a cost + bsin t)~.,.
=a +O - a+ O=O
(b) Yes. F (x, y) = k x = (kx, ky) and
W = I c F · d r =I;,. (k cost, k s~ t} · (-sin t , cost} dt = J;" (- k sin t cost+ k sin t cost) dt = I~.,.. 0 dt = 0.
49. Let r(t) = (x(t), y(t), z(t)) and v = (v1, v2, va). Then
I c v · dr = I: (v1, v2, va) · (x' (t), y' (t) , z' (t)) dt = I: [v1 x' (t} + v2 y' (t) + va z' (t)] dt
= [v1 x(t) + v2 y(t) + va z(t)) : = [v1 x (?) + V2 y(b) + va z(b)J - [v1 x(a) + V2 y(a) + vs z(a)]
= v 1 [x(b)- x(a)J + v2 [y(b)- y(a)] + va [z (b) - z(a)]
=: (v1, v2, va) · (x(b)- x(a), y(b) - y(a) , z(b)- z(a))
= (v1, v2, va) · [(x(b) , y(b) , z(b)}- (x(a), y(a), z (a))] = v · (r(b)- r (a)]
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310
0
CHAPTER 16
VECTOR CALCULUS
51. The work done in moving the object is fc F · dr = fc F · T ds. We can approximate this integral by dividing C.into
7 segments of equal length t:.s
= 2 and approximating F · T , that is, the tangential component of force, at a point (xi , yi) on
each segment. Since C is composed of straight line segments, F · T is the scalar projection of each force vector onto C.
If we choose (xi, yi) to be the point on the segment closest to the origin, then the work done is
.
7
fc F · T d.s ~ E (F (xi, yi) · T(xi, yi)] t:.s = (2 + 2 + 2 + 2+ 1 + 1 + 1)(2) = 22. Thus, we estimate the work done to
i=l
.
be approximately 22 J.
16.3 The Fundamental Theorem for Line Integrals
1. C appears to be a smooth curve, and since \Jf is continuous, we know f is differentiable. Then Theorem 2 says t hat the value
of fc \Jf · dr is simply the difference of the values off at the terminal and initial points of C. From the graph, this is
50- 10 = 40.
3. 8(2x- 3y) j8y
= - 3 = 8( -3x + 4y- 8)j8x and the domain of F is IR2 which is open and simply-connected, so by
Theorem 6 F is conservative. Thus, there exists a function f such that \J f = F , that is, f:r:(x, y) = 2x - 3y and
f 11 (x, y) = -3x + 4y - 8. But f,.(x, y) = 2x - 3y implies f(x, y) = x 2 - 3xy + g(y) and differentiating both sides of this
equation with respect to y gives f 11 ( x, y) = - 3x + g' (y). Thus - 3x + 4y - 8 = -3:~.: + g' (y) so g' (y ) = 4y - 8 and
g(y) = 2y2 - By+ [(where I< is a constant. Hence f(x, y) = x 2 - 3xy + 2y2 - 8y +I< is a potential function for F .
5. 8(e"' cosy)j8y := - e"' sin y, 8(e"' siny)f8x = e" sin y. Since these are not equal, F is not conservative.
7. o(ye"' + sin y)j8y =ex+ cosy= o(e"' + X cosy)f8x and the domain ofF is IR2 • Hence F is conservative so there
exists a function f such that \J f =F. Then fx (x, y)
= ye" +sin y implies f( x, y) = ye" + x sin y + g(y) and
f v(x, y) = e"' + xcos y + g'(y). But f 11 (x, y) = e"' + x cosy so g(y) = K and f(x, y) = ye" + xsin y + K is a potential
function for F.
9. 8(lny + 2xy 3 )j8y = 1/y + 6xy 2 = 8(3x 2 y 2 + xfy)j8x and the domain ofF is {(x, y) I y> 0} which is open and simply
connected. Hence F is conservative so there exists a function f such that \J f = F . Then f:z:(x, y) =In y + 2xy3 implies
f(x,y) = xlny + x 2 y 3 + g(y) and ! 11 (x , y) = xfy + 3x2 y 2 + g'(y). But f 11 (x,y) = 3x 2 y 2 + xjy so g'(y) = 0
g(y) = [(and f( x, y) = xlny + x y
2
3
=}
+ [(is a potentia l function for F .
11. (a) F has continuous first-order partial derivatives and
8
8
2xy = 2x = X (x2 ) on JR 2 , which. is open and simply-connected.
8
8y
Thus, F is conservative by Theorem 6. Then we know that the line integral ofF is independent of path; in particular, the
value of fc F · dr depends only on the endpoints of C. Sin.c e all three curves have the same initial and terminal points,
J~ F · dr will have the same value for each curve.
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SECTION 16.3
THE FUNDAMENTAL THEOREM FOR LINE INTEGRALS
0
311
(b) We first find a potential function f, so that \l f = F. We know ! :r:(x, y) = 2xy and j y(x, y) = x 2 • Integrating
fx(x , y) with respect to x, we have f(x , y) = x 2 y + g(y). Differentiating both sides with respect toy gives
· J11 (x, y) = x 2 + g'(y), so we must have x 2 + g'(y) = x 2
=?
g'(y) = 0 =>
g(y) = K, a constant.
Thus f(x, y) = x 2 y + K. All three curves start at (1, 2) and end at (3, 2), so by Theorem 2,
fe F · dr = !(3, 2) - !(1, 2) = 18 - 2 = 16 for each curve.
2
13. (a) f r(x , y) = xy 2 implies J(x, y) = ~x y
2
+ g(y) and fy (x, y) = x 2 y + g'(y). But /y(x , y) = x 2 y so g'(y) = 0 =>
g(y) = K, a constant. We can take K = 0, so f( x, y) = ~x 2 y 2 .
(b) The initial point of C is r (O) = (0, 1) and the terminal point is r(1) = (2, _1 ), so
fe F · dr = /(2, 1) - f(O, 1) = 2 - 0 = 2.
15. (a) f x(x , y, z ) = yz implies f(x , y, z) = xyz + g(y, z) and so jy(x, y , z) = xz + g11 (y, z). But jy(x, y, z) = xz so
gy(y, z ) = 0 => g(y, z) = h(z) . Thus f( x, y, z) = xyz + h(z) and f:(x , y, z) = X'!f + h'(z). But
f:(x, y, z) = xy + 2z, so h'(z) = 2z => h(z ) = z2 + K.. Hence f(x, y, z) = xyz + z 2 (taking J( = 0).
(b) fe F · dr = !(4,6,3) - !(1, 0, - 2) = 8 1 - 4 = 77.
17. (a) f:r:(x, y, z) = yze"'= implies f(x, y, z) = ye"'= + g(y, z) and so fv(x, y, z) = e"'= + gy(y, z) . But jy(x, y, z ) = e"'= so.
gy{y,z) = 0
=?
g(y,z) = h(z). Thus f( x,y,z)·= ye"'= + h(z) and f z (x , y,z) = xyex= + h'(z). But
f:.:(x,y,z) = xyex=, so h'(z) = 0
(b) r (O) = (1, -1, 0), r(2)
=?
h(z ) = K. Hence f(x,y ,z) = ye"'= (taking!(= 0).
=(5, 3, 0) so fe F · dr = /(5, 3, 0) - /(1 , - 1, 0) = 3e + e = 4.
0
0
19. The functions 2xe-Y and 2y - x 2 e- v have continuous first-order derivatives on JR2 and
~ (2xe-v) = -2xe- Y = vx
~ (2y- x 2 e:-v), so. F(x, y) = 2xe-Y i + (2y- x 2 e-Y) j is a conservative vector field. by
Ulj
Theor~m 6 and hence the line integral is independent of path. Thus a potential function f exists, and fx(x , y) = 2xe-v
implies f(x, y) = x 2 e- 11 + g(y) and fv(x, y) = - x 2 e-Y + g'(y). But f v(x, y) = 2y- x 2 e-v so
g'(y) = 2y
=?
g(y) = y 2 + K. We can take K = 0, so f(x,y) = x 2 e- 11 + y 2 • Then
.fe 2xe-v dx + (2y - x 2 e- v) dy = /(2, 1) - f(l , 0) = 4e- 1 + 1 - 1 = 4/e.
21. IfF is conservative, then fe F · dr is independent of path. This means that the work done along all piecewise-smooth curves
that have the described initial and terminal points is the same. Your reply: It doesn't matter which curve is chosen.
23. F (x, y) = 2y3 12 i + 3x Jyj , W = .fe F · dr. Since 8(2y3 12 )/8y = 3 y'Y = 8(3x Jy )/ ax, there exists a function f
suchthat\lf=F.Infact,fx(x, y)='2y 3 12
=> f(x,y)=2xy 312 +g(y) =>
/y (x, y) = 3x y'Y so g'(y) = 0 or g(y) = K. We can take I< = 0
=?
fv(x,y)=3xy 1 12 +g'(y). But
f(x, y) = 2xy3 12 • Thus
W = fe F · dr = j(2, 4) :.... j(I,·l) = 2(2)(8)- 2(1) .= 30.
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312
0
CHAPTER 16
VECTOR CALCULUS
25. We know that if the vector field (call it F) is conservative, then around any closed path C, fc F · dr = 0. But take G to be a
circle centered at the origin, oriented counterclockwise. All of the field vectors that start on G are roughly in the direction of
motion along C, so the integral around G will be positive. Therefore the field is not conservative.
27.
r1
'
.. '
I
'
I
1I I \
tf rrI
'
' ' ' '
I
.
'
-2'11'
I I
l I
\
I
'
'
'
I
paths, the number and size of the field vectors pointing in directions similar
I
I
to that of the path seem to be roughly the same as the number and size of the
r 1
I
I
vectors pointing in the opposite direction. To check, we calculate
.
.
° (~
I
' ' I I
'
UI I '
' I
I
. ' ' , 1r it
t 1 I ,I I '
rI I
From the graph, it appears that F is conservative, since around all closed
\:!l
I
\
'
'
'
I
I
:
o (sin y) = cos y =
ay
0X
~ ~
+ x cos y). Thus F is conservative, by
Theorem 6.
29. Since F is conservative, there exists a function f such that F = \1 f, that is, P = /z, Q = ! 11 , and R = f= · Since P ,
Q, and R have continuous first order partial derivatives, Clairaut's Theorem says that 8P / EJy = fx 11 = fvx = EJQf EJx,
EJP/8z = fxz = /zx = 8 R /.8x, and 8Q/8z = fyz = f zv = 8 Rf8y.
y
31. D = {(x , y) I 0 < y < 3} consists of those points between, but not
3
on, the horizontal lines y = 0 andy = 3.
----------- -----------
(a) Since D does not include any of its boundary points, it is open. More
formally, at any point in D there is a disk centered at that point that
lies entirely in D.
0
---------·--.:
X
'(b) Any two points chosen in D can always be joined by a path that lies
entirely in D , so D is com1ected. (D consists of just one "piece.")
(c) Dis connected and it has no holes, so it's simply-connected. (Every simple closed curve in D encloses only points that are
in D.)
33. D
= {(x, y) II ::; x + y
2
2
::; 4, y
y
2: 0} is the semiannular region
in the upper half-plane between circles centered at the origin of radii
l and 2 (including all boundary points).
(a) D includes boundary points, so it is not open. [Note that at any
boundary point, (1, 0) for instance, any disk centered there cannot lie
entirely in D .]
(b) .The region consists of one piece, so it's connected.
(c) D is connected and has no holes, so it's simply-connected.
y
8P
35. (a)P= - x 2 +y2 , .A..=
VIJ
y2 - x2
x
8Q
y2 -
x2
8P
8Q
and Q = - - - - - =
.Thus - =,.-.
(x2 + y2) 2
x2 +y2' OX
(x2 +y2 )2
UX
8y
(b) G 1: X = cost, y = sin t, 0 ::; t ::; ;rr' c 2: X = cost, y = sin t, t = 21!' to t = 1l'. Then
{ F ·dr = ("'(- sin t)(-s~n t)+.(~ost)(cost)dt= ("' dt = 7rand { F· dr = ("' dt= - 1!'
}0
cos t + sm t
./0
lc2
J 21r
Jc,
© 2012 Ccngogc Lcoming. All RlgbLS Restl'\'ed. Moy no1be scanned, copied. or duplicalcd, or poSicd lo o publicly occessiblc wcbsilc, in whole or in p.1n.
SECTION 16.4
GREEN'S THEOREM
D
313
Since these aren' t equal, the line integral ofF isn't independent of path. (Or notice that J~ F · dr = f 2 1r dt = 21r where .
2
Cs is the circle x + y
2
0
3
= 1, and apply the contrapositive of Theorem 3.) This doesn't contradict Theorem 6, since the
domain of F , which is IR 2 except the origin, isn' t simply-connected.
16.4
Green's Theorem
1. (a) Parametric equations for C are x = 2 cos t, y = 2 s in t, 0 :::; t :::; 27r. Then
f c(x - y) dx + (x + y) dy = J;" [(2 cost- 2sin t )( -2 sin t) + (2 cos t+ 2 sin t)(2 cost)] dt
= J;"(4sin t+ 4 cos 2 t)dt = J~" 4dt = 4t]~"' = 81r
2
(b) Note that Cas given in part (a) is a positively oriented, smooth, simple closed curve. Then by Green's Theorem,
f c(x - y)dx + (x + y) dy= ffv [: ., (x+y) - : 11 (x - y)] dA = ffv [1- (-1)] dA = 2ffvdA
= 2A(D) = 27r(2) 2 = 81r
3. (a)
y
Ct: x = t
(1,2)
o
T hus
c1 (I, OJ
=> dx = dt, y = D => dy = Ddt, D :::; t :::; 1.
C2: x = 1 =>
dx =Ddt, y = t
=> dy = dt, D:::; t $ 2.
C3: x= 1 - t
=> dx = -dt, y=2-2t => dy = - 2dt, D$ t$ 1.
X
fcxydx + x 2 y 3 dy =
xydx+x 2 y3 dy
f
C1 + C2 + C3
= f 01 Ddt + f 02 t 3 dt + J; [-(1- t)(2 - 2t)- 2(1- t?(2 - 2t) 3 ] dt
= D+ [~t 4 ]~ + [t(1 - t? + ~(1- t) 6]~ = 4- ¥ = ~
2 3
(b) f c xydx +x y dy = ffv
[:x (x y
2
3
) -
/J~ (xy)] dA = f01 J02 ~(2xy 3 - x)dydx
y=2z d
rt (8xs - 2X2) dX = 4f - 32 = 3'2
= Jort [21 xy4 - xy] y=O
X= Jo
5.
The region D enclosed by Cis given by {(x, y) I D :::; x:::; 2, x :::; y $ 2x }, so
y
4
(2,4)
f c xy2 dx + 2x2 ydy = ffv [ :, (2xiy) - : 11 (xy2 )] dA
~
I: I:"' (4xy - 2xy) dy dx
= Jr2
[xlY2] y=
y=2x d~
o
x
""
2
2
4
= Jf o 3x 3 dx = !!.x
] = 12
4
o
7. fc
(v + e-fi) dx+ (2x +cosy ) dy = Jfv [:x(2x + cosy
2
2
) -
:v·
(v + e-fi)] dA
= fol Jyfi (2 - 1) dx dy = fol (yl/2 - y2) dy = ~
@ 2012 Ccngage Learning.. AU Rights Reserved. May nor be sc:a.nncd, copied, or dupliCDtctl, or posted to n publicly .:JCC("SSiblc website. in whole or in JUrt.
314
CHAPTER 16
0
VECTOR CALCULUS
9. fc y dx- x dy = ffv [ :., ( -x ) - :U (y ) ] dA = ff0 (- 3x
3
3
3
= - 3
3
2
-
3y2 ) dA = J:"" J:(-3r2 ) rdrdO
J:" dO. .r: r 3 dr = -3(27!)(4) = - 2471"
11. F (x, y) = (y cosx - xysinx, xy + xcos x) and the region D enclosed by Cis given by
{(x, y) I 0 ~ x ~ 2, 0 ~ y ~ 4- 2x}. Cis traversed clockwise, so-C gives the positive orientation.
f c F · dr = - J_ 0 (ycosx- xys.i nx )dx + (x~ + xcosx) dy = - ffv [ :., (xy + xcosx) - :Y (ycosx- xysinx)] dA
xsin x + cosx -: cosx + xsin x)dA = - J0 .{0 - :r ydydx
2
= - ffv(Y -
4 2
4 2
2
2
2
2
2
2
=- Jo
f [ ly ] v= - "' dx =- f 1(4 - 2x) dx - - f (B - Bx + 2x2) dx- - (Bx- 4x 2 + ~x3 )
2
v=O
Jo 2
Jo
a
o
= - (16 - 16 + 1: - o) = - 136
13. F (x, y) = (y- cosy, x sin y) and the region D enclosed by Cis the disk with radius 2 centered at (3, -4).
C is traversed clockwise, so -C gives the positive orientation .
.f0 F · dr = - J_ 0 (y - cosy)dx + (xsiny) dy = - JJ0 [ :., (xsiny)- :u(y- cosy)] dA
= - ffv(siny - 1 - sin y) dA = ffv dA = area of D .= 7r(2? = 47r
y
15. Here C = C1 + C2 where
01 can be parametrized as X = t, y = 1, - 1 ~ t ~ 1, and
C2 isgivenby x=-t, y=2 - t 2, -1 ~t~ l.
Then the line integral is
f
. 2
2
y e"' dx + x ev dy
~~
1
= J_ 1 [1 · et + t 2 e · 0] dt
.
+ J~ 1 [(2- t 2? e-t( - 1) + (- t) 2e2 - t· ( - 2t)] dt
-I
0
X
= J~ 1 [et - (2 - t 2 ) 2 e- t - 2t3 e 2 - t ] dt =-Be+ 4Be- 1
2
according to a CAS. The double integral is
.! r (aQ aP)
. JD
ax - 8y
dA =
;·-1111r2_,2 (2xeY -.2ye"') dy dx = - Be + 4Be- 1' verifying Green's Theorem in this case.
17. By Green's Theorem, W = f c F · dr = f c x(x + y) dx + xy 2 dy = J(0 (y 2 - x) dA where C is the path described in the
question and D is the triangle hounded by C . So
W = J0 J;-"'(y2 - x) dy dx = f 0 [ty 3 - xy]~ = ~-"' dx =
1
1
• = [- f.I (1 - x)
4
-
J; (i(1 - x)
3
-
x(1 - x)) dx
~ x 2 + ~ x 3 ] ~ = (- ~ + ft ) - (- f2) = - 112
19. Let C1 be the arch of the cycloid from (0, 0) to (27r, 0), which corresponds to 0 ~ t ~ 271", and let C2 be the segment from
(271", 0) to (0, 0), so 02 is given by X = 271"..:. t, y = 0, 0 ~ t :::; 271". Then c = cl u 02 is traversed clockwise, so - C is
oriented positively. Thus -C encloses the area under one arch of the cycloid and from (5) we have
A = - f_0 y dx = .{0 1 y dx + f 0 2. y dx = f~" (1 -cos t )(l - cost) dt + f~" 0 (- dt)
= J~" (1 - 2 cost+ cos 2 t) dt + 0 = (t- 2 sin t + ~t + ~sin 2t] ~" = 371"
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SECTION 16.4
GREEN'S THEOREM
0
315
21. (a) Using Equation 16.2.8, we write parametric equations of the lin_e segment as x = (1 - t)x1 + tx2, y = (1- t)y1 + ty2 ,
0 ~ t ~ 1. Then dx = (x2- xl) dt and dy = (y2- yl) dt, so
f c x dy- y dx = f01 [(1 - t)x1 + tx2](y2- Yl) dt + [(1 - t)y1 + ty2](x2- x1) dt
=
J; (x1(Y2 - Yl) - y1(x2- xl) + t((y2- Y1)(x2- x1)-:- (x2 - x1)(y2- Y1)]) dt
= f0 (x1y2- x2yl) dt = x~y2- X2Y1
1
(b) We apply Green 's Theorem to the path C = C1 U C2 U · · · U Cn, where Ci is the line segment that joins (xi , Yi) to
(xi+l,Yi+ l) fori= 1, 2, ... , n -1, and Gn is the linesegmentthatjoins (x,.,yn) to (xl,Yl). From (5),
~ J~ x dy - y dx = JJ0 dA , where D is tite polygon bounded by C. Therefore
area of polygon= A(D)= ffv dA = ~ J~ xdy- ydx
= ~ (J~ x dy- ydx + f c 2 xdy - ydx + .... + J~n-l xdy- ydx +fen xdy- ydx)
1
To evaluate these integrals we use the formula from (a) to get
A(D) = ~[(x1y2- X2Yl) + (X2Y3 - XaY2) + ' · · + (Xn-;-lYn- XnYn- 1) + (XnYl - XlYn)].
(c) A = ~ [(0 · 1 - 2 · 0) + (2 · 3 - 1 · 1) + (1 · 2 - 0 · 3) + (0 · 1 - ( -1) · 2) + ( ~ 1 · 0- 0 · 1)]
= Ho+5+2+2) = ~
23. We orient the quarter-circular region as shown in the figure.
y
fc
a
2
A = {7ra sox =
1l'a! /2
2
x dy and y = -
Here c = c l + c2 + Cs where cl: X = t,
1ra!12
fc
2
y dx.
y = 0, 0 ~ ·t ~ a;
C2: x = a cos t , y =a sin t, 0 ~ t. ~ ~;and
c.
c3: X = 0, y = a - t, 0 ~ t ~ a. Then
f cx 2 dy = fc, x 2 dy
a
X
+f c x2 dy +.fc x2 dy = foa Odt + J 12 (acost) 2(acost) dt + J; Odt
2
3
0"
2
= J~"/ a 3 cos3 tdt = a 3 0"/\1- sin 2 t) costdt = a 3 [sin t
J
-.i s in3 t];12 = ~a~
J
_
1
4a
2
so x = 7ra2 /2 Jc x dy = 311'.
fcy 2 dx = fc, y 2 dx + )~2 Y2 dx + fc 3 Y2 dx = foa 0 dt + f 0"
12
(a sin t) 2( - a sin t) dt + J0a 0 dt
J 12 ( - a 3 sin 3 t) dt = - a 3 J0" 1\ 1 - cos2 t) sin tdt = -a3 [~ cos3 t - cos t]; 12 = - ~a3 ,
= 0"
so 'ii = -
1l'a!/2
£y = ~;.Thus
2
dx
(x,y) = ( ::. :: )·
25. By Green's Theorem, - ~pfcy 3 dx = - kp J.f'v( - 3y 2 ) dA = ffv y 2pdA = I ., and
tPfcx 3 dy = ~p ff0 (3x 2 ) dA = ffv x 2 pdA = Iu .
27. As in Example 5, let C' be a counterclockwise-oriented circle with center the origin and radius a, where a is chosen to
be small enough so that C' lies inside C, and D the region bounded by C and C'. Here
2
22
2
2
2
3
EJP = 2x(x + y ) . - 2xy · 2(x + y ) · 2y = 2x - 6x~ and
=> oy
(x2 + y2)4
(x2 + y2)3
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316
0
CHAPTER 16
VECTOR CALCULUS
'* .. -{)Q
{)X = -2x(x + y ? -(X(y +- y.x
2
2
2
2
2
) • 2(x
2) 4
2
L +
P dx
Q dy + j_ c, P dx + Q dy =
+ y 2 ) · 2x
2:~;3 - 6xy 2
.
= (x2 + y2) a . Thus, as m the example,
IL(~~ -~:) = IL
0 dA = 0
dA
and J c F · dr = .fc, F · dr. We parametrize C' as r (t) =a cost i +a :sin t j , 0:::; t:::; 21r. Then
1
c
F · dr =
1
2
F · dr =
C'
2
1
"
o
2
2
a o
2
l 2" (-cost sin t- cost (1 - sin t))
=. -1 l2~ (-cost sin2 t - cos3 t) dt = -1
a o
a o
11 ~ costdt
= --
2
2(acost)(asint)i+ (a sin. t - a cos t) j (
. .
·)
·
- a sm t 1 + a cost J dt
2
(a2 cos2 t + a 2 sin2 t)
2
2
dt
2
= - -1 sint] " = 0
a
o
.29. Since C is a simple closed path which doesn' t pass through or enclose the origin, there exists an open region that doesn't
contain the origin but does contain D. Thus P
= -y/(x2 + y 2 ) and Q = xj(x 2 + y 2 ) have continuous partial derivatives on
this open region containing D and we can apply Green's Theorem . But by Exercise 16.3.35(a), aPjay = aQ j8x, so
fc F · dr = ffv OdA ~ 0.
.
.
31. Usmg the first part of(S), we have that I In dx dy = A(R) = Ion X dy. But X = g(u, v), and dy =
&
au du + M
8v dv,
and we orient as by taking the positive direction to be that which corresponds, under the mapping, to the positive direction
alongaR, so
{
i on
8
X dy = { g(u, ·u) ( ah du + {)ah dv) =
l as
u
v
= ± .ffs [ :v. (g(u, v) g~) = ±
r g(u, v) {)ahu du + g(u, v) Bahv dv
.los
:v (g(u, v) ~~) ) dA
[using Green's Theorem in the uv-plane]
02
82
81
" - !l.!l
ffS (!L9..
" ) dA
Uu !ll!
Dv + g(u v) l:lul:lv
l:l·v Vv.' - g(u v) 8vUu
>
>
[using the Chain Rule]
= ± .ffs (~~ ~- Z~ ~ ) dA _[by the equality of mixed partials] = ± Jj~ ~~::~\ dudv
The sign is chosen to be positive if the orientation that we gave to 8S corresponds to-the usual positive orientation, and it is
negative othe~ise. In either case, since A(R) is positive, the s ign chosen must be the same as the sign of ~i:: ~?.
ThereforeA(R) = l l dxdy=
16.5
lfsl~~::~~ ~dudv.
Curl and Divergence
1. (a) curlF = 'il x F =
8j 8x
j
k
8 / 8y
8 /8z
x + yz y + xz z + xy
= [:Y(z + xy) ...:. :z(y + xz)] i - [:x(z + xy)- :z(x + yz)] j + [:x (y ~ xz) -
:y
(x + y z)] k
= (x- x) i - (y - y) j + (z- z) k = 0
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SECTION 16.5
8
0X
CURL AND DIVERGENCE
D
317
8
8
(y+xz)+ - (z + xy)=1 + 1+1 =3
0Z
0y
(b)cliv F = 'V · iF=- (x+yz) + -
k
j
8/8x 8/8y 8/8z = (ze"- 0) i - (yze"' - xye• )j + (0 - xe") k
xyez
0
yze"'
3. (a) curlF = \1 x F =
= ze"' i + (xye" - yze"' ) j - xez k
8
8
8
(b) clivF = 'V · F =ox (xye") + oy (0) + oz (yze'" ) = ye• + 0 + ye"' = y(e• + e"' )
5. (a) curlF = \1 x F =
= ( 2
X
?l
)
j
k
8/8x
8/8y
8/ 8z
X
y
z
.jx2 + y2 + z2
.jx2 + y2 + z2
+y- + Z 2 312
(b) div F = \1. F = ..E._ (
ox
.
[(- yz+ yz) i -(-xz +xz)j + (- xy+xy) k ]=O
x
.jx2 + y2 + z 2
) +
~ ( .jx2 + Yy2 + z2 ) +..E._
(
az .jx2 + y2 + z2
Z
8y
)
x2 + y2 + z2 _ x2 x2 + y2 + z2 _ y2 x2 + y2 + z2 _ z 2
2x2 + 2y2 + 2z2
2
= (x2 + y2 + z2)3/ 2 + (x2 + y2 + z2)3/ 2 + (x2 + y2 + z2)3/ 2 = (x2 + y2 + z 2)3/ 2 = -..jr-x=;2;=+=y:;:=
2 =+=z=;;:
2
j
7. (a) curlF = \1 x F =
k
8 / ax
8/8y
8/ 8z
e"' sin y eY sin z ez sin x
= {0 - eV cos z) i - (e"' cosx - O)j + (0- e'" cosy) k
= (-e11 cosz, - e" cosx, -e"' cosy)
(b) cliv F = \1 · F =
!(
e"' sin y) + ~ (eVsin z) +
!(
e" sin x) = e'" sin y + eYsin z + e" sin x
9. If the vector field is F = .Pi+ Q j + Rk, then we know R = 0. ln addition, the x -component of each vector ofF is 0, so
8P 8P aP 8R 8R aR
.
.
8Q
·
P = 0, hence - = -a = -a = - = - = -a = 0. Q decreases as y mcreases, so - < 0, but Q doesn't change
X
y
Z
X
y
Z
y
0
0
0
0
.
h
In t eX- Or Z ·
.
d'
.
. oQ
oQ 0
1rectJons, SO ax = az = .
8P 8Q 8R
8Q
+ - +-a =0 + - + O < O
Z
0X
8y
0y
(a)cliv F = -
(b) curlF =(. oR - aQ) i + (oP- oR) j + (oQ - ap) k = (0 -O) i+ (0 - O)j + (0 - O)k = 0
8y
8z
az
OX
ax
ay
11. If the vector field is F = Pi + Q j + R k,' then we know R = 0. ln addition, the y-component of each vector ofF is 0, so
8Q 8Q 8Q 8R aR 8R
.
.
8P
.
Q = 0, hence ox = 8y = az = ox = oy = az = 0. P mcreases as y mcreases, so oy > 0, but P doesn't change m
. th
d'
.
ap ap O
e X- or Z - lreCtiOns, SO ax = az = .
.
ap aQ aR
+ -8y + -8z = 0 + 0 + 0 = 0
ax
(a) d1v F = -
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318
0
CHAPTER 16
(b) curlF =
VECTOR CALCULUS
({)R
- {)Q) i + ({)P
- {)R)
j + ({)Q - {)p) k = (0- 0) i -t (0 - O) j + (o- {)P) k =- {)P k
8y .{)z
{)z
8x
ax
8y
8y
8y
fJP 0 8P k .
. . . th
.
d"
.
:
S mce -;:;- > , - .q,, IS a vector porntmg m e negative z- 1rect1on.
uy
13. curiF
vy
• ·
= "il x F = 8f8x
j
k
8f8y
8f8z
= (6xyz
2
-
6xyz 2 ) i - (~y 2 z2 - 3y2 z 2 )j + (2yz3 - 2yz 3 ) k = 0
and F is defined on all oflR3 with component functions which have continuous partial derivatives, so by Theorem 4,
F is conservativ~. Thus, there exists a· function f such that F = "il f. Then f., (x, y , z) = y 2 z 3 implies
f(x,y,z) = xy2 z 3 + g(y,z) and f 11 (x,y,z) = 2xyz3 + g11 (y,z). But f 11 (x,y,z) = 2xyz3 ,sog(y,z) = h(z) and
f(x, y,z) = xy 2 z 3 + h(z) . Thus fz(x, y, z) = 3xy2 z 2 + h'(z) but f:(x, y, z) = 3xy2 z 2 so· h(z) = K, a constant.
Hence a potential function for F is f (x, y, z) = xy 2 z 3 + K.
15. curl F = "il X F =
8f8x
j
k
8f8y
8f8z
3xy2 z 2 2x2 yz3 3x2 y2 z 2
= (6x2 yz 2 - 6x 2 yz 2 ) i - (6xy2 z 2 - 6xy 2 z) j .+ (4xyz3 - 6xyz 2 ) k
= 6xy2 z(I- z) j + 2xyz2 (2z- 3) k =1- 0
so F is not conservative.
17. cur!F = "il x F = 8f8x
j
k
8f8y
8f8z
F is defined on all oflll3 , and the partial derivatives of the component functi~ns are continuous,'so F is conservative. Thus
there exists a function f such that "il f =F. Then f,(x, y, z) = e11 = implies f(x, y, z) = xe11=+ g(y, z)
=>
fu(x, y, z) = xzev= + g11 (y, z). But / 11{x, y, z) = xze11=, so g(y, z ) = h(z) and f(x, y, z) = xe11z + h(z).
Thus f:(x, y, z) = xye11z + h'(z) but f:(x, y, z) = xye11 = so h(z) =Kanda potential function for F is
f(x , y, z) = xe11=+ K .
.19. No. Assume there is such a G. Then div(curl G )
='! (x sin y) + ~ (cosy) + ~ (z - xy) =sin y- siny + 1 f 0,
uX
uy
uz
which contradicts Theorem 11 .
j
k
21 . curlF = 8f8x 8f8y 8f8z
f(x)
g(y)
= {0- 0) i + {0- O)j + {0 - 0) k = 0. Hence F = f(x) i + g(y) j + h(z) k
h(z)
is irrotational.
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SECTION 16.5
CURL AND DIVERGENCE
0
319
ForExercises23-29, 1etF(x,y,z) = P1 i+ Qd + R1 ~and G(x,y,z) = P2 i + Q2j + R2 k.
23. div(F +G)= div(Pt + P2 , Q1 + Qz,R1 + R 2 ) = o(P1 + P2) + o(Q 1+ Qz )" + o(R1/ R2 )
•
0X
0y
Z
= oP1 + oPz + oQ1 + oQ2 + oR1 + oR2 = (oP1 + oQ1 + oR1) + ( oP2 + oQ2 + oR2 )
ox
ox
oy
oy
oz
oz
ox
oy
oz
ox
oy
oz
= div(P1 ,,Q1, R1 ) + div(Pz , Qz , Rz ) = div F + div G
27. div(F X G) ='iJ · (F x G)=
=
[Q oR2
+ R oQ1 _ Q2 oR1 _ R1 oQ2] _ [p oRz + Rz oP1 _ Pz oR1 _ R oPz]
ox
ox
ox
.oy
oy
ay
ay
ox
2
1
1
1
+ [Pt oQ2 + Q2 oP1 _ p2 oQ1. _ Q1 oP2 ]
oz
oz .
oz
oz
= [p2 (oR1 _ oQ1) +Q 2 (oP1 _ oR1) +Rz(oQ1 _ 8P1 )]
oy
oz
oz
ox
OX
ay
_[p (oRz
_ oQz) -f_ Q (oP2 _ oRz) + R1(oQz ~ aP2 )]
oy
az
az
ox
ox
1
1
8y
= G · curl F - F · curl G
29. curl{curl F) = 'iJ x ('iJ x F) =
ajax
j
k
a;ay
a;az
Now let's consider grad(div F ) - 'iJ2 F and compare with the above.
(Note that 'iJ 2 F is defined 'on page J J 19 [ET I 095).)
[continued]
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320
0
CHAPTER 16
VECTOR CALCULUS
(8
2
2
2
2
2
2
. F) n 2p - [ ( EPH+fiQ1+fiR1)·1 +( 8 P1 +8 Ch
8 R1)·
8 R1)k]
P1. 8 Q1
d(d
gra
IV
- V
OX2
OXOY OXOZ
oyox
oy 2 + [)y[)z J + , OZOX + Ozoy + OZ 2
2
2
2
2
2
2
_ [( 8 P1 + 8 P1 + B P1) i + (8 Q1 + 8 Q1 + 8 Q1) j
8x2
8y2
8z 2
8x 2
8y2
8z 2
Then applying Clairaut's Theorem to reverse the order of differentiation in the second partial derivatives as needed and
comparing, we have curl curl F = grad div F - \l 2F as desired.
31. (a)\lr =V.Jx2+y2 + z2=
x
i+
y
j +
z
k = xi + yj+z k =~
.
.Jx2+y2+z2
.Jx2+y2+z2
.Jx2+y2+z2
.Jx2+y2 + z2
r
j
k
(b)\lxr= :x
:V
:z =l:V(z) - !(v)]i + [:z(x) - :x(z)]j + [:x(y)-:V(x)]k = O
X
y
Z
(c) \l
(.!) =
r
\l (
1
) .
.Jx2 + y2 + z2 ·
1
1
1
(2x)
·
(2y)
(2z)
.
2 .Jx2 + y2 + z2
.
2 Vx2 + y2 + z2
2 .Jx2 + y2 + z2
--~~~~~---1 Jk
x2 + y2 + z2
x2 + y2 + z2
x2 + y2 + z~
-
x i+ y j +z k
r
(x2 + y2 + z2)3/2 =- r3
(d) \lin r = \lln(x2 + y2 + z2)1/2 = ~ \lln(x2 + y2 + z2)
x
.
--::----::-----::- 1
x2 + y2 + z2
y
•
z
k
x i + y j +z k
r
+ x2 + y2
+ z2 J + x2 + y2 + z2 = x2 + y2 + z2 = -7.2
33. By (13), fc f(\lg) · n ds = ffv div(f\lg) dA = JJ'o!f div(\lg) + \lg · \l f] dA by Exercise 25. But div(\lg) = \l2 g.
Hence ffv f\l 2 gdA = fc f(Vg) · nds - ffv \lg · \lf dA.
35. Let f(x, y) = 1. Then V f = 0 and Green's first identity (see Exercise 33) says
ffv \l2 gdA = f c (Vg) · n ds - ffv 0 · \lgdA =>
V 2g = 0
ffv \l 2 gdA = J~ \lg · n ds. But g is harmonic on D, so
=> f c\lg· nds = OandfcDngds = fc(\lg· n)ds =O.
37. (a) We know thatw = vjd, and from the diagram sin (;I= d/r
=> v = d!JJ = (sinO)rw = lw x rl. But vis perpendicular
to both wand r, so that v = w x r.
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SECTION 16.6 PARAMETRIC SURFACES AND THE;IR AREAS
0
321
k
j
(b) From(a), v=w x r= 0 0 w =(O ·z -wy)i + (wx~ O ·z) j+ (O · y -x· O) k=- wyi+ wx j
X
y
Z
k
j
(c) cur] v = "il X v = ajax a jay ajaz
-wy
wx
0
= [~
(0) - ~ (wx) ] i + [~ (- wy ) - ~ (o)] j + [~ (wx )- ~ (-wy )] k
ay
az
az
ax
ax
ay
= [w - (- w)] k = 2wk = 2w
39. For any continuous function f on iR3 , define a vector field G (x , y, z) = (g(x , y , z) , 0, 0) where g( x, y, z ) = fox f (t, y, z) dt.
a
Then div G = -a
X
a
a
a x
(g(x ,.y, z)) +a- (0) +a- (0) = ~ f0 f (t , y , z) dt = f (x, y, z) by the Fundamental Theorem of
y
Z
vX
.
3
Calcu lus. Thus every continuous function f on R is the divergence of some vector field.
16.6
Parametric Surfaces and Their Areas
1. P (7, 10, 4) lies on the parametric surface r (u , v) = (2u + 3v , 1 + 5u - v , 2 + u + v) if and only if there are values for u
and v where 2u + 3v = 7, 1 + 5u - v = 10, and 2 + u + v = 4. But solving the first two equations simultaneously gives
u = 2, v = 1 and t~ese values do not satisfy the third equation, so P does not He on the surface.
Q(5, 22, 5) lies on the surface if2u + 3v = 5, 1 + 5u- v = 22, and 2 + u + v = 5 for some values ofu and v. Solving the
first two equations simultaneously gives u = 4, v = - 1 and these values satisfy the third equation, so Q lies on the surface.
3. r(u , v) = (u + v ) i + (3 - v) j + (1 + 4u + 5v) k = (0,3, 1} + u (1,0, 4) + v(1, -1, 5). From Example 3, we recognize
this as a vector equation of a pl.ane through the point (0, 3, 1) and containing vectors a = {1, 0, 4) and b = (1, - 1, 5). If we
j
wish to find a more conventional equation for the plane, a normal vector to the plane is ~ X b =
1
k
0 4 =4 i - j - k
1- 1 5
and an equation of the plane is 4(x- 0)- (y- 3) :._ (z- 1) = 0 or 4x- y- z = - 4.
5. r(s, t ) = (s, t ,
e-s
2
) , so the corresponding parametric equations for the surface are X =
s, y = t, z = t 2 - s 2 . For any
point (x , y , z) on the surface, we have z = y 2 - x 2 . With no restrictions on the parameters, the surface is z = y 2 we recognize as a hyperbolic paraboloid.
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x which
2,
322
0
CHAPTER 16
VECTOR CALCULUS
The surf~e has parametric equations x = u , y = v , z = u + v, --; 1 ~ u ~ 1, -1 ~ v ~ 1.
2
2
In Maple, the surface can be graphed by entering
pl ot3d([u"2,v"2,u+v ) ,u=-l .. l,v=-l .. l);.
In Mathematica we use the Paramet r icPlot3D command.
If we keep u constant at u 0 , x = uij, a constant, so the
corresponding grid curves must be the curves parallel to the
yz-plane. If vis constant, we have y
= vij, a constant, so these
grid curves are the curves parallel to the xz-plane.
9. r (u,v) = (ucosv,usinv,u5 ).
The surface has pararnetrid equations x = u cos v, y = u sin v,
z = u 5 , -1 ~ u ~ 1, 0 ~ v ~ 21r. Note that ifu = u 0 is constant
then z· = u8 is constant and x = uo cos v, y = Uo sin v describe a
vconstant
circle in x, y of radius luo 1. so the corresponding grid curves are
circles parallel to the xy-plane. If v = vo, a constant, the parametric
equations become x = u cos vo, y
= u sin vo, z = u 5 . Then
y = (tan v0 )x, so these are the grid curves we see that lie in vertical
planes y
= kx through the z-axis.
11. x = sin v, y = cosusin4v, z = sin2usin4v, 0 ~ u ::; 21!', - ~ ::; v::; ~ Note that if v = vo is constant, then x = sin vo is constant, so the
corresponding grid curves must be parallel to the yz-plane. These
are the vertically oriented grid curves we see, each shaped like a
" figure-eight." When u = uo is held constant, the parametric
equations become x
= sin v, y = cos u0 sin 4v,
z = sin 2u0 sin 4v. Since z is'a constant multiple of y, the
corresponding grid curves are the curves contained in planes
z = ky that pass through the x-axis.
13. r( '1.1;• v) = u cos v i + u sin v j + v k. The parametric equations for the surface are x = u cos v, y = u sin v, z = v. We look at
the grid curves first; if we fix v, then x andy parametrize a straight line in the plane z = v which intersects the z-axis. If u is
held constant, the projection onto the xy-plane is circular; with z = v, each grid curve is a helix. The surface is a spiraling
ramp, graph IV.
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SECTION 16.6 PARAMETRIC SURFACES AND THEIR AREAS
0
323
15. r (u., v) = sin v i+ cos U. sin 2v j +sin u. sin 2v k. Parametric equations for the surface are x = sin v, y = cos u. sin 2v,
z = s in u. s in 2v. lfv = vo is fixed, then x = sin vo is constant, andy= {sin 2vo) cosu and z = (s in 2v0 ) sin u. describe a
c ircle of radius !s in 2vo l, so each-corresponding grid curve is a circle contained in the vertical plane x =sin v 0 parallel to the
yz-plane. The only possible surface is graph II. The grid curves we see running lengthwise along the surface correspond to
holding u constant, in which casey = (cosu.o) sin 2v, z =(sin uo) sin 2v
~
z =(tan uo)y, so each grid curve lies in a
plane z = ky that includes the x-axis.
17. x = cos3 u cos3 v, y·= sin3 u cos3 v, z = sin3 v. lf v = vo is held constant then z = sin3 v 0 is constant, so the
corresponding grid curve lies in a horizontal plane. Several of the graphs exhibit horizontal grid curves, but the curves for this
surface are neither circles nor straight lines, so graph Ill is the only possibility. (ln fact, the horizontal grid curves here are
3
3
members of the family x = acos u , y = asin u. and are called astroids.) The vertical grid curves we see on the surface
3
3
correspond to u = uo held constant, as then we have x = cos uo cos v, y = sin3 u.o cos3 v so the corresponding grid curve
lies in the vertical plane y = (tan3 uo)x through the z-axis.
19. From Example 3, parametric equations for the plane through the point (0, 0, 0) that contains the vectors a = (1 , -1, O) and
b = (0, 1, -1 ) are x = 0 + u(1) + v(O) = u., y = 0 + u.( - 1) + v(1) = v - u., z = 0 + u.(O) + v( - 1) = - v.
2
~
2
21. Solving the equation for x gives x = 1 + y + -!z2
x = J1 + y 2 + tz 2 • (We choose the positive root since we want
the part of the hyperboloid that corresponds to x ~ 0.) lfwc let y and z be the parameters, parametric equations are y = y,
z=z, x= J1+y2 +i·z 2 •
23. Since the cone intersects the sphere in the circle x 2
can parametrize the surface as x = x, y = y, z =
+ y 2 = 2, z = .../2 and we want the portion of the sphere above this, we
v'4 - x 2 - y 2 where x 2 + y 2 ~ 2.
Alternate solution: Using spherical coordinates, X = 2 sin ¢ cos e, y = 2 s in ¢ sine, z = 2 cos¢ where 0 s; <P s; f and
o s; e s; 21r.
25. Parametric equations are x = x, y = 4cos8, z = 4sin0, 0 s; x s; 5, 0 :5 () :5 21r.
27. The surface appears to be a portion of a c ircular cylinder of radius 3 with axis the x -axis. An equation of the cylinder is
. y 2 + z 2 = 9, and we can impose the restrictions 0 :5 x :5 5, y :5 0 to obtain the portion shown. To graph the surface on a
CAS, we can usc parametric equations x = u, y = 3 cos v , z = 3 s in v w ith the parameter domajn 0 ~ u ~ 5, ~ ~ v :5 3; .
Alternatively, we can regard x and z as parameters. Then parametric equations are x = x, z = z , y = -.J9=?, where
0 s; x s; 5 and -3 s; z s; 3.
29. Using Equation.s 3, we have the parametrization X =
X,
y = e-:r. cos e,
z = e-"' sin8, 0 s; x s; 3, 0 s; 8 s; 27r.
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324
0
CHAPTER 16
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31. (a) Replacing cos u. by sin u. and sin u. by cos u. gives parametric equations
x = (2 + sin v ) sin u., y = (2 +sin v) cos u., z = u. +cos v. From the graph, it
appears that the direction of the spiral is reversed. We can verify this observation by
noting that the projection of the spiral grid curves onto the xy-plane, given by
x = (2 + sin v) sin u., y = (2 + sin v) cos u., z = 0, draws a circle in the clockwise
direction for each value of v. The original equations, on the other hand, give circular
proj ections drawn in the counterclockwise direction. The equation for z is identical in
both surfaces, so as z increases, these grid curves spiral up in opposite directions for
the two surfaces.
(b) Replacing cos u. by cos 2u. and sin u. by sin 2u. gives parametric equations
x = (2 + sin v) cos 2u., y = (2 +sin v) sin 2u., z = u. + cos v. From the graph, it
appears that the number of coils in the surface.doubles within the same parametric
domain. We can verify this observation by noting that the projection of the spiral grid
curves onto the xy-plane, given by x = (2 +sin v) cos 2u., y = (2 + sin v) sin 2u,
z = 0 (where v is constant), complete circular revolutions for 0 ~ u. ~ 1r while the
original surface requires 0 ~ u. ~ 21r for a complete revolution. Thus, the new
surface winds around twice as fast as the original surface, and since the equation for z
is identical in both surfaces, we observe twice as many circular coils in the same
z-interval.
2
33. r (u., v) = (u. + v) i + 3u. j + (u. - v) k.
r u = i + 6u.j + k and r , = i - k , so r ,. x r , = - 6u. i + 2 j - 6u k. Since the point (2, 3, 0) corresponds to u. = 1, v = 1, a
normal vector to the surface at (2, 3, 0) is -6 i + 2 j - 6 k , and an equation of the tangent plane is -6x + 2y- 6z = - 6 or
3x- y + 3z = 3.
35. r (u.,v)=u.cosv i +u.sinv j +v k
=> r(1, ~) = (~.4. i)·
r u = cos v i + sin v j and r , = -u sin v i + u. cos v j + k , so a normal vector to the surface at the point ( ~ ,
r u ( 1, f) x r , ( 1, i') = ( ~ i +
4 ,~) is
4 j) x (- 4 i + ~ j + k) = 4 i - ~ j + k. Thus an equation of the ta~gent plane at
(t .4,f)is4(x·- t) - ~(y-4) + 1 (z -~ )=0 or ~x - h +z =~ .
37. r (u,v)=u 2 i +2u.sinv j +ucos v k
=> r (1,0)=(1,0, 1).
r u = 2u. i + 2sinv j + cosv k and r , = 2u.cosv j - usinv k ,
2
so a normal vector to the surface at the.point (1 , 0, 1) is
r u(1, 0) X r ,(1, 0) = (2 i + k ) X (2j) = - 2 i + 4 k.
Thus an equation of the tangent plane at (1, 0 , 1) is
- 2(x- 1) + O(y- 0) + 4(z - 1) = 0 or -x + 2z = 1.
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SECTION 16.6 PARAMETRIC SURFACES AND THEIR AREAS
D
39. The surfaceS is given by z = f(x , y) = 6- 3x - 2y which intersects the xy-plane in the line 3x + 2y = 6, soD is the
I
triangular region given by { (x, y ) 0:::; x:::; 2, 0 :::; v $ 3 - ~x}. By Formula 9, the surface area of S is
I
A (S)=
!fv
1+
(~Y + (~Y dA
= ffv J1 + (- 3) 2 + (-2)~ dA = Ji4J.(v dA = v'I4A(D ) = v'I4 (~ · 2 · 3) = 3 y'l4.
41 . Here we can write z = f(x, y) = i --:- i x- jy and Dis the disk x
A(S) =
=
1~v 1 + (:~r + (:~r
dA =
2
+ y 2 $ 3, so by Fonnula 9 the area of the surface is
1~v J1 + (-t/ + (-~)
2
:lP A(D) = fl ·1r(y'3) = ViA1r
2
dA =
-41 1~v dA
43. z = f (x, y) = ~(x 3 1 2 + y 3 12 ) and D = {(x , y) I 0 $ x $ 1, 0 $ y $ 1 }. Then f, = x 112 , f v = y 112 and
A(S) = ffv J1 + (..fi)
2
+ (.jfj) 2 dA = f01 J; y'1 + x + ydydx
= fo [tCx + y + 1)312
1
J::: = j f
1
0 [(x+ 2)
dx
312
3 2
- (x + 1) 1 ] dx
= j [Hx+ 2)5/2 _ ~(x + 1)5/ 2]: = Ji(35/2 _ 2s; 2 _ 2 s~2 + 1) = A-( 31>/2 _ 2112 + 1)
45. z = J(x, y) = xy with x 2 + y 2 $ 1, so f , = y, fu = x
A(S ) = ffv Jl + Y + X dA
2
2
=>
= J;"' J; -.fr2TI rdrd(} = J:"' [~ (r2 + I?' 2
J:::
d(}
= J;"' i(2 V2 - 1) d8 = 2; (2 .j2 - 1)
47. A parametric representation of the surface is x = x, y = 4x + z 2 , z = z with 0 :::; x :::; 1: 0 :::; z :::; 1.
Hence r :x: x r : = (i + 4 j ) X (2z j + k ) = 4 i - j + 2z k.
Note: In general, ify = f(x , z ) then r ,
A(S) =
=
~ r : = :~ i - j + ¥z k and A (S) =
JL + ( :~y + ( ~~y
1
dA. Then
J; J; ) 17 + 4z dxdz = J ) 17 + 4z dz
1
0
2
2
%(z ) 17 + 4z2 + ¥ lni2z + v'4z2 + 17 I)] ~ = 4I + 1,t(ln(2 + v'2I) - ln v'I7]
49. r ., = (2u, v , 0), r , = (0, u, v), and r u x r ,. = (v2 , - 2uv, 2u2 ). Then
J; .../v + 4u v + 4u dv du = J; J; J(v + 2u2)2 dv du
= J J; (v + 2u dv du = J; (tv + 2u vJ ::~ du = J (J + 4u du = [~u + ~u ]~ = 4
1
A(S) = ffv lr u x r vl dA = J0
1
0
2
2
)
2 2
4
3
2
4
2
1
0
2
3
)
51. From Equation 9 we have A(S) = ffv J l + (1, ) 2 + Uu) 2 dA. But if Jf, J :::; 1 and If vi :::; 1 then 0 :::; (f.,? :::; 1,
2
0 $ U u) $ 1
=> 1 $ 1 + U:~Y + Cfu)2 $ 3 => 1 $ J 1 + (!:,) 2 + Uu) 2 $ v'3. By Property 15.3 .1 1,
ffv1dA $ ffvJl+(f,.)2+(f11 )2 dA $ ffv ../3dA =>
A(D) $ A(S)$../3A( D ) =>
.
1rR2 $ A(S) $ ../31rR2 •
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325
326
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CHAPTER 16
VECTOR CALCULUS
2
2
53. z = f( x , y) = e- "' - " with x 2 + y 2 :=:; 4.
IIoJ 1 + ( -2xe->= 2 + ( - 2ye-"' - v 2 dA = IIo y'1 + 4(x2 + y2)e-2(>= +u > dA
= I:1< I: v'1+ 4r e - r r dr dB = I:"dB I: r y'1 + 4r2e- 2r dr = 27r I: r V1+ 4r2e- 2r dr ~ 13.9783
2
A(S) =
2
2
-Y )
2
2
2
)
2 2
2
2
4x
2
2
+ 4y 2
1 + (1 +x 2 + y2)4 dydx.
Using the Midpoint Rule with f( x, y) =
3
A(S) ~ 2:
1+ (
.
+ 4y2
) , m
2
1 +x +Y2 4
4x
2
= 3, n = 2 we have
2
2: f(xi ,'il; ) ~A = 4 (!(1, 1) + /(1, 3) + /(3, 1) + / (3, 3) + /(5, 1) + / (5, 3)] ~ 24.2055
i = lj = l
11 + +x +y
6
(b) Using a CAS we have A(S) =
4
1
4 2 4
(l x : Y:) 4 dy dx ~ 24.2476. This agrees with the estimate in part (a)
.
-
to the first decimal place.
57. z = 1 + 2x + 3y + 4y2 , so
Us ing a CAS, we have
o r !!!
v'I4 + ll
Jn n3 ~+
8
16
0+
V'J!
70 .
59. (a) x = a sin u cos v, y = bsin u sin v, z = ccos u
=>
~~
IZfi
(b)
71.
2
z 0
and s ince the ranges of u and v ~ suffic ient to generate the entire graph,
-2
'lii! ~
the parametric equations represent an ellipsoid.
-2
(c) From the parametric equatio ns (with a = 1, b = 2, and c = 3),
we calculate r u
y
0
-
= cosucosv i + 2cos usinv j - 3sin u k and
r, =- s in usin v i + 2sin ucos v j. So r ,. x r, = 6 sin 2 t! cos v i + 3 sin2 'u sin vj + 2 sin u cos u k , and the surface
61. To find the region D : z = x 2 + y 2 implies z + z 2 = 4z or z 2 - 3z = 0. Thus z = 0 or z = 3 are the planes where the
surfaces intersect. But x 2 + y 2 + z 2 = 4z implies x 2 + y 2 + (z - 2) 2 = 4. so 'z = 3 intersects the upper hemisphere.
Thus (z - 2) 2 = 4 - x 2 - y 2 or z = 2 +
J 4 - x 2 - y 2 • Therefore D is the region inside the c~cle x2 + y 2 + (3 - 2) 2 = 4,
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SECTION 16.6 PARAMETRIC SURFACES AND THEIR AREAS
0
2
that is, D = {(x, y) I x 2 + y :<:; 3}.
A(S) =
Jl
.)1 + ((- x)(4 - :z;2- y2)-l/2j2 + [(-y)(4 _ x2 _ y2)-1/ 2]2 dA
1 ""1,(3 J1+ -4 -~2- r rdrd8= 1
2
2
=
" 1
o
2
o
o
o
..f:i ~dB=
2r dr
4- r2
1
2
o
" [ - 2(4 -
2 1 2
r ) I ) r = ..f:i d8
r=O
2
2
= Jo
f " (-2+4)d8
= 28] o" =411"
.
63. Let A(Si) be the surface area of that portion of the surface which lies a9ove the plane z = 0. Then A(S) = 2A(Si).
Following Example 10, a parametric representation of sl is X = asin <,bcosO, y = asin ¢ sin8,
z = a cos ¢ and Jrq. x ro.l = a 2 sin¢. ForD, 0 :<:; <,b :<:; ~and for each fixed ¢, (x- ~a) 2 + y 2 5 (~a) 2 or
2
[asin ¢ cos0 - ~a] + a2 sin2 <,bsin2 0 :<:; (a/ 2) 2 implies a 2 sin2 ¢ - a 2 sin ¢ cos8 :<:; Oor
sin <,b (sin <,b - cos 8)' 5 0. But 0 5 <,b 5 ~ . so cos 0 ~ sin¢ or sin(~ + 0) ~ sin <,b or <,b- ~ 5 0 5 ~ - <,b.
Hence D = { (¢, 0) I 0 5 <,b 5 ~. <,b - ~ 58 5 ~ - <,b }. Then
12
12
2
A(Si) = f 0.,. J~"'}~l;;j a sin <,b dO d<,b = a 2 f 0" (1r- 2¢ ) sin <,b d¢
= a 2 [(- 1r cos¢)- 2( - ¢ cos¢+ sin¢)]~ 1 2 = a 2 (1r - 2)
Thus A(S) = 2a2(7r- 2).
Alternate solution: Working on 81 we could parametrize the portion of the sphere by x = x, y = y, z =
Then Jrz X r11 l =
a
Ja2- x2 - y2
!!
A(S1) =
0 ~ (x- (a/2))2 + y2 ~ {a/ 2)2
= f-rr/ 2
-1f/ 2
and
j "/21acoa8
a
dA =
Ja2-:z;2-y2
J a2 - :z;2 - y2.
a
~
_ .,./2 o
r dr dO
]r=acoso dO= J" 12 a [1- (1- cos 8?12] dO
-a(a2 - r 2 ) 1 12
2
r= O
2
- 71'/2
= f~~~2 a 2 (1 - lsin OI) dO = 2a2 J0..-12(1 - sinO) d8 = 2~2 ( ~- 1)
Thus A(S) = 4a 2 (~ - 1) = 2a2 (1r- 2).
Notes:
(1) Perhaps working in spherical coordinates is the most obvious approach here. However, you must be careful
in setting up D.
(2) In the alternate solution, you can avoid having to use Isin 81 by working in the first octant and then-
multiplying by 4. However, if you set up S1 as above and arrived at A(S1 ) = a 21r, you now see your error.
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327
328
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CHAPTER 16
VECTOR CALCULUS
16.7 Surface Integrals
1. The faces of the box in the planes x = 0 and x = 2 have surface area 24 and centers (0, 2, 3), (2, 2, 3)0 The faces in y = 0 and
y = 4 have surface area 12 and centers (1, 0, 3), (1, 4, 3), and the faces in z = 0 and z = 6 have area 8 and centers (1, 2, 0),
(1, 2, 6)0For each face we take the point P;j to beth~ center of the face and f(x , y, z) = e -ool (x+u+z), so by Definition I,
ffs f(x, y, z) dS ~ [f(O, 2, 3)](24) + [!(2, 2, 3)](24) + [f(l, b, 3)](12)
+ [f(l, 4, 3)](12) + [f(l , 2, 0)](8) + [f(l, 2, 6)](8)
0
30 We can use the xz- and yz-planes to divide H into four patches of equal size, each with surface area equal to ~ the surface
area of a sphere with radius VSO, so D.S = H4)7r( v'50)
2
= 257l'. Then (±3, ±4, 5) are sample points in the four patches,
and using a Riemann sum as in Definition l , we have
ffH f(x , y , z) dS ~ j(3, 4, 5) D.S + f(3, -4, 5) !:!.S + !( -3, 4, 5) !:!.S + f( -3, -4, 5) D.S
= (7 + 8 + 9 + 12}(257!') = 9007l' ~ 2827
So r(u, v) = (u + v) i + (u - v) j + (1 + 2u + v) k, 0 ~ u ~ 2, 0 ~ v ~ 1 and
0
r ,. x r 11 = (i + j + 2k) X (i - j + k ) = 3 i+j - 2 k
==?
iru X rvl = y'3 2 + 12 + (- 2)2 = -1140 Then by Formula2,
ff8 (x + y + z) dS = ffv(u + v + u- v +1 + 2u + v) lru x r vl dA = f 0 f~(4u + v + 1) -/I4dudv
1
0
= .;I4
J; [2u + uv + u] ::~ dv = .;I4 J; (2v + 10} dv = .;I4 [v + lOv] ~ = 11 v'l4
2
2
7o r (u,v) = (ucosv,usinov,v), 0 ~ u ~ 1, 0 ~ v ~ 7l' and
ru X rv = (cosv,sin v, O) X (-'-usinv, ucosv, 1) = (sin v, - cosv, u ) · =?
!ru X rv! = Vsin2 v + cos 2 v + u 2 = ..)u2 + 1. Then
ffsoY dS = ffv (usin v) lru x r vl dA = J; j~" (usinv) vu2 + 1 dv du = J; u..Ju2 + 1 du J0" sin v dv
0
= [t(u2 + 1?/ 2 ] : [- cosv]~ = t(2 3 / 2 - 1) 02 = H2v'2 - 1)
9. z = 1 + 2x + 3y so
0
8z
ox = 2 and 8z
oy = 30 Then by Fonnula 4,
= -/14 J:j J;(x 2y + 2X3?i + 3x2y2) dy dx = -/14 J; [~ x2y2 + x3y2 + x2y3]~=~ dx
= v'l4 J;(10x
2
+ 4x 3 ) dx = v'l4 [4fx 3 + x 4 ] ~ = 171 v'f.1
110 An equation of the plane through the points (1, 0, 0), (0, -2, 0), and (0, 0, 4) is 4x- 2y + z = 4, so Sis the region in the
plane z = 4 - 4x + 2y over D = {(x, y) I 0 ~ x ~ 1,2x- 2 ~ y ~ 0}. Thus by Formula4,
J; J °x_ x dy dx = v'2f f [xy]~~~x- 2 dx
= v'2f J (- 2x + 2x) dx = v'2f [-ix + x ]~ = v'2f (- i + 1) = 4J.
ffs X dS = ffv x y'( - 4) 2 + (2) 2 + 1 dA = ...J2f
1
0
2
2
3
2
1
0
2
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SECTION 16.7
SURFACE INTEGRALS
0
329
13. S is the portion of the cone z 2 = x 2 + y 2 for 1 $ z $ 3, or equivalently, Sis the part of the surface z = ...jx2 + y2 over the
region D = { (x, y) Jl $ x 2 + y 2 $ 9 }. Thus
r---------~----------~--
2
· Jfs x2z2dS= Jfv x (x +y )
2
2
(~Y +(~Y + l dA
15. Using x and z as parameters, we have r (x, z) = x i+ (x 2 + z 2 )j + z k 1 x 2 + z 2 :::; 4. Then
r x x r : = (i + 2xj ) X (2zj + k ) = 2x i - j +2z kand Jr, X r , J = J4x2 + 1 + 4z2 = ) 1 +4(x2 +z2). Thus
JJ (x 2 + z 2))1 + 4(x2 + z 2) dA = J;,. J; r 2 J 1 + 4r 2rdrdO = J;,. dB J; r 2 v'1 + 4r2 r dr
ffs ydS =
·
x2 +z2~ ~
.
[let u = 1 + 47'2 =>
7'
2
= t(u- 1) and idu = 1' dr]
= 27T J:7 t(u- l )JU. id·u = ft1T f t17 (u3/2 ...:. ul/2) du
= ...!..7T (1u5/2 - 1u3/2)17 = ...!..7T(1(17)5 /2- 1(17? / 2 - 1 + 1) = 2:.. (391 Jl7 + 1)
16.
5
3
1
16
:;
3
5
60
3
17. Using spherical coordinates and Example 16.6.1 0 we have r (¢,B) = 2 sin 4> cos Bi + 2 sin 4> sin Bj + 2 cos 4> k and
2
Jrq, x r~l = 4 sin¢. Then ffs(x 2 z + y 2 z) dS = .{0 "
J; 12 (4sin2 1p)(2 cos¢)(4 sin 4>) d¢ dB = 167T sin4 ¢] ~ 12 = 167T.
19. S is given by r (u,v) = ui + co~vj + sinv k , 0 $ u $ 3, 0 $ v $ 1Tj2. Then
r u X rv = i X (- sin v j +cosv k ) = - cosv j- sin v kand Jr,. X rvl = ) cos2 v +sin2 v = 1, so
.[[5 (z + x
2
y) dS =
J; 12 J;(sinv + u co~v)(l) dudv = f~.,. (3sin v + 9 cos v) dv
12
2
2
= [- 3cosv + 9sin v]~/ = 0 + 9 + 3-0 = 12
21. From Exercise 5, r (u, v) = (u + v) i + (u - ·u) j + (1 + 2u + v) k , 0 $ u $ 2, 0 $ v $ 1, and r u x r ., = 3 i + j - 2 k.
Then
F (r (u, v)) = (1 + 2u + v)e<u+v)(u- v) i- 3(1 + 2u + v)e<u+u)(u- v) j + (u + v)(u- v) k
2
= (1 + 2u + v)e" - " i - 3(1 + 2u + v)e" - 112 j + (u 2 - v2 ) k
2
2
Because the z-component ofru x r v is negative we use - (r ,. x r v) in Formula 9 for the upward orientation:
J; J; [-3(1 + 2u + v)e" -~ + 3(1 + 2u + v) e"'
2
2
= ./~ f 2(u - v dudv = 2 .{ [tu
uv J:::~ dv = 2 J; (~- 2v 2) dv
ffs F · dS = J/~ F · ( -(r .. X r u)) dA =
1
2
0
)
= 2 [~v - ~v3 ] ~ = 2
2
1
0
3
-
2
2
2
- "
+ 2(u2 - v 2)) d·1~ dv
2
(! - j) = 4
23. F (x, y, z) = xy i + yz j + zx k, z = g(x, y) = 4 - x 2 - y 2, and Dis the square [0, 1) x [0, 1], so by Equation 10
Jfs F · dS = ff0 [-xy( -2.7J) - yz( - 2y) + zx]dA = .{01 J; [2x2 y + 2y2 (4- x2 - y2 ~ + ~(4 - x2 - y 2 )] dy dx
= J; (t x2 + 1} x - x3 + ~~) dx = i~~
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330
0
CHAPTER 16 VECTOR CALCULUS
25. F(x, y, z) =xi - z j + y k , z = g(x, y) = .j4- x 2 - y 2 and Dis the quarter disk
{ (x , y) I0 ~ x ~ 2, 0 ~ y ~ v'4 - x 2 }. 8 has downward orientation, so by Fonnula 10,
=-
!1( v'
x2
4 - x2 - y2
D
y
v'4 - x 2 - . y 2 · v'4 - x2
+ y ) dA
- y2
= - IIv x 2 (4- (x 2 +y 2 ))- 112 dA = - I 0..,.12 I 02 (r cos B?(4- r 2 )- 112 rdr dB
2
·= - .[0..,.1 cos2 Bd8 I~ r 3 (4- r· 2 )- 1 12 dr
= - I0..,. 12 (~ +
= -
[let u = 4- r 2
=?
r 2 = 4 - u and-~ du = rdr]
t cos 28) d8 I~ -~(4 - u)(u) - 112 du
[~8 + t sin28]~12 (-~) [8v'U - iu312]: = -t(-~)(-16 + ¥) = -~1!'
27. Let 8 1 be the paraboloid y = x 2 + z 2 , 0 ~ y ~ 1 and 82 the disk x 2 + z 2 ~ 1, y = 1. Since 8 is a closed
surface, we use the outward orientation.
On S 1 : F (r (x, z)) = (x2 + z 2 ) j - z k and r , x r = = 2x i - j + 2z k (si~ce the j-component must be negative on S 1 ). Then
fis 1 F · dS =
o;2
II [-(x 2 .+ z2 ) - 2z2 ] dA = -I~..,. I;(r2 + 2r2 sin2 8) r dr d8
+=2 $1
= -I~..,. I; r 3 (1 + 2sin2 8) dr d() = - ]~..,. (1 + 1- cos 28) dB-I 0 r 3 dr
1
t sin 28]~..,. [~r· ]~ = - 41!' . ~ = -1r
OnS2 : F(r(x, z))= j -z kandr.., x r , =j. Thenifs F · dS=
JI (1)dA=1r.
4
= - [28-
2
.,2
+ ;:2 $ 1
Hence'Jis F · dS = - 1r + 1r = 0.
29. Here 8 consists of the six faces of the cube as labeled in the figure. On 81:
F = i + 2yj + 3z k, rv x rz = i and IIs 1 F · dS = I~ 1 }~ 1 dydz = 4;
82: F =xi + 2j + 3z k, r= x r, = j and IIs 2 F · dS = I~ 1 I~ 1 2dxdz = 8;
8s: F = x i +2yj + 3k, r , x r v ·= kand fis3 F · dS = I~ 1 I~ 1 3dxdy = 12;
8 4 : F = -i +2y j +3zk,r: x r v = -i and IIs 4 F · dS = 4;
85: F = x i- 2j+3zk, r., x r: ~ - j andiJ~ 5 F · dS = 8;
86: F =x i+ 2yj- 3k, r 11 x r ., = -k and J/~0 F · dS = J~ 1 j~ 1 3dxdy = 12.
6
Hence ffs F · dS = I: Jfs. F · dS = 48.
i=l
1.
31. HereS consists of four surfaces: S 1 , the top surface (a portion ofth~ circular cylinder y 2 + z 2 = 1); S2, the bottom surface
(a portion of the xy-plane); Ss, the front half-disk in the plane X= 2, and s 4 . the back half-disk in the plane X = 0.
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SECTION 16.7
SURFACE INTEGRALS
0
331
On 8 1 : The surface is z = J1 - y 2 for 0 S x S 2, - 1 S y S 1 with upward orientation, so
!"ls1f
1~o ! [-x (0) - y
1
F . dS =
2
h) + z
2
2
( -
-1
1 -y
2
]
.
1
2
dy dx =
·
o
1
/
(
- 1
h
1-y2
+ 1 -. y 2 ) dy dx
On 8 2 : The surface is z = 0 with downward orientation, so
On 8 3 : The surface is x
= 2 for - 1 S y S 1, O.S z S yf1- y 2 , oriented in the positive x-direction. Regarding y and z as
parameters, we have r 11 x r.., = i and
On 8 4 : The surface is x = 0 for -1 :::; y :::; 1, 0 $ z $ yf1 - y 2 , oriented in the negative x-direction. Regarding y and z as
parameters, we use - (r 11 x r z) = -i and
ff54 F · dS = f~ 1 fo~ x2 dzdy = f~ 1 fo~ (0) dz dy = 0
Thus ffs F · dS =
33. z = xeY
1+ 0 + 211" + 0 = 211" + ~·
=> 8zj8x = e11 , 8zj8y = xe11 , so by Formula 4, a CAS gives
JJ5 (x 2 +y2 +z2 )d8 = J01 f 01 (x2 +y2 +x2 e211 )Je2 Y +x2 e2v + 1dxdy ~ 4.5822.
35. We use Formula 4 with z = 3- 2x 2 - y 2
3- 2x2 - y 2 ~ 0 are
-/i $ s ji
X
=> 8zj8x = ·- 4x, 8z j8y = -2y. The boundaries of the region
and -J3 - 2x2 $ y $ J3 - 2.7:2 • so we use a CAS (with precision reduced to
seven or fewer digits; otherwise the calculation may take a long time) to calculate
37. If 8 is given byy = h(x,z), then 8 is also the level surface f(x,y,z) = y - h(x,z) = 0.
Vf(x,y,z))I = - h., i+j-h: k , and - n .IS the umt· norma 1 t hat pomts
·
h 1ft N
·
tot e e . ow we proceed as m the
n = I~/(
v
x,y,z
y'h~+l+h~
derivation of (1 0), using Formula 4 to evaluate
where D is the projection of 8 onto the xz-plane. Therefore
IIs
F · dS
=
Ii ( ~~
P
- Q + R ~~) dA.
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332
0
CHAPTER 16
VECTOR CALCULUS
39. m =· ffs }( dS = }( · 47r(~a 2 ) = 21ra 2 K; by symmetry M xz = lvf11z = 0, and
M:cv = Jj~ z }( dS = }( f~" f 01f 12 (a cos ¢)(a2 sin ¢) d¢ d() = 2711<a 3 [-~cos 2¢] ~ 12 = 1r }(a 3 •
Hence (x,y,z) = (0,0, ~ a).
I
(b) 1:: = ffs(x + y
2
2
)(10- Jx
2
+y2 )
dS =
.
1
ff
2
s :r2 + v2 s 16
43. The rate of flow through the cylinder is the flux JJ~ pv · n dS
(x + y
2
)(10- jx +y2 ) ../2dA
2
•
= ffs pv · dS . We use the parametric representation
r (u,v) = 2cosu i + 2 sinuj + v k for S, where 0 $ u $ 27r, 0 $ v $ 1, so r ,. = - 2sinu i + 2cosuj, r v = k , and the .
outward orientation is given by ru x r v = 2 cos u i + 2 sin u j . Then
ffs pv · dS = p ]~" f01 (vi+ 4sin2 u j + 4 cos2 u k) · (2cosu i+ 2sinu j ) dvdu
= p f02" . J: (2v casu+ 8sin3 u) dv du = p J~" (casu+ 8 sin3 u) du
= p[sin u + 8(-i)(2 + sin2 u) casu] ~" = 0 kg/s
45. S consists of the hemisphere 8 1 given by z = ja2 - x 2 - y 2 and the disk 8 2 given by 0 $ x 2 + y 2 $ a 2 , z = 0.
On 81: E = asin¢ cosB i +a sin ¢ sinBj + 2acos¢ k ,
T q, x To = a 2 sin 2 ¢ cosB i +a2 sin2 </> sinl'.lj +a2 sin ¢ costpk. Thus
ffs 1 E · dS =
.r;" f0"1\a3sin q) + 2a3sin cos ¢) dq) dB
3
<J;
2
= J:" J~-rf (a3 sin ¢+ a 3 .sin ¢ cos2 <P) d¢d8 = (27r)a3 (1 + .i) = ~1ra
2
3
On S2: E =X i + y j , and r y X r x = -k so Jfs2 E . dS = 0. Hence the total charge is q = co ffs E . dS = ~7ra 3 co.
47. IC9u = 6.5(4yj + 4z k) . S is given by r (x, 1'.1) = x i + y'6 cos(} j + v'6 sin 8 k and since we want the inward heat flow, we
use r x x ro = - v'6 cos 0 j -
ffs (-K \i'u) · dS = f
2
"'
0
v'6 sin(} k. Then the rate of heat flow inward is given by
f04 - (6.5)( - 24) dx d() = (27r)(l56)(4) = 12487r.
49. LetS be a sphere of radius a centered at the origin. Then ir l =~and F (r ) = cr / lrl 3 = (c/a3 ) (x i + y j + z k). A
parametric representation for S is r( ¢, 0) = a sin¢ cos 0 i +a sin¢ sin 0 j + a cos <P k~O $ <P $ 1r, 0 $ 0 $ 27r. Then
r .p = a CDS</> CDS 0 i + a cos <P sin(} j - a sin ¢ k, r o = -a sin ¢ sin () i + a sin ¢ cos 0 j , and the outward orientation is given
by r 4> x ro = a 2 sin2 ¢ cos 0 i + a 2 sin2 ¢ sinO j + a 2 sin <P cos </J k. The flux ofF across Sis
ffs F · dS = g Jt'~ c3 (asin </J cosO i +a sin ¢ sin Oj + acos¢k)
a
·. (a2 sin2 <P cosO i + a2 sin2 ¢> sin8 j +a2 sin </J cosif> k) dOd¢
= : 3 f0"' .{0 " a 3 ( sin3 ¢J + sin </> cos2 ¢) dOd¢ = c .{0" f021r sin <P dOd¢ = 47rc
2
Thus the flux does not d.epend on the radius a.
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SECTION 16.8
STOKES' THEOREM
0
333
16..8 Stokes' Theorem
1. Both Hand Pare oriented piecewise-smooth surfaces that are bounded by the simple, closed, smooth curve x 2 + y 2 = 4,
z = 0 (which we can take to be oriented positively for both surfaces). Then Hand P satisfy the hypotheses of Stokes'
Theorem, so by (3) we know JJH curl F · dS = fc F · dr = Jj~ curl F · dS (where Cis the boundary curve).
3. The paraboloid z = x 2 + y 2 intersects the cylinder x 2 + y 2 = 4 in the circle x 2 + y 2 = 4, z = 4. This boundary curve C
should be oriented in the counterclockwise direction when viewed from above, so a vector equation of C is
r (t) = 2 cost i + 2sin t j + 4k, 0 ~ t ~ 271". Then r '(t) = -2 sin t i + 2 cos tj,
F (r (t)) = (4cos2 t)(16) i + (4 sin2 t)(16) j + (2 cos t)(2 sin t)(4) k = 64 cos2 t i + 64sin2 t j + 16 sin t cost k ,
and by Stokes' Theorem,
Jf5 cur1F · dS = fc F · dr = J:"lr F (r (t)) · r '(t) dt = J:"(-128cos2 t sint + 128sin2 t qost + O)dt
= 128 [~ cos3 t + t sin3 t] ~,. = 0
5. C is the square in the plane z = -1. Rather than evaluating a line integral around C we can use Equation 3:
Jfs, curl F · dS = fc F · dr = jf52 curl F · dS where 81 is the original cube without the bottom and 8 2 is the bottom face
of the cube. curlF = x 2 z i + (xy- 2xyz) j + (y- xz) k. For 82, we choose n = k so that C has the same orientation for
both surfaces. Then curl F · n = y - xz = x +yon 82, where z = - 1. Thus ff52 curl F · dS = f~ 1
.C1 (x + y) dx dy = 0
so ffs, curlF · dS = 0.
7. curl F = - 2z i - 2x j - 2y k and we take the surfa<;e 8 to be the planar region encfosed by C, so 8 is the portion of the plane
x + y + z = 1 over D = {(x, y) I 0 ~ x ~ 1, 0 ~ y ~ 1 - x}. Since C is oriented counterclockwise, we orient 8 upward.
Using Equation 16.7. 10, we have z = g(x, y) = 1- x - y, P = -2z, Q = - 2x, R = -2y, and
fc F · dr = ffs curlF · dS = JJ'v [-( - 2z)( - 1) - ( -2x) (-1) + (- 2y)] dA
=
f 01 J01- x( -2) dydx = -2 J;(1 - x) dx = - 1
9. curl F = (xex 11 - 2x) i - (ye"' 11 - y) j + (2z- z) k and we take 8 to be the disk x 2 + y 2 ~ 16, z = 5. Since C is oriented
counterclockwise (ftom above), we orientS upward. Then n = k and curl F · n = 2z - z on S , where z = 5. Thus
fc F · dr = ffs curlF · n d8= Jj~ (2z- z) d8 = Jj~(10- 5) dS = 5(area of S) = 5(7r · 4 2 ) = 801r
11. (a) The curve of intersection is an ell ipse in the plane x
+ y + z = 1 with unit normal n = ~ (i + j + k ),
curlF = x 2 j + y 2 k , and curlF · n = ~(x 2 + y 2 ). Then
2
fc F · dr = ffs -ja (x 2 + Y2 ) dS = JJ~2 + v2 ~ 9 (x2 + y 2 ) dx dy = J0 7r
J; r dr df) = 21r(!!]-) = ~7<
3
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8
334
D
CHAPTER 16
VECTOR CALCULUS
(c) One possible parametrization is x = 3 cost, y = 3 sin t,
(b)
z = 1 - 3 cos t - 3 sin t, 0 ~ t ~ 271'.
2
-2
13. The boundary curve Cis the circle x 2 + y 2 = 16, z = 4 oriented in the clockwise direction as viewed from above (since Sis
oriented downward). We can parametrize C by r(t) = 4cost i - 4sintj + 4k, 0 ~ t ~ 21r, and then
r'(t) = - 4sint i - 4costj. Thus F (r(t)) = 4sint i + 4cos t j - 2 k , F (r (t)) · r ' (t) = -16sin2 t- 16 cos2 t = -16, and
fcF · dr = J;.,. F(r(t)) · r '(t ) dt = J;"(-16) dt = -16 (27r) = -327!'
Now curl F = 2 k, and the projection D of Son the xy-plane is the disk x 2 + y2 ~ 16, so by Equation 16.7.10 with
z = g(x, y) = .Jx2 + y 2 [and multiplying by - 1 for the downward orientation] we have
ffs curl F · dS = - ffv( -0 - 0 + 2) dA = -2 · A(D) = -2 · 7r(42 ) = - 327!'
15. The boundary curve C is the circle x 2 + z 2 = 1, y = 0 oriented in the counterclockwise direction as viewed from the positive
y-axis. Then C can be described by r(t) = cost i - sin t k, 0 ~ t ~ 21r, and r' (t) = -sin t i - cost k. Thus
F (r (t)) = -sin t j +costk, F (r (t )) · r ' (t) = - cos 2 t, and .fc F · dr = J;"(-cos2 t).dt = -~t- ~ sin2t]~.,. = -1r.
Now curl F = -i- j - k , and S can be parametrized (see Example 16.6.10) by
r (rfJ, 9) = sin rfJ cos9 i+ sin rfJ sinBj + cosq'> k,O ~ B ~ 7!',0 ~ rp ~ 7!'. Then
+ sin 2 4> sin Bj +sin 1p cos rp k and
JJ5 curlF·dS= JJ cur!F·(r,.;x ro)dA=f0" J0" (-sin2 ¢cosB - sin2 ¢sinB - sin¢cos¢)dBd¢
r .p x ro = sin2 4> cos Bi
.
~ +~ S l
17. lt is 'easier to use Stokes' Theorem than to compute the work directly. LetS be the planar region enclosed by the path of the
particle, so Sis the portion of the plane z =
h for 0 ~ x ~ 1, 0 ~ y ~ 2, with upward orientation.
curl F =By i + 2z j + 2y k and
fcF · dr = ff5 curlF · dS = ff0 [-By (0) - 2z (~ ) + 2y] dA = J0 J0 (2y- h) dydx
1
2
= f 01 J; ~Y dy dx = J; [1Y ]~=~ dx = f01 3 dx = 3
2
19. Assume S is centered at the origin with radius a and let Ht and H 2 be the upper and lower hemispheres, respectively, of S.
I
Then .ff5 curlF · dS =
JJH 1curlF · dS + JJH2curlF · dS = J~1 F · dr + J~2 F · dr by Stokes' Theorem. But C1 is the
circle x 2 + y 2 = a 2 oriented in the counterclockwise direction while C 2 is the same circle oriented in the clockwise direction.
Hence fc2 F · dr = - fc 1 F · dr so f fs curl F · dS = 0 as desired.
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SECTION 16.9
THE DIVERGENCE THEOREM
0
335
16.9 The Divergence Theorem
1. div F = 3 + x + 2x = 3 + 3x, so
JII Ediv F dV = J: J: f01 (3x + 3) dx dy dz = ~ (notice the triple integral is
three times the volume of the cube plus three times X).
To compute Jfs F · dS , on
S 1 : n = i, F = 3 i +y j + 2z k , and Jf51 F · dS = ffsi 3 dS = 3;
X
= 3x i + X j + 2xz k , n = j and I fs2 F dS = I f s2 X dS = ~;
S3: F = 3x i + xy j + 2x k , n = k and f f 53 F · dS = fj53 2xdS = 1;
s2: F
0
S 4 : F = 0 , ffs,, F · dS = 0; S:;: F = 3x i +2x k , n = - j and ff56 F · dS = f fs 5 OdS = 0;
S 6 : F = 3x i + xy j , n = - kand JJ56 F · dS = JJ56 OdS = 0. Thus ff5 F · dS = ~ 3. div F = 0 + 1 + 0 = 1, so
JJJ E div F dV = JJJE l dV = V(E) = ~ 7T · 43 = ~ 7T. S is a sphere of radius 4 centered at
2 6
the origin which can be parametrized by r (¢>, 0) = (4sin ¢>cos 0, 4 sin </>sin 0, 4 cos¢>), 0 $ ¢ $ 1r, 0 $ 0 $ 27T (similar to
Example 16.6. 10). Then
r .p x r 9 = (4 cos ¢>cos 0, 4 cos </>sin e, - 4sin </>) x (- 4sin <f>sin8,4 sin </>cos 0, 0)
= (16sin2 ¢>cos 0, 16 sin2 ¢sin 0, 16 cos ¢>sin¢)
and F (r (¢,0)) = (4cos¢,4sin ¢sin 0,4sin</>cos0). Thus
F · (r .p x r o) = 64 cos ¢sin </> cos 0 + 64sin3 ¢sin2 B + 64cos ¢sin 2 ¢>cos() = 128 cos ¢sin 2 q> cos () + 64 sin 3 ¢> sin 2 ()
2
and
'
f fs F · dS = JJ.; F · {r ,p x r o) dA = J~" J; (128cos ¢sin2 ¢>cosO + 64sin3 <f>sin2 8) d<f>dB
= J~" [ 1 ~ 8 sin3 ¢cos0+64(- ~{2+sin2 <f>)cos¢)sin2 8J !:~ dO
f27r 256 ° 2 0 d()
-- Jo
3 Sill
2 3
= ill
(l.
() - l4 Sill 28] 02 " -- ill
3
2
3 7T
5. div F = /}., (xye") + !Ju (xy z ) + //:: ( - ye" ) = ye" + 2xyz
0
3
-
ye" = 2xyz 3 , so by the Divergence Theorem,
3 2
Jfs F · dS = JJJ E div F dV = f 0 j 0 f 012xyz3dzdydx = 2 J~ xdx J02 ydy J01 z 3 dz
= 2 (~x 2]~ [h2 ]~ (tz"]~ = 2 (~) (2) {i) = ¥
7. div F = 3y2 + 0 + 3z 2 , so using cylindrical coordinates with y = r cos B; z = r sin B, x = x we have
ffs F · dS = J/J E(3y2 + 3z2 ) dV = J~" J: f~ 1 {3r2 cas2 B + 3r2 sin2 B) r dx dr dB
~ 3 f 0 " dB ] 0 r 3 dr f~ 1 dx = 3{27r) (i) (3) = 92"
2
1
9. div F = 2x s in y - x sin y - x s in y = 0, so by the Divergence Th~orem, Jf s F · dS = JJJEO dV = 0.
11. div F = y 2 + 0 + x 2 = x 2 + y 2 so
2
Jj~ F · dS = JJJ E(x + y 2 ) dV = J;" f~ .J;~ r 2 • r dz dr d(J ~ J~" .[0 r 3 (4- r 2 ) dr dB
3
5
4
= J~" dB J~ (4r - r ) dr = 27r [r -
2
'kr 6 ) ~ = ~7T
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336
0
CHAPTER 16
VECTOR CALCULUS
13. F (x, y , z) = x J x 2 + y 2 + z 2 i + yJx 2 + y 2 + z 2 j + z J x 2 + y 2 + z 2 k , so
div F = x. ~(x2 + y2 + z2)-112( 2x) + (x2 + y2 + z2)112 + y. t(x2 + y2 + z2)-112( 2y) + (x2 + y2 + z2)1/2
+ z . ~(x2
= (x2 + y2 + z2)-112 [x2 + (x2 + y2
+y2 + z2) - 112(2z) + (x2 + y2 + z2)1/2
!- z2) + y2 + (x2 + y2 + z2) + z2 + (x2 + y2 + z2)]
= 4(x2 + y2 + z2) = 4Jx2+y2+z2.
Jx2 +y2 +z2
Then
=
!!L
4J.x2 + y2 + z2 dV =
•E
= f 01r
j ·Tr/7:12,-11 4.J(li . 2 sin¢ dp dB d¢
p
0
0
0
12 sin¢d¢J;"' d1J J;4p3 dp = [-cos¢]~ 12 [B]~Tr (p4) ~ = (1) (27r) (1) = 21r
'
1
1
2 4
4
341
1
15. .[Js F ·dS=JJJ E -/3-x2 dV=J- 1 J- 1 Jo
f -"' -u· v'3-x2dzdydx = ao .J2+lllsin20
(:4a
)
17. For 81 we haven = - k, so F · n = F · ( -k) = - x 2 z- y 2 = -y 2 (since z = 0 on 81). So if Dis the unit disk, we get.
.fIs1 F · dS = .ff51 F · n dS = IJ'v (-y2 ) dA = - f 02 1r It r·2 ( s in 2 fJ) r dr dB = - {1r. Now since Sz is closed, we can use
the Divergence Theorem. Since div F = :., (z 2 x) + tu ( iv3
coordinates to g~t
+ tan z) + tz (x 2 z + y2 ) = z 2 + y 2 + x 2 , we use spherical
JJ52 F · dS = I.JJ EdivF dV = J021r Io1r! 2 I: p2 · p2 sin</>dpd<{>dO =~'IT. Finally
19. The vectors that end near P1 are longer than the vectors that start near P1, so the net flow is inward near Pt and div F(Pl) is
negative. The vectors that end near P2 are shorter than the vectors that start near P2, so the net flow is outward near P2 and
div F(P2) is positiye.
,,,,,,,,,/
5
21.
'
'
'
\
1
I
f
I
I
/
From the graph it appears that for points above the x-axis, vectors starting near a
particular point are longer than vectors ending there, so divergence is positive.
The opposite is true at points below the x-axis, where divergence is negative.
-5~-----+------45
I
~
/
I
I
I
/'/If
\
'
'
'
\
\
\
'
F(x,y) = (xy, x +y 2 )
divF = : , (xy)+ :Y (x+y 2 ) =y+2y =Jy.
=}
Thus div F > 0 for y > 0, and div F < 0 for y < 0.
\\\'\
-5
.
23. Smcc -
x i+yj+ z k
X
lxl
for ..!!_ (
8y
3
8 (
= ( 2
and 2
2)
8X
X + y + Z 312
( 2
X
X
+ y + Z 2) 31 .,2
)
=
(x 2 +y2 +z 2 )-3x 2 .
. .
.
( 2
.,
With Similar expressiOns
2)
X + y- + Z 512
y
) and ..!!_ (
z
) , we have
(x2 + y2 + z2)3/2
8z (x2 + y2 + z2)3/2
x )
3(x2 + y2 + z2)- 3{x2 + y2 + z2) .
div ( - l3 =
..
· = 0, except at {0, 0, 0) where it is undefined.
x
(x2+y2+z2)" 12
1
25.
.fIs a · n dS = JIIE diva dV = 0 since div a = 0.
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CHAPTER 16 REVIEW
27.
0
337
JI5 curlF · dS =III Ediv(curlF) dV = 0 by Theorem 16.5. 11.
2
29. ff5 (f'Vg) · n dS = JJJ E div(f'Vg) dV = fff E(f'V g + 'Vg · 'Vf) dV by Exercise 16.5.25.
, 31. If c = c 1 i + c2 j + C3 k is an arbitrary constant vector, we define F = fc = fc1 i + fc2 j
.
.
af
af
af
.
·
.
= dtv
fc = ax Ct + f)y C2 + f)z C3 = 'V f c and the Divergence Theorem says .ffs F dS = f f f E dJV F dV =>
dtv F
0
Ifs F · n dS = IIJ E 'V f ·
Jfs
+ fc3 k. Then
0
cdV.In particular, ifc = i then Jfs f i · n dS = JJJ 'V f · i dV =>
E
jjL:~ dV (where n ~ n 1 i + n2 j + n3 k). Similarly, ifc =j we have Jfs fn 2 dS = JJL :~ dV,
fn 1 d~ =
and c = k gives
/Is
f na dS =
j JL:~ dV. Then
.fJ~ f n dS = (ff8 /n1 dS) i + (Jf8 fn2 dS) j + (ff5 jn3 dS) k
=
(/IL:~ dV) + (/IL~; dV) + (/IL~~ dV) k IIL(:~ + ~ + ~~ k) dV
= JJJ E 'Vf dV
16
i
j
=
i
j
as desired.
Review
CONCEPT CHECK
1. See Definitions 1 and 2 in Section 16.1. A vector field can represent, for example, the w ind velocity at any location in space,
the speed and direction of the ocean current at any location, or the force vectors of Earth's gravitational field at a location in
space.
2. (a) A conservative vector field F Is a vector field which is the gradient of some scalar function f.
(b) The fun ction f in part (a) is called a potential function for F , that is, F = 'Vf .
3. (a) See Definition 16.2.2.
(b) We normally evaluate the line integral us ing Formula 16:2.3.
(c) The mass ism = fc p (x, y) ds, and the center of mass is (x, Y) where x = ~ fc xp (x, y) ds, y = ~ fc yp (x, y) ds.
(d) See (5) and (6) in Section 16.2 for plane curves; we have similar definitions when 0 is a space curve
· [see the equation preceding (10) in Section 16.2 ).
(e) For plane curves, see Equations '16.2.7. We have similar results for space curves
[see the equation preceding (10) in Section 16.2 ).
4. (a) See Definition 16.2. 13.
(b) If F is a force field, J~ F · dr represents ti1e work done by Fin moving a particle along the curve 0 . .
(c) Ic F ·dr=J0 Pdx+Qdy+Rdz
5. See Theorem 16.3.2.
,
0
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338
0
CHAPTER 16 VECTOR CALCULUS
6. (a) fc F · dr is independent of path if the line integral has the same value for any two curves that have the same initial and
tenninal points.
(b) See Theorem 16.3.4.
7. See the statement of Green's Theorem on page.ll08 [ET 1084].
8. See Equations 16.4.5.
8R - 8Q)
. ( -8P - -8R) J. + ( -8Q - -8P) k=V x F.
9. (a)curlF= ( - •+
8y
8z
8z
8x
ax
8y
.
(c) For curl F , see the discussion accompanying Figure 1 on page 111 8 [ET 1094] as well as Figure 6 and the accompanying
discussion on page 1150 [ET 1126]. For div F, see the discuss ion following Example 5 on page 1119 [ET 1095] as well as
the discussion preceding (8) on page 1157 [ET 1133].
10. See Theorem 16.3.6; see Theorem 16.5.4.
11. (a) See (I) and (2) and the accompanying discussion in Section 16.6; See Figure 4 and·the accompanying discussion on
page 1124 [ET 1100].
(b) See Definition 16.6.6.
(c) See Equation 16.6.9.
12. (a) See (1) in Section 16.7.
(b) We normally evaluate the surface integral using Fonnula 16.7.2.
(c) See Formula 16.7.4.
(d) The mass ism =
Jf5 p(x,y,z)dS and the center of mass is (x,y,z) wherex = ~ JJ8 xp(x,y,z) dS,
y = ~ Ifs yp(x, y, z ) dS, z = ~ .ffs zp(x, y , z) dS.
13. (a) See Figures 6 and 7 and the accompanying discussion in Section 16.7. A Mtibius strip is a nonorientable surface; see
Figures 4 and 5 and the accompanying discussion on page 1139 [ET 111 5].
(b) See Definition 16.7.8.
(c) See Formula 16.7.9.
(d) SeeFonnula 16.7.10.
14. See the statement of Stokes' Theorem on page 1146 [ET I J22].
15. See the statement of the Divergence Theorem on page 11 53 [ET J 129].
16. In each theorem, we have an integral of a "derivative" over a region on the left side, while the right side involves the values of
the original function only on the boundary of the region.
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CHAPTER 16 REVIEW
0
TRUE-FALSE QUIZ
1. False; div F is a scalar field.
3. True, by Theorem 16.5.3 and the fact that div 0 = 0.
5. False. See Exercise 16.3.35. (But the assertion is true if D is simply-connected; see Theorem 16.3.6.)
7. False. For example, div(y i) = 0 = div(xj) but y i =I x j.
9. True. See Exercise 16.5.24.
11. True. Apply the Divergence Theorem
and use the. fact that div F = 0.
..
EXERCISES
1. (a) Vectors starting on C point in roughly the direction opposite to C, so the tangential component F · T is negative.
J
Thus c F · dr = j~ F · T ds is negative.
(b) The vectors that end near Pare shorter than the vectors that start near P, so the net flow is outward near P and·
div F(P) is positive.
3. fc y zcosx ds = J071' (3cos t) (3sin t) cost .j(1)2 + ( -3sint)2 + (3cos t)2 dt = J0rr (9cos2 t sin t)v'IO dt
- t cos3 t)]~ = - 3 VfO (-2) = 6 VfO
= 9 JlO (
5. f c y 3 dx + x2 dy = J~1 [y3( -2y) + (1 - y2)2] dy = J~1 (-y4 - 2!/ + 1) dy
= [- !y5
- l3 y3 + y ]1- 1 = _ !5 - 13 + 1-!
- 13 + 1 = ~
5
5
15
7.C:x=1+2t => dx=2dt,y = 4t
=> dy = 4dt,z=- 1+3t => dz = 3dt,O~t~l.
fc xydx + y 2 dy + yzdz = f 0 [(1 + 2t)(4t)(2) + (4t) 2 (4) + (4t)(-1 + 3t)(3)] dt
1
9. F (r(t)) = e - t i + t 2 ( -t) j + (t2 + t 3 ) k, r '(t) = 2t i + 3e j - k and
1
6 - !t3 - !t4 ] = ll - 1
f.c F · dr = Jort (2te-t - 3t 5 - (t2 + t 3 ) ) dt = [-2te-t - 2e-t - !t
2
3
4
o
12
c·
11. : 11 [(1 + xy)e"'"] = 2xe"'11 + x 2 ye"' 11 = :, [e" + x 2 e"' 11 ] and the domain ofF is JR?, so F is conservative. Thus there
exists a function f sue~ that F = 'V f. Then fu(x, y) = e11 + x 2 e"' 11 implies f(x , y) = e11 + xe"'ll + g(x) and then
f ,(x, y) = xye"'11 + e"'11 + g'(x) = (1
Thus f (x, y) =
e 11
tu (4x y
-
13. Since
3 2
+ xy)e"'ll + g'(x). But J,(x, y) = (1 + xy)e"' so g'(x) = 0 => g(x) = K.
11
,
+ xe"'" + [(is a potential function for F .
2xy3 ) = Bx 3 y- 6xy 2 = :, (2x 4 y- 3x2 y 2 + 4y3 ) and the domain ofF is IR?, F is conse~ative.
Furthermore f( x, y) = x 4 y 2 - x 2 y 3 + y 4 is a potential fu nction for F . t = 0 corresponds to the point (0, 1) and t = 1
corresponds to (1, 1), so f c F · dr = /(1 , 1) - /(0,.1) = 1 - 1 = 0.
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339
340
CHAPTER 16 VECTOR CALCULUS
0
15. C1: r(t)=t i +t2 j, -l ~t~ l;
y
cl
C2: r (t) = :-ti + j , - 1 ~ t ~ 1.
. (- l , l ) ot--~~+---..., (1 . 1 )
Then
Ic xl dx- x 2 ydy =
f (t
5
1
2t5 )dt + J~ 1 tdt
-
X
~ [-.!t6)1
+ [.!t2]1
= '0
6
-1
2
-1
Using Green's Theorem, we have
j~ xy dx - x ydy =
2
2
Ji [
2
2
:x (-x y) - :y (xy ) ] dA =
Ji
1
(-2xy- 2xy) dA = /_
= I~ 1 (-2xy 2 J~:: 2 dx = I~ 1 (2x5 - 2x)dx = [~x 6 - x 2
2
2
17. Ic x ydx- xy dy. =
:r2
1:
- 4xydydx
1
t =0
1
II [ tfx (-xy 2 ) - %v (x 2 y)] dA =
.f[ ( - y 2 - x 2 ) dA = - I;" I; r 3 dr dB= -81r
+ y2 ~ 4
:z:2 + y2 ~ 4
19. If we assume there is such a vector field G, then div( curl G) = 2 + 3z - 2xz. But div(curl F) = 0 for all vector fields F.
Thus such a G cannot exist.
.21. For any piecewise-smooth simple closed plane curve C bounding a region D, we can apply Green's Theorem to
F (x, y) = f(x) i + g~y) j to get Ic f(x) dx + g(y) dy = IIo [ /!"' g(y) .
/! f(x)] dA = Jj 0 dA = 0.
0
11
'
23. \7 2 f = 0 means that~:{ + ~:; = 0. Now ifF= f v i - f x j and C is any closed path in D , then applying Green's
Theorem, we get
Ic F · dr= Ic f vdx- f x dY = IIo [:x (-fx)- /!11 Uv)]dA
= -.f[0 (fxx + f.uv) dA = - IJ'v 0 dA = 0
Therefore the line integral is independent of path, by Theorem 16.3.3.
25. z
= f (x, y) = x 2 + 2y with 0 ~ x ~ 1, 0 ~ y ~ 2x. Thus
27. z = f(x, y) = x2 + y 2 with 0 ~ x2 + y 2 ~ 4 so r., x r 11 = -2x i - 2y j + k (using upward orientation). Then
II5 z dS=
II
(x 2 +y2 )J4x 2 +4y2 +ldA
;r2 + y2 ~ 4
(Substitute u = 1 + 4r 2 and use tables.)
29. Since the sphere bounds a simple solid region, the Divergence Theorem applies and
IIsF · dS =IIIE divFdV = f.[f E(z- 2) dV = I.ff Ez dV- 2III E dV
= O [oddfu~ction in =. ]
and E os symmctnc
_
? = _Q17r'
3
2 . V(E) = _ 2 . h( 2
3
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CHAPTER 16 REVIEW
D
Alternate solution: F(r(¢,B)) = 4sin¢ cosB cos ¢ i - 4 sin¢ sinBj + 6sin ¢ cosB k,
r q, x r o = 4sin2 ¢ cos .B i + 4sin2 <P sin Bj + 4sin¢ cos ¢ k, and
F . (r q, x r o) = 16sin3 ¢ cos2 B cos¢ - 16sin3 ¢ sin2 B + 24sin2 ¢ cosq> cos B. Then
Jfs F · dS = f02" f 0" (16 sin3 ¢ cos¢ cos2 B - 16 sin3 ¢ sin2 B + 24 sin2 ¢ cos¢ cos B) d¢ dB
64
= J:" ~( - 16sin B) dB = - 3 1r
2
31 . Since curlF = 0 ,
JJ8 (curlF) · dS = 0. We parametrize C: r (t) = costi +sin t j , 0::; t::; 21r and
f c F · dr = J:" (- cos2 t sin t + sin2 t cost) dt =
33. The surface. is given by x + y
1cos3 t + t sin3 t] ~,. = 0.
+z = 1 or z = 1 - x- y, 0 ::; x::; 1, 0 ::; y::; 1- x and r, x r = i + j + k. Then
11
fcF · dr = ffs curlF · dS = ffv(- y i - z j - x k ) · (i + j + k) dA = ffv(-1)dA =- (area of D) =- ~.
35.
JJJ E divFdV =
JJJ
,2 + 112 + z2.,; 1
3dV = 3(volume of sphere)= 47r. Then
F (r(¢, B)) · (r .p x r o) = sin3 </J cos2 B + sin3 ¢ sin2 B + sin <P cos2 </J = sin </J and
ffs F · dS =
J:"J; sin ¢d¢ dB = (27r)(2) = 47r.
37. Because curl F = 0, F is conservative, so there exists a function f such that \1 f = F . Then f,(x, y, z) = 3x 2 yz - 3y
implies f(x , y, z) = x 3 yz - 3xy + g(y, z)
=> / 11 (x, y , z) = x 3 z - 3x + g11 (y, z). But f 11 (x, y, z) = x 3 z- 3x, so
g(y, z) = h(z) and f(x, y , z) = x 3 yz - 3xy + h(z). Then fz(x, y, z) = x 3 y + h' (z) but fz(x, y, z) = x 3 y + 2z,
so h(z) = z2 + Kanda potential function for F is f (x, y , z ) = x 3 yz - 3xy + z 2 . Hence
f c F · d~ = f c 'V f · dr = f(O, 3, 0) - f(O, 0, 2) = 0 - 4 = - 4.
J
39. By the Divergence Theorem, f s F · n dS =
JJJ~ div F dV = 3(volume ?f E) = 3(8 - 1) = 21.
and ffs 2a · dS = ffs curlF · dS = f c F · dr = f c(a x r) · dr by Stokes' Theorem.
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341
-0
PROBLEMS PLUS
1. Let 81 be the portion ofD(8) between 8(a) and 8, and let 881 be its boundary. Also let 8L be the lateral surface of 81 [that
is, the surface of 8 1 except 8 and 8(a)]. Applying the Divergence Theorem we have/" {
r · n d8 = { { { \l. r dV.
J8S1 r 3
JJJs 1
r3
But
"il ·
:3 = \ :x 1:y ' ! ) · \ (x2 + y2x+ z2)3/2' (x2, + y2y+ z2)3/2' (x2 + y./+ z2//2)
3y2) + (~2 + y2 + z2 - 3z2)
-- (x2 +1/ + z2 - 3x2) + (x2(x2++y2y2++z2-- o
z2)5/2
j"l {os , r r' ° d8 = !"J{l {s, 0 dV = 0. On'the other hand, notice that for the surfaces of 881 other than S(a) and 8,
=>
3
r · n = O =>
0=
!"{
!Is
s
r ·3n dS =
l os, r
r · n d8 = -
- .3
!"{
!"1
S(a)
1
that - / "
!" {
!"{
r ·3n dS +
r ·an d8 +
r ·sn d8 =
Js r
J s(a) r
JsL r
-r ·3-n d8. Notice that on S(a), r =a =>
r
!"{
r ·3n
=
!"{
Js(a)
r ·4r
=
J s(a)
=
=>
n = - -r =--r and r · r = r 2 = a 2 , so
r
a
.
r r d8 .!"r a d8 !"r a:a d8 ~a !"r dS =
J s(a)
!"{
r ·3n d8 +
r ·3n d8
is r
Js(a) r
Js(a)
area of2S (a) = ID(8) 1.
a
r ·n
Therefore ID(S) I = Js ~ dS.
3. The given line integral ~ .{0 (bz - CIJ) dx +(ex- az ) dy + (ay - bx) dz can be expressed as fc F · dr if we define the vector
field F by F (x, y, z) =P i + Q j + R k = ~(bz- CIJ) i +
Hex- az) j + ~(ay- bx) k. Then define 8 to be the planar
interior ofC, so Sis an oriented, smooth surface. Stokes' Theorem says .fc F · dr = .ffs curl F · dS = JJ~ curl F . n d8.
Now
curl F = (
~: - ~~) i + (a:; - ~=) j + ( ~~ - ~:) k
= (~a+ ~a) i + (~b+ tb) j + (~c+ tc) k = a i +bj + ck = n
so curlF · n = n · n = lnl = 1, hence Jfs curl F · n dS = Jfs dS w hich is simp ly the su rface area of S. Thus,
2
J~ F · dr = ~ j~(bz- cy) dx +(ex- az) dy + (ay .- bx) dz is the plane area.enclosed by C.
5. (F ·\l) G=
(p~:.rc+Ql:y+Rl !)cP2 i +Q2j+R2 k)
= (p1 8P2 + Q1 8P2 + R 1 8P2) i + (PI 8Q2 + Q1 8Q2 + R 1 8Q2) j
ax
By
8z
OX
8y
az
= (F · \JP2) i
+ (F · \JQ2) j + (F · \JR2) k.
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343
344
0
CHAPTER 16 PROBLEMS PLUS
Similarly, (G · V) F = (G · V Pt) i + (G · VQt)j + (G · V Rl) k. Then
k
F xcurlG =
8R2j8y - 8Q2/8z 8P2/Bz - 8R2/8x 8Q2j8x- 8P2/8y
=
(Q 8Q2
_ Q8P2 _ Rt 8P2 + R t BR2) i + (R BR2 _ Rt 8Q2 _ Pt 8Q2 + Pt BP2) j
ax
By
Bz
Bx
By
Bz
Bx
By
1
1
1
+
(p 8P2
_ Pt 8R2 _ Q BR2 + Q BQ2 ) k
8z
ox
By
Bz
1
1
1
and
Then
and
Hence
(F · V ) G + F x curl G + (G · V ) F + G x curl F
=
[(p 8P2 + pBPt ) + (Q BQ2 +Q BQt) + (Rt BR2 + R 2 BRt)]
1
~
2
~
1
~
2
~
~
~
i
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17
D SECOND-ORDER DIFFERENTIAL EQUATIONS
17.1 Second-Order Linear Equations
=> (r- 3)(r + 2) = 0 => r = 3, r = -2. Then by {8) the general s?lution
1. The auxiliary equation is r 2 - r- 6 = 0
isy = c1e3"' +cze- 2 "'.
=> r = ± 4i. Then by ( 11 ) the general solution is
3. The auxiliary equation is r 2 + 16 = 0
y = e 0 "'(c1 cos4x + cz sin4x) = c1 cos4x + c2 sin4~.
5. The auxiliary equation is 9r2 - 12r· + 4 = 0
=>
(3r- 2)2 = 0
=>
r = ~·Then by (10), the general solution is
_1· The auxiliary equation is 2r·2 - r = r(2r - 1) = 0 => r = 0, r = t, soy = c1e 0 "' + cze"/ 2 = c1 + c2 e"'12 .
9. The auxiliary equation is r
2
-
4± v'-36
.
~ 2 ± 3t, soy = e 2 "' ( c1 cos 3x + c 2 sin 3x ).
2
4r + 13 = 0 => r =
11. The auxil iary equation is 2r2 + 2r - 1 = 0
=> r = -
2
~ v'I2 = -~ ±
v;,
so
y = c1e( - l/2+VJ/2)t + cze( - l /2 - VJ/2)t.
13. The auxiliary equation is 100r 2 + 200r + 101 = 0
=> r =
- 200 ± V - 400
1 .
::;= -1 ± Tiit, so
200
P = e-t (c 1 cos ( iot) + c2 sin U0 t)] .
15. The auxiliary equation is 5r2 - 2r- 3
= (5r + 3)(r - 1) = 0 => r = _ 15,
10
r = 1, so the general solution is y = c 1 e- 3 x/S + c 2e"' . We graph the basic
solutions f(x) = e - S:r/ 5 , g(x) = e"' as well as y = e- 3 x/S + 2e"',
y = e- 3 x /f> -
e"', and.y = - 2e- 3 "'/5 - e"'. Each solution consists of a single
continuous curve that approaches either 0 or ±oo as x _, ± oo.
17. r 2 - 6r + 8
= (r - 4)(r- 2) = 0, so r = 4, r· = 2 and the general solution is y = c1e4"' + c2 e2"'. Then
·y' = 4c1e4 "' + 2cze2 "', so y(O) = 2 =>
c1
+cz = 2 and y'(O) = 2 => 4cl + 2cz = 2, giving c1 = - 1 and c = 3.
2
Thus the solution to the initial-value problem is y = 3e "' - e "'.
2
19. 9r2
4
+ 12r + 4 = (3r + 2? = 0 => r =- ~and the general solution is y = c1e- 2"'/ 3 + c2 xe- 2"'13 . Then y(O) = 1 =>
c1 = 1 and, since y' = - fr cle- 2 "'13 + Cz (1 - ~x) e- 2"'1 3, y'(O) = 0
=> - ~c1 + cz = 0, so c2 = t and the sol ution to
the initial-value problem is y = e - 2"'13 + ~xe- 2"'/3.
CD 2012 Cc:nJ>ase l earning. All RighL'i RI!SCn·cd. Muy not be scanned. copied, or duplic::ttcd. or posted to a publicly accessible website. in whole or in part.
345
3.c6
0
CHAPTER 17
SECOND-ORDER DIFFERENTIAL EQUATIONS
21. r 2 - 6r + 10 = 0 =>
r = 3 ± i and the_general solution is y = e 3 "'(c1 cosx + c2 sinx). Then 2 = y(O)·= c1 and
3 = y' (0) = c2 + 3ct
=> c2 = -3 and the solution to the initial-value problem is y = e3 "' (2 cos x - 3 sin x ).
23. r 2 - r- 12 = (r- 4)(r + 3) = 0
=> r = 4, r = - 3 and the general solution is y = c1e 4"' + c 2 e- 3"'. Then
0 = y(1) = c1e4 + c2e- 3 and 1 = y' (1) = 4cte4 - 3c2e- 3 so Ct = te- 4 , c2 = - te3 and the solution to the initial-value
25. r 2 + 4 = 0
=> r = ±2i and the general solution is y = c1 cos 2x + c 2 sin 2x. Then 5 = y(O) = c1 and 3 = y(1r / 4) = c2,
so the solution of the boundary-value problem is y = 5 cos 2x + 3 sin 2x.
27. r 2 + 4r + 4 = (r + 2) 2 = 0
=> r = -2 and the general solution is y = c1e- 2"' + c2xe- 2"' . Then 2 = y(O) = c1 and
0 = y(1) = c1e- 2 + c2e- 2 so c2 = -2, and the solution of the boundary-value problem is y = 2e- 2"' - 2xe- 2"'.
29. r 2 - 7' := r(r- 1) = 0
=> r = 0, r = 1 and the general solution is y = c1 + c2e"'. Then 1 = y(O) = c1 + c2
e -2
1
e-2
~
and 2 = y(1) = Ct + c2e so c1 = - - , c2 = - - . The solution of the boundary-value problem is y = - - + --.
e-1
e- 1
.
e-1
e- 1
31. r 2 + 4T + 20 = 0 =>
2 = y(1r) = c1 e _ 2,.
r = - 2 ± 4i and the general solution is y = e- 2"' (ct cos4x + c2 sin 4x). But 1 = y(O) = c1 and
=> Ct = 2e2,., so there is no solution.
33. (a) Case 1 (A = 0): y" + AY = 0
=>
y"
= 0 which has an auxiliary equation r 2 = 9 =>
where y(O) = 0 and y(L) = 0. Thus, 0 = y(O) = Ct and 0 = y(L) = c2L
Case 2 (A< 0): y" + Ay = 0 has auxiliary equation r 2 = ->..
r = 0 =>
y = Ct + c2x
=> Ct = c2 = 0. Thus y = 0.
=> r = ±..J=X [distinct and real since A< 0] =>
y = c 1 e.;::x"' + c2e- ..r-x"' where y(O) = 0 and y(L) = 0. Thus 0 = y(O) = Ct + c2 (*)and
0 = y(L) = Cte.;=xL + c2e - .,..CXL {t).
Multiplying(*) b~ e.;=xL and subtracting (t) gives c2 ( e.,..CXL - e - ..r-xL) ·= 0
=> c2 = 0 and thus Ct = 0 from(*).
Thus y = 0 for the cases >.. = 0 and A < 0.
(b) y" + A'I/ = 0 has an auxiliary equation r 2.+ A = 0
=> r = ±i ,;>.. =>
y = c 1 cos ,;>.. x
+ c:2 sin,;>.. x where
y(O) = 0 and y(L) = 0. Thus, 0 = y(O) = Ct and 0 = y(L) = c2 s in JXL since Ct = 0. Since we cannot have a trivial
solution, c2 i' 0 and thus sin,;>.. L = 0
=>
,;>.. L = n1r where n is an integer =>
A = n 21r2JL 2 and
y = c2 sin(n1rx / L) where n is an integer.
2
35. (a) r - 2r + 2 = 0
=>
r = 1 ± i and the general solution is y = e"' (c1 cosx + c2 sin x). lfy(a) = c and y(b) = d then
ea. (c1 cos a+ c2 sin a) = c =>
Ct cos
a+ c2 sin a= ce-a and eb (c1 cosb + c2 sin b)= d =>
Ct cosb + c2 sin b = de- b. This gives a linear system in c1 and c2 which has a unique solution if the lines are not parallel.
If the lines are not vertical or horizontal, we have parallel lines if cos a = .k cos band sin a = k sin b for some nonzero
@) 2012 Cengage Le:uning. All Rights Rcscn'cd. Mny noc. be sc:umcd. copied. or dupJicatL-d, or posted 10 a pub,h::ly accessible \vebsile, in whole or in JXUt.
SECTION 17.2 NONHOMOGENEOUS LINEAR EQUATIONS
cos a
sin a
constant k or --b = k = ---:--b
'cos
SID
=>
sina
sinb
cosa
cosb
347
=> tan a = tan b => b - a = mr, n any integer. (Note that
none of cos a, cos b, sin a, sin b are zero.) If the lines are both horizontal then cos a = cos b = 0
similarly vertical lines means sin a = sin b = 0 =>
0
::::?
b - a = mr, and
b- a = mr. Thus the systerit has a unique solution if b - a t= mr.
(b) The linear system has no solution if the lines are parallel but not identical. From part (a) the lines are parallel if
b - a = mr. If the lines are not horizontal, they are identical if ce-a = kde-b ::::?
ce-a =k=~
de-b
cosb
=>
c
b cos a
.
sin a
.
- = ea- - b. (If d = 0 then c = 0 also.) If they are honzontal then cos b = 0, but k = ---:-b also (and SID b t= 0) so
d
~
~
. -d
c = e a - bsina
. 1'fb - a = n1r and -dc r..J. e a- bcosa
we reqmre
---:--b. Thus th. e system has no soIullon
-b unless cos b = 0, .m
SID
COS
'
. h
c _J. a- bsin a
whIC case d r e
sin b .
(c) The linear system has infinitely many solution if the lines are identical (and necessarily parallel). From part (b) this occurs
a-bcosa
ni
b 0
hi h
c
a- bsina
.c
when b - a = n1r and -d = e
-b u ess cos = , m w c case -d = e
---:--b.
cos
sm
0
17.2
Nonhomogeneous Linear Equations
1. The auxiliary equation is r 2 - 2r - 3
= (r- 3)(r + 1) = 0
::::?
r = 3, r = - 1, so the complementary solution is
Yc(x) = c1e 3"' + c2e-"'. We try the particular sc;>lution yp(x) = Acos2x + Bsin2x, so
y~ = - 2A sin 2x + 2B cos 2x andy~ = - 4A cos 2x - 4B sin 2x. Substitution into the differential equation gives
( -4Acos 2x - 4Bsin2x)- 2( -2Asin2x + 2B~os2x)- 3(Acos2x + Bsin2x) = cos2x
=>
(-7A- 4B) cos2x + (4A -7B) sin2x = cos2x. Then -7A - 4B = 1 and 4A - 7B = 0 =>
B = - 6~. Thus the general solution is y( x) = Yc(x) + YP (x) = c1 e3 "' + c2e - x -
7
6 5 cos 2x -
6;
A =- :,., and
sin 2x.
3. The auxi liary equation is r 2 + 9 = 0 with roots r = ± 3i, so the complementary solution is Yc(x) = c1 cos 3x + c 2 sin 3x.
Try the particular solution yp(x) = A e- 2 "', soy~= - 2Ae- 2 "' andy~ = 4Ae- 2 "' . Substitution into the differential equation
gives 4Ae- 2 "' + 9(Ae- 2 "') = e- 2 "' or 13Ae- 2 "' = e- 2 "'. Thus 13A = 1
;>
A= f3 and the general solution is
5. The auxiliary equation is r 2 - 4r + 5 = 0 with roots r = 2 ± i, so the complementary ·solution is
Yc(x) = e2 "'(c1 cosx + c2 sin x). Try yp (x) = Ae-"', soy~= - Ae-"' andy~ = Ae-"'. Substitution gives
Ae-:r- 4( -Ae-"') + 5(Ae-x) = e-"'
=> lOAe-"' = e-"'
::::?
A = 110 • Thus the general solution is
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348
0
CHAPTER 17
SECOND-ORDER DIFFERENTIAL EQUATIONS ,
7. The a~xiliary equation is r
2
+ 1 = 0 with roots = ±i, so the complementary solution is Yc(x) = C1 cosx + c2 sin x.
1'
For y"-+: y = e"' try yp 1 (x) = Ae"'. Then y~ 1 =
y; = Ae"' and substitution gives Ae"' +'Ae"' = e"' => A=~.
1
so yp 1 (x) = ~ e"'. For y" + y = x 3 try y 112 (x) = Ax 3 + Bx2 + Cx + D. Then y~2 = 3Ax2 + 2Bx +
y;2 = 6Ax + 2B. Substituting, we have 6Ax + 2B
6A + C.= 0
f:! and
+ Ax + Bx + Cx +D = x 3 , so A = 1, B = 0,
3
2
=> C = - 6, and 2B + D = 0 => D = 0. Thus Yv2 ( x) = x 3 - 6x and the general solution is
y(x) = Yc(x) + Yp 1 (x) + YP2 (x) = c1 cosx + C2 sin x + ~e"' + x 3 - 6x. But 2 = y(O) = c1 +
c1 = ~ and 0 = y' (0) = c2 + ~ - 6
y(x) = ~ cosx +
¥ sinx + %e"' + x
=>
3
-
c2 .=
4 =>
¥. Th11s the solution to the initial-value problem is
6x.
9. The auxiliary equation is r 2 - r = 0 with roots r = 0, r = 1 so the complementary solution is Yc(x) = c1 + c2 e"'.
Try yp(x) = x(Ax + B)e"' so that no term in YP is a solution of the complementary equation. Then
y~ = (Ax
2
+ (2A + B)x + B)e"' andy;= (Ax 2 + (4A + B )x + (2A + 2B))e"'. Substitution into the differential equation
+ 2B))e"'- (Ax 2 + (2A + B )x + B)e"' = xe"' => (2Ax + (2A + B))c"' = xe"' =>
A =~. B = -1. Thus yp(x) = (!x 2 - x)e"' and ~he general solution is y(x) = c1 + c2e"' + Gx2 - x)e"'. But
gives (Ax 2 + (4A + B)x + (2A
2 = y(O) = c1 + c2 and 1 = y' (0) = c2 - 1, so c2 = 2 and c1 = 0. The solution to the initial-value problem is
y(x) = 2e"' + (~x2 - x)e"' = e"'(!x 2 - x + 2)'.
11. The auxiliary equation is r 2 + 3r + 2 = (r + 1) (r + 2) = 0, so r = - 1, r = - 2 and Yc(x) = c1e- "' + c2e- 2 "'.
Try YP = A cos x + B sin x
=>
y~
= - A sin x + B cos x, y; = - A cos x - B sin x . Substituting into the differential
equation gives (- A cosx - B sin x) + 3( - A sin x + B cos x) + 2(Acosx + B sin x) = cosx or
(A + 3B) cos x + ( -3A + B) sin x = cos x. Then solving the equations
fo and the general
cos x + fo sinx. The graph
A + 3B ·= 1, - 3A + B = 0 gives A = 110 , B =
solution is y(x ) = c1e-"' -j- c2e- 2"' + 110
-3 ~~~7-?~~~~~ s
Yp
shows yp and several other solutions. Notice that all solutions are
asymptotic to YP as x -+ oo. Except for y 11 , all solutions approach either oo
-3
or -oo as x -+ - oo.
13. Here Yc(x) = cre2"' + c2e-"', and a trial solution is 1h•(x) = (Ax+ B)e"' cosx + (Cx + D )e"' sin x.
15. Here Yc(x) = c1e 2"'
+c2e"'. For y"- 3y' + 2y = e"' try y (x) = Axe"' (since y = Ae"' is a solution of the complementary
1, 1
equation) and fo~ i/' - 3y' + 2y = sin x try yp2 (x) = B cos x + C sin x. Thus a trial solution is
Yp (x) = Yv 1 (x) + Yp2 (x) =Axe"' + B cosx + Csin x.
17. Since Yc(x) = e- x (c1 cos 3x + c2 s in 3x) we try Yv(x) = x( Ax 2 + Bx +C) e- x cos 3x + x(Dx 2 +Ex+ F) e-x sin 3x
(so that no term ofyp is a solution of the complementary equation).
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2
SECTION 17.2 NONHOMOGENEOUS LINEAR EQUATIONS
D
349
.Note: Solving Equations (7) and (9) in The Mettiod of Variation of Parameters gives
and
We will use these equations rather than resolving the system in each of the remaining exercises in this section.
19. (a) Here4r 2 + 1 = 0 => r = ±tiand Yc(x) = Clcos(tx) + czsin(~x). Wetryaparticularsolution ofthe form
=> y~ = -Asinx + Bcosx and
Yv(x) = Acosx + Bsinx
v;: = - Acosx - Bsin x. Then the equatio~
4y" + y = cosx becomes 4( - Acosx- Bsinx) + (Acosx + Bsinx) = cosx or
=>
-3Acosx- 3Bsinx = cosx
A=
- 1, B = 0. Thus, Yv(x) = - %cosx and the general solution is
(b) From (a) we know that Yc(x) = c1 cos~+ cz sin~· Setting Yl =cos~. Yz = sin~, we have
·2x )cos "' . Then
anduI2 = cosxcos~
= 21 cos (2 · 2"' )cos 2" '=12 (1 -2sill
1
2
2
4 ·2
• 2 - COS2 2 Sill 2 dx = - COS 2
( ) = J (2} Sill
X
U1 X
X
•
"')
X +
2 COS3 Z and
'j
2
· 2x COS x) dX = SID
· 2:r. - ~2 SID
· 3x . Tl lUS
() = J( 2l COS 2x - Sill
2
2
2
U2 X
i sin ~)sin~ = - (cos2 ~ - sin ~) + j (cos ~
2
= - cos {2 · ~) + i (cos2 ~ + sin2 ~ ) ( cos ~ - sin2 ~ ) = -cos x + j cos x = - i cos x
and the general solution is y(x) = Yc(x) + y ,(x) = c1 cos~. + sin~- t cos x.
yp(x) = (-cos~ + ~ cos ~)cos~ + (sin~ 3
1
21. (a) r 2 - 2r + 1 = (r -1) = 0
2
3
2
4
-
sin4 ~)
Cz
=> r· = 1, S? the complementary solution is Yc(x) = c1e'" + czxex. A particular solution
is of the form Yv(x) = Ae2"'. Thus 4Ae2 " ' - 4Ae2"' + Ae2"' = e2 "'
=> Ae2"' = e2 :r: => A= 1 => Yv(x) = e2 "'.
So a general solution is y(x) = Yc(x) + Yv( x) = c1ex + c2xe"' + e2"'.
(b) From (a), Yc(x) = c1e"' + czxex, so set Yl = ex, Yz = xex. Then, Y1Y~ - YzY~ = e2 "' (1 + x) - xe 2 "' = e 2"' and so
=> u1 (x) = - J xe:r: dx = - (x - 1)e"' [by parts] and u~ = e"' => u2(x) = J e"' dx = e"'. Hence
u~ = - xe"'
yp (x ) = (1- x)e "' + xe "' = e "' and the general solution is y(x) = Yc.(x) + yp(x) =
2
23. As in Example 5, Yc(x)
sec 2 x cosx
so ui =
_
and u21 =
1
2
2
c 1 e "'
+ czxe"' + e 2 "'.
= c1 sin x + cz cosx, so set Y1 = sinx, Yz = cos x . Then Y1Y~ - YzYi =- sin2 x- cos2 x = - 1,
= secx =>
ul(x) = J sccxdx =ln(secx + tan x)for. O <x<~ ,
sec2 x sin x
_1
= -secx tanx => uz(x) = - secx. Hence
Yv(x) = ln(sec x +tan x) · sin x - sec x · cos x = sin x ln(sec x + tan x) - 1 and the general solution is
y(x) = c1 sin x + Cz cosx +sin x ln(secx + t a n x)- 1.
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350
0
CHAPTER 17
SECOND-ORDER DIFFERENTlAL EQUATIONS
2
o:
2o:
1
1
3x
1
-e x
25. Yt = e , Y2 = e and Yt Y2 - Y2Y1 = e . So Ut = (
) 3 =
·
1 +e- o: e"'
U2 (x) = I
yp(x) =ex
3
e- x .
- - - and
1 + e-x
1
ex 2 dx = m( e"' + ) -e-o: = ln(1 + e - "') - e-x. Hence
ex
e"'+e"'
1nn .+e-x) + e "'[ln(1 +e-x)- e-"'J and the general solution is
2
y(x) = (c1 + ln(1 + e - "')]e"' + [c2 - e- x + ln(1 + e- ")]e2"'.
27. r 2 - 2r + 1 = (r -1) 2 = 0
=> r = 1 so Yc(x) = Ct e"' + c2xe"'. Thus Yt =ex, Y2 = xe"' and
YIY~- y2y~ = e"'(x + 1)ex- xe"'e"' = e 2 "'. Sou~=-
Ut = -
I
xe'c · e"' /(1 + x 2 )
2
ex
2
2)
1
x
1 (
e"' · e"' /(1 + x )
2
"'
+
x
dx
=
In
1
+
x
,
u
=
e
= 1 +1 x 2
2
2
1
2
x
= - - -2
1+x
~
U2 =
~
I
1 d
-I
d
1 + x 2 x = tan x an
yp(x) = - 4e"' ln(1 +x2) +xe"'tan- 1 x. Hence the general solution isy(x) = e"' [c1 + c2x- ~ ln(1 + x 2) +xtan- 1 x].
17.3 Applications of Second-Order Differential Equations
1. By Hooke's Law k(0.25) = 25 so k = 100 is the spring constant and the differential equation is 5x" + 100x = 0.
The auxiliary equation is 5r 2 + 100 = 0 with roots r = ±2 J5 i , so the general solution to the differential equation is
x(t) = c1 cos(2 V5 t) + c2 sin(2 V5 t). We are given that x(O) = 0.35 ~
2 J5 c2 = 0 =>
c1 = 0.35 and x (0) = 0
~
1
c2 = 0, so the position of the mass after t seconds is x(t) = 0.35 cos(2 J5 t).
3. k(0.5) = 6 or k = 12 is the ·spring constant, so the initial-value problem is 2x" + 14x 1 + 12x = 0, x(O) = 1, x 1 (0) = 0.
The general solution is x(t ) = c1e-Gt + c2e-t. But 1 = x(O) = c1 + c2 and 0 = x 1 (0) = -6c1 - c2. Thus the position is
given by :i:(t) = - ~e- 6 t + ~e-t.
5. For critical damping we need c 2 - 4mk = 0 or m = c2 /(4k) = ·142 /(4 · 12) = ~ kg.
2
.
7. We are given m = 1, k = 100, x(O) = -0.1 and x (0) = 0. From (3), the differenti al equation is ~t~ + c
1
d
d; + 100x =
0
with auxiliary equation r 2 + cr + 100 = 0.
If c = 10, we have two complex roots r = - 5 ± 5 v'3 i, so the motion is underdamped and the solution is
x =e-st [ct cos(5 v'3t) + c2 sin(5 v'3 t)J. Then -0.1 = x(O) = c1 and 0 = x 1 (0) = 5 v'3c2 - 5ct
~
C2. --
sox=e-st [--;0.1cos(5v'3t)- 10J:isin(5v'3t)].
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-
1
"i:ii"7s
,
SECTION 17.3
APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS
x = e - 15t/'2 [c1 cos(fL:{f-t) + c2 sin( 2flt) ], so - 0.1 = x (O) = c1 and 0 = x'(O) = fL:{f-c2 -
[-o 1 cos( .fit) - ~
10 sin(!b(J_t)]
351
- lf ± fL:{f-i. The general solution is
If c = 15, we again have underdamping since the auxi liary equation has roots r =
Thus x -- e -lSt/2
0
lf c1 =>
c2 = - 10377 .
5
·
2
v7
2
·
For c = 20, we have equal roots r 1 = r2 = -10, so the oscillation is critically damped and the solution is
x = (c1 + c2t)e- 10 t_ Then -0.1 = x(O) = c 1 and 0 = x'(O) = -10c1 + c2
=> c2 = -1, sox= ( -0.1- t)e- 101 .
If c = 25 the auxiliary equation has roots r1 = --,-5, r2 = ...:.20, so we have overdamping and the solution is
X= c1e-st + C2e-
20t. Then -0.1 = x(O) = Cl + C2 and 0 = x'(O) = -5cl- 20c2
=> C1 =- {5 and C2 = fo•
If c = 30 we have roots r = -15 ± 5 v's, so the motion is
0.02
overdamped and the solution is x = c1e( -IS+ 5 v'5 )t + c2e( - 15 - 5 v'5 )t.
c = 10
c = 15
Then -0.1 = x(O) = C1 + c2 and
O=x'(O) = (-15+5v's)c1+(-15-5v's)c2
c1 =
-5 - 3 y'5
100
_ ( - s- 3
xwo
=>
+ 3 v'5 , so
and c2 = -5 100
fi) e(
- 15+ s v'S)t
+ ( -s 100
± 3 v'5) e< -1s - & fi)t
·
·
9. The differential equation is mx" + kx
-0.11
= Fo coswot and wo "1- w = .jkfm. Here the auxil iary equation is mr2 + k = 0
with roots ±.jklmi = ±wi so Xc(t) = c1 coswt + C2 sinwt. Since wo "1- w, try x 1,(t) = Acoswot + B sinwot.
Then we need (m) ( -w5) (A coswot + B sinwot) + k(Acoswot + Bsinwot) = Fo coswot or A(k- mw~) = Fo and
B(k - ~w5) = 0. Hence B = 0 and A = k Fo 2 =
( ;"
) since w2 = k. Thus the motion of the mass is given
· - mw
m w - w2
m
0
0
0
11. From Equation 6, x(t) = f(t) + g(t) where f(t) = c1 coswt + c2 sinwt and g(t) =
is periodic, with period
~ = !..
'
1
wo
J
( ;o
) cosw0 t. Then f
mw - w 02
:.:r, and ifw "1- wo, g is periodic with period :,~. If :' is a rational number, then we can say
0
=> a = wo
bw where a and b are non-zero integers. Then
'x(t +a·:;-)= f(t +a· 2; ) + g(t +a· :.:r) = f(t) + g(t + :~ · :.:r) = f(t) + g(t + b - :,~) = j(t) + g(t) = x(t)
so x(t) is periodic.
13. Here the initial-value problem for the charge is Q" + 20Q' + 500Q = 12, Q(O) = Q' (0) = 0. Then
Qc(t) = e- lOt(c1 cos 20t + c2 sin 20t) and try Qp (t) = A
=> 500A = 12 or A = 1 ~ 5 .
The general solution is Q(t) = e -IOt(c1 cos 20t + c2 sin 20t) + 1 ~5 • But 0 = Q(O) = c1 + 1 ~ 6 and
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352
D
CHAPTER 17
SECOND-ORDER DIFFERENTIAL EQUATIONS
Q' (t) = I (t) = e-lot[( -10c1 + 20c2) cos 20t + ( - lOc2- 20ct) sin 20t) but 0 = Q'(O) = - 10c1 + 20c~. Thus the charge
is Q(t) = - 2 ~ 0 e- tot(6 cos 20t + 3 sin 20t) + 1 ~ 5 and the current is J(t) = e-lOt (~)sin 20t.
= 12sin lOt so try
15. As in Exercise 13, Qc(t) = e-tot(ci. cos 20t + c2 sin20t) but E(t)
Qp(t) = A cos lOt+ B sin lOt. Substituting into the differential equation gives
=>
(- lOOA + 200B + 500A)cos lOt+ (-lOOB- 200A + 500B) sin lOt = ·12sin lOt
400A + 200B = 0 and 400B - 200A = 12. Thus A= - 2 ~ 0 , B =
1 ~ 5 and the general solution is
Q(t) = e- lOt(c1 cos 20t + c2 sin 20t) -
2 ~ 0 cos lOt+ 1 ~ 5 sin lOt. But 0 = Q(O) = c1 - 2 ~ 0 so c1 = 2 ~0 .
fs sin lOt+
lOc1 + 20c2) cos 20t + ( -lOcz - 20ct) sin 20t] and
Also Q' (t) =
6
2 5 cos lOt+ e-lOt[( -
0 = Q' (0) = fg - lOc1 + 20c2 so c2 = - 5 ~0 . Hence the charge is given by
Q(t) = e-wt ( 2 ~ 0 cos 20t- 5 ~ 0 sin 20t] -
11. x(t)=Acos(wt +8)
¢:?
2 ~ 0 cos lOt + 1 ~ 5 sin lOt.
x(t)=A[coswtcos8-sinwtsin5]
cos 8 = c1 / A and sin 8 = -c2/ A
x(t)=A(~coswt+~sinwt)where
¢:?
x(t) = c1 cos wt + c2 sin wt. [Note that cos2 8 + sin 2 8 = 1
¢:?
c~ + c~ = A 2 .)
=>
17.4 Series Solutions
00
1. Let y(x) = 2: C,:.x" . Then y'(x) =
00
2: nenx"- 1 and the given equation, y' - y = 0, becomes
n=l
n=O ·
00
00
00
00
n= l
n=O
n=O
n=O
2: nenxn-l- 2: CnX" = 0. Replacing n by n + 1 in the first sum gives 2: (n + l)cn+tX"- L enx" = 0, so
00
2: [(n + l)Cn+I- en]x" = 0. Equating coefficients gives (n + l )cn+l - en= 0, so the recursion relation is
n=O
oo
oo
·
oon
in general, en= ~.Thus, the solution is y(x) = 2: cnx'' = 2: c~ x" = eo 2: ; = eoe"' .
n.
n=O
n=O n.
n=O n.
00
00
00
n =l
n=O
= 2: enx", we have y'(x) = 2: ncnx"- 1 = 2: (n + l)Cn+tx" and
3. Assuming y(x)
n= O
~
00
00
n=O
n=2
n=O
-x 2 y = - I:: Cnx"+2 = - I:: Cn- 2x". Hence, the equation y' = x 2 y becomes I:: (n + l )cn+tx"or Ct + 2c2x +
f: [(n + l)cn+I - Cn-2) x" = 0. Equating coefficients gives
n=2
Ct
00
2: Cn - 2x" = 0
n=2
= c2 = 0 and Cn+t = Cn-21
n+
for n = 2, 3, ... . But c1 = 0, so C<1 = 0 and C7 = 0 and in general C3n+l = 0. Similarly C2 = 0 so C3n+2. = 0. Finally
C3 =
Co
C3
CO
C6
CO .
Co
Co
Co
.
3' C6 = ·5 = 6 . 3 = 32 . 21 , cg = 9 = 9 . 6 . 3 = 33 . 31 , ... , and C3n = 3n . n!. Thus, the solution
00
isy(x) =
00
00
c
00
x3n
L CnXn = 2: C3nX3 n = 2:::: -3 O 1x 3" =Co L -3
n=O
n= ,O
n=O
n. n .
00
I = Co
n=O nn.
(x3/3) n
L - - 1-
n = O
n.
=
3
coe'" 13.
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SECTION 17.4
~
I: c.,x" '*
5. Let y (x) =
y' (x) =
n= O
00
~
~
n=l
n =O
[since
0
353
I: nenx"- 1 and y" (x) = I: (n + 2){n + l)cn+2X". The differential equation
.
00
00
becomes L: (n + 2){n + 1)cn+2x" + x I: nc.,.x"- 1 +
n=O
SERIES SOLUTIONS
00
L.: CnX = 0 or L.: [(n + 2){n + 1)cn+2 + nen + en]x" = 0
11
n= O
n=l
n=O
n~1 nenx" = n~O ncnx"] . Equating coefficients gives (n + 2)(n + 1)Cn+2 + (n + 1)Cn = 0, thus the
.
I . .
- (n + l )c,.
Cn
recurs1onreat1ontscn+2= (n+ 2 )(n+l) =--n-+-
.
.
b
Co
coeffictents are gtven y c2 = , C4 = - 4C2 =
2
c2n = (- 1)
Co
n
2 · 4 · · · · · 2n
I
.
00
( l )n
(- 1)"
~
0 1 2
,n=,,
, .... Then theeven
co
C4
Co
d.
. , co= -6 =-~,an m genera,1
2 4
( - 1)"Co
.
C1
C3
C1
C5
C1
, c 7 = - - = - - -- ,
• Theoddcoeffi.ctents are C3 = - - , cs = - - = 3
3·5
2 n n.1
5
7
3·5·7
=
and m genera, C2n+l = -
2
Cl .
(-2)nn!ct Th
I . .
(Z
) = (2
3 · 5 · 7 ····· n+ 1
n+ l )l. . e so utton IS
00
{-2)n n l
"' - 11- x-" + c1 "'
·1 x 2"+ 1
L- 2 n.'
n=O
nL=O {2n + 1)·
1 {x) = c0
y
00
L.: CnX"
7. Let y (x) =
00
:::}
y' (x) = I.: ncnx"- 1 =
n=O
n=l
00
00
I.: (n + l)Cn+lx" .and y" (x) = L.: (n + 2){n + l )c,+ 2xn. Then
n=O
n=O
00
00
00
00
n=O
n=O
n =l
n =O
(x -1)y"(x) = L (n+2){n+ l )Cn+2Xn+l _ L.: (n+2)(n+l)cn+2X" = L n(n+ 1)Cn+1x"- I.: (n+2) (n+ l )cr•+ 2x".
DO
Since L.: n(n + 1)cn+ 1x" =
n=l
00
L.: n(n + 1)Cn+J.' t", the differential eq uation becomes
n=O
DO
00
00
n=O
n=O
n =D
L n(n + l)Cn+lX"- L.: (n + 2)(n + 1)Cn+2X" + 2: (n + 1)Cn+!X" = 0
00
00
u=O
n=O
:::}
L: [n(n + 1)Cn+l - (n + 2)(n + l )Cn+2 + (n + 1)cn+1]x" = 0 or 2: [(n + 1)2cn+ I - (n + 2)(n + 1)Cn+2]x" = 0.
Equating coefficients gives (n + 1) 2c,,+I - (n + 2)(n + 1)cn+2 = 0 for n = 0, 1, 2, .... Then the recursion relation is
in general en = Ct , n = 1, 2, 3, .... Thus the solution is y(x) =Co + c1
n
f xn.
Note that the solution can be expressed as
n
n =1
co- c1 ln{l - x) for Jx J < 1.
00
00
n=O
n=l
2::.: c.. x".-Then -xy'(x) = - x L.: nc,.xn- I
9. Let y(x) =
~
~
n= l
n =O
2: nc.,x" = - L.: ncnx",
~
y"(x) =
L.: (n + 2){n + l )cn+2X" , and the equation y"- xy' - y = 0 becomes
n= O
00
L: [(n + 2)(n + 1)Cn+2- nen - en]x" = 0. Thus, the recursio11 relation is
n= O
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354
0
CHAPTER 17
SECOND-ORDER DIFFERENTIAL EQUATIONS
c,..(n + 1) .
nc,.. + Cn
~
Cn
·oneoftheg•vencon•tlons
·
d' ·
· {0) 1 8
•sy
= . ut
0 12
Cn+2=(n+ 2 )(n + 1)=(n+ 2 ){n+ 1 )=n+ 2 .orn = , , , .. . .
y(O) =
n~o c,..(O)" = ~ + 0 + 0 + .. · =eo •.so Co= 1. Hence, C2 = ~ = ~· C4 = ~ = / 4 , C6 = ~ = 2 . ~ . 6 , ... ,
C2n =
! The other given condition is y'(0) = 0. But y'{0) = n=f l ncn(O)n-l =
2 n.
/
1
•
c1 + 0 + 0 +
· · · = C1, so C1 = 0.
I
By the recursion relation, ca =
problem is y(x) =
~ = 0~ cs = 0, .. . '. C2n+l = 0 for n = 0, 1, 2, . . .. Thus, the solution to the initial-value
00
00
n =O .
n=O
x2n
00
00
(x2/2)n
n = O
n.
L; c,..xn = L; C2nX2n = L; - ,.. 1 = L; - - 1-
11. Assuming that y(x) =
n=O 2
n.
2
2
= e"' / .
00
00
00
00
00
n=O
n=O
n=O
n= l
n=O
L; cnx", we have xy = x L; c,..xn = L; c,..xn+l, x 2y' = x 2 L; nc,,xn- l = L; ncnxn+t,
y"(x)=
f: n(n-1)c,..x"- = n=-f: 1(n+3){n + 2)Cn+ax"+1
2
[replace n with n + 3]
n=2
= 2c2 +
f (n + 3){n + 2)Cn+ax"+l,
n =O
and the equation y" + x 2 y' + xy = 0 becomes 2c2 +
00
L; [(n + 3)(n + 2)Cn+a + ncn + c,..J xn+l = 0. So c2 = 0 and the
n =O
(n + l)c,..
)(
) , n = 0, 1, 2, .. . . But Co = y(O) = 0 = c2 and by the
n+3 n+2
(
recursion relation, Can = can+2 ~ 0 for n = 0, 1, 2, .... Also, c1 = y' (0) = 1, so C4 = - :~~ = -
~ ,
4 3
17 Review
CONCEPT CHECK
1. (a) ay" +by' + cy = 0 where a, b, and care constants.
(c) If the auxiliary equation has two distinct real roots r 1 and r· 2, the solution is y = c1 ert"' + c2er2 "'. If the roots are real and
equal, the solution is'!/ = c 1 er"' + c2xer"' where r is the common root. If the roots are complex, we can write r1 = a+. i/3
and r2 = a- i/3, and the solution is y = ea"'(c1 cos/3x + c2 sin/3x).
2. (a) An initial-va lue problem consists of finding a solution y of a second-order differential equation that a lso satisfies given
conditions y(xo) =Yo andy' (xo) = '!/I. where yo and '!/1 are constants.
© 2012 CcnCJige Lcnming. All RighiS Rcscr.'Cd. Mfty nol bo: sciUUICd, copied. or duplicated. or posted to a publicly accessible website, in "itolc or in pan.
CHAPTER 17 REVIEW
0
355
(b) A boundary-value problem consists. of finding a solution y of a second-order differential equatio n that al~o satisfies given
boundary conditions y(xo) =Yo and y(x1) = Yl·
3. (a) ay" +In/ + CIJ = G{x) where a, b, and care constan ts and G is a continuous function.
(b) The complementary equation is the related homogeneous equation ay" +by'+ cy = 0. If we find the general solution Yc
of the complementary equation and YP is any particular solution of the original differential equation, then the general
solution of the original differential equation is y(x) = Yr>(x) + Yc(x).
(c) See Examples 1-5 and the associated discussion in Section 17.2.
(d) See the discussion on pages 11 77- 1179 [ ET 1153-11 55].
4. Second-order linear differential equations can be used to describe the motion of a vibrating spring o r to analyze an electric
circuit; see the discussion in Section 17 .3.
5. See Example I and the preceding discus.sion in Section 17 .4..
TRUE-FALSE QUIZ
1. True. See Theorem 17.1.3.
3. True. cosh x and sinh x are linearly independent solutions of this linear homogeneous equation.
EXERCISES
·1. The auxi liary equation is 4r2 - 1 = 0
3. The auxiliary equation is r
5. r
2
-
4r + 5 = 0
2
=>
(2r + 1}(2r - 1) = 0
=> r = ±t. Then the general solution
+ 3 = 0 => r = ±J3i . Then the general solution is y = c1 cos ( J3 x) + c 2 sin ( J3 x).
=> r = 2 ± i , so Yc (x) = e 2"' {c 1 cosx + c 2 sin x ). Try YP (x) = Ae2"' => y~ = 2Ae2"'
and y~ = 4Ae 2"' . Substitution into the differential equation gives 4Ae 2 "' - 8Ae 2"' + 5Ae2 "'
the general solution is y( x)
=-e 2"'
=> A = 1 and
= e 2 "' ( c1 cos x + c2 sin x) + e2"' .
7. r 2 -21·+ l=O => r =landyc(x) = cle"' + c2xe"'. Tryy1,(x)=( Ax +B)cosx + (C x+D) sin x =>
y~ = (C- Ax- B) sinx +(A+ Cx +D) cos x andy~= (2C - B- Ax)-cosx + (-2A- D- Cx) sinx. Substitution
gives (-2Cx + 2C- 2A- 2D)cosx + (2Ax The general solution is y( x) = c1 e"' + c2 xe"' 9. r
2
-
2A + 2B - 2C}sin x =xcosx => A= 0, B = C = D = - t.
! cos x - !(x + 1) s in x.
r- 6 = 0 . => r = -2, r = 3 and Yc(x) = c 1e __:_ 2 "' + C2 e 3 "' . For y" - y'- 6y = 1, try Yr>t (x) = A. Then
y~ 1 (x)
= y~1 (x) = 0 and substitution into the differential equation gives A= -i- For y"- y' - 6y = e- 2 :r try
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356
CHAPTER 17
D
yp 2 (x)
SECOND-ORDER DIFFERENTIAL EQUATIONS
= Bxe- zx [since y = B e- 2 x satisfies the complementary equation). Then y~ = (B- 2Bx)e- 2"' and
y~ = (4Bx- 4B)e-2 x, and substitution gives - 5Be- 2 "' = e-2 "'
'* · B = -k· The general solution then is
y(x) = c1e- 2x + Cze 3 "' + yp1 (x) + yp 2 (x) = c1e- 2x + cze3 x - ~- ixe- 2"'.
11. The auxiliary equation is r 2 + 6r = 0 and the general solution is y(x) = c 1 + c2 e-6 x = k 1 + k2 e- 6 <z- l ). But
3 = y(l) = k 1 + kz and 12 = y'(l) = -6k 2 . Thus k2 = - 2, k1 = 5 and the solution is y(x) = 5- 2e- G(:z:-l) .
.13. The auxiliary equation is r 2 - 5r + 4 = 0 and the general solution is y(~1;) = c1 e'" + c2 e 4 '". But 0 = y(O) = c 1 + c2
and 1
= y'(O) = C1 + 4c2, so the solution is y(x) = ~(e "' - c"' ).
4
15. r 2 + 47' + 29 = 0
r = - 2 ± 5i and the general solution is y = e-2x (c1 cos 5x + c2 sin 5x). But 1 = y(O) = c 1 and
::::?
'* c1 = e2" , so there i~ n~ solution.
2
- 1 = y(1r) = -c1e - "
17. Let y(x)
~
=
=
n=O
n=O
n=O
= E CnXn. Then y" (x) = 2.: n(n- l )enx"- 2 = .E (n + 2)(n + 1)Cn+2Xn and the differentia l equation
00
becomes E [(n + 2)(n + l )cn+2 + (n + l )en)xn
n=O
= 0. Thus the recurs ion relation is Cn+2 = -c,.f(n + 2)
•
1
•
for n = 0, 1,2, .... But eo = y ( 0 ) = 0, soc2" = Oforn = 0,1 , 2, .... Also c1 = y (0) = 1, so C3 = C7 =
(- 1) 3
(-1) 3 23 3!
· (- 1t2nnl
. . =
, ... , C2n+l = ( n + l) ! for n
71
2
3 5 7
.
1
(- 1?
3, C5 = ""3-5'
. ..
= 0, 1, 2, .... Thus the solut10n to the m1t1al-value problem
~
(- 1)"2nnl 2
=E
CnXn = E
· x n+l.
00
is y(x)
n =O
n=O
(2n + 1)!
19. Here the initial-value problem is 2Q" + 40Q' + 400Q = 12, Q (0) = O.Dl, Q' (0) = 0. Then
Qc(t) = e- 101 (c1 cos lOt + cz sin lOt) and we try Qp(t) = A. Thus the general solution is
Q(t) = e -lOt(c1 cos lOt+ cz sin lOt)+ 1 ~0 . But 0.01 = Q'(O) = c1 + 0.03 and 0 = Q"(O) = - lOc1 + lOcz,
so c 1 = - 0.02
= c 2 • Hence the charge is given by Q(t) = - 0.02e-10t (cos lOt + sin lOt) +0.03.
21. (a) Since we are assuming that the earth is a solid sphere of uniform density, we can calculate the density pas follows:
P=
M · If V..
. of the eart h w h'1ch i 1es
' WJt
. h'm a d Jstancc
'
massofef arth = ~
r IS th e vo Iume o fth e portJon
r 01J: t he
vo 1ume o cart11
3 1rR
.4
3
Mr 3 .
Gl\IJ,.rn
GMm
center, then Vr = 31rr
and M r = pV,. = R 3 . Thus Fr = - -r-2= -----w-r.
(b) The particle is acted upon by a varying gravitational force during its motion. By Newton's Second Law of Motion,
d2y ·
GMrn
GM
11
2
2
m dt 2 = F11 = -----w- y, soy (t) = - k y (t) wh~re k = R 3 • At the surface, - mg = Fn =
g=
·..
GM
2
GMm
--w,
so
•g
R2 . Therefore k = R.'
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CHAPTER 17 REVIEW
D
357
(c) The differential equation y 11 + k 2 y = 0 has auxiliary equation r 2 + k 2 = 0. (This is the r of Section 17.1,
not the r measuring distance from the earth's center.) The roots of the auxiliary equation are ±ik, so by ( II ) in
Section 17.1' the general solution of our differential equation for t is y( t) := Cl cos kt + C2 sin kt. It follows that
y' (t) = -c1k sin kt + c2k cos kt. Now y (0) = Randy' (0) = 0, so c1 = Rand c2k = 0. Thus y(t) = Rcos kt and
y'(t) = - kRsinkt. This is simple harmonic motion (see Section 17.3) with amplitude R, frequency~. and phase angle 0.
The period is T = 21r j k. R Rj 3960 mi = 3960 · 5280 ft and g = 32 ft/s 2 , so k = v'i(R Rj 1.24 x 10- 3 s- 1 and
T = 27r/k Rj 5079 s Rj 85 min.
(d) y(t) = O #
coskt = O
#
kt = !+1rnfor someintegern
=> y'(t) = - kRsin(f+7rn)=±kR.Thusthe
particle passes through the center of the earth with ~peed kR Rj 4.899 mi/s Rj 17,600 mi/ h.
© 20 12 Ccngage L""ming. All Rights Resc" ·ed. May not be scanned. copied. or duplicnlcd, or posted lo a pubficly accessible wcbsilc, i n whole or in p:lrt.
0 APPENDIX
Appendix H Complex Numbers
1. (5- 6i) + (3 + 2i) = (5 + 3) + (-6 + 2)i = 8 + (-4)i = 8 - 4i
3. (2 + 5i)(4- i) = 2(4) +2(-i) + {5i)(4) + (5i)(- i) = 8 - 2i + 20i - 5i2 = 8 + 18i- 5(-1)
= 8 + 18i + 5 = 13 + 18i
5. 12 + 7i = 12 - 7i
1+ 4i
1 +4i 3 - 2i
3 - 2i+ 12i-8(-1)
11 + 10i
11
10 .
7· 3 + 2i = 3 + 2i . 3 - 2i =
3 2 + 22
=
13
= 13 + 13 ~
1
1
1-i
1 -i
1- i
1 1.
9 -- = -- . - - =
= -- = - - - t
. 1 +i
1+i 1- i
1 -(-1)
2
2
2
11 . i 3 = i 2 . i = ( - 1)i = -i
13. yC25 = J25 i = 5i
15. 12 - 5i = 12 + 15i and 112- 15il = J12 2 + (-5)2 = )144 + 25 = Jl69 = 13
17. - 4i = 0 - 4i = 0 + 4i = 4i and l-4il = J0 2 + (-4) 2 = VW = 4
19. 4x
2
+9 =0
#
4x
2
=- 9 #
X 2-- _Q
4
2
21. By the quadratic formula, x + 2x + 5 = 0
·
·
2
23. By the quadratic formula, z + z + 2 = 0 #
25. For z = -3 + 3i, r =
x=
#
z=
- 2 ± J2 2 - 4(1)(5)
-2 ± v'-16
-2 ± 4i
.
=
( )
=
= -1 ± 2t.
2 1
2
2
-1 ± y'12 - 4(1)(2)
( )
21
=
- 1 ±A
1 · v'7
= - 2 ± 2 i.
2
J( -3) 2 + 32 = 3 V2 and tan 8 = ! 3 = -1 => 8 = 3; (since z lies in the second quadrant).
Therefore, - 3 + 3i = 3 J2 (cos 34,. + i sin 3; ) .
27. For z = 3 + 4i, r = )32 + 42 = 5 and tan 8 = ~
=> 9 = tan- l ( ~) (since z lies in the first quadrant). Therefore,
3 +4i = 5[cos(tan- 1 ~) +isin(tan- 1 1)].
29. For z = .J3 -H, r =
J(v'3) + 1 = 2 and tan 9 = -ja => 8 = % =>
2
2
Forw=l+v'3i, r=2andtanB=v'3
z = 2 (cos -if + i sin -if).
=> 8 =~ => -w =2(cosi + isin~).
Therefore, zw = 2 · 2 [cos ( ~ + ~) + i sin ( i + ~)] = 4 (cos i + i sin i),
z/w =
Hcos ( i - i) + i sin (% - i)] = cos (- %) + i sin (-f), and 1 = 1 + Oi = 1(cos 0 + i sin 0) =>
® 2012 Ccngnge Learning. All Rights Reserved. May not be SCDJmed, copied, or duplicated, or posaed loa publicly ac.cc.ssiblc website. in "hole or in p3r1..
359
360
APPENDIX H COMPLEX NUMBERS
D
1/ z = ~ [cos(O- ~) + i si~(O- ~ )] = ~[cos( - i) + i sin(-~) ] . For 1/z; we could al~o use the fonnula that precedes
Example 5 to obtain 1/z =
Hcos "if - i sin i).
31. For z = 2¥'3- 2i, r = J(2 v'3) + (-2) = 4 and tan6 = '27a = -~
2
z = 4(cos( --jr} + isin(
2
-i)J. Forw = - 1 + i, r = -/2, tanB = ! 1 = -1
=> 6 = -~ =>
=>
6 = 3;
=>
w = ..J2 (cos 34" + isin 3; ) . Therefore, zw = 4 V2 [cos( -{f + 3; ) + i sin(-~+ 3; ) ] = 4 ..J2 (cos
4
3
z/w- 72 [cos(-2I6
4")+isin(-2I
6 -
i; + isin i;),
4
11
13
13
71' )] -- 72
[cos(- 11
1 271')+isin(- 1271' )] -- 2 vf2(cos
~
12"+isin
. .
12") ' and
3
4
1/z = i(cos(-{f) - isin(-{f)] = Hcos "if+ isin"lf).
33. For z = 1 + i, r = ..j2 and tan B =
(1 +_i)
20
t = 1 => B = "i => z = ..j2 (cos "i + isin f). So by De Moivre's Tlworem,
= [..J2 (cos "i +isinf)J
20
= (2 1 12 ? 0 (cos 20~71' + isin 20~") = 210(cos51l' + isin51l')
= 210 [-1 +i(O)] = -2 10 = - 1024
So by De Moivre's Theorem,
5
(2 v'3 + 2i)
= [4(cos "if+ isin ~)) 5 = 45 (cos 56"+ i sin ~G11') = 1024[-4 + ~i] = -512 v'3 + 512i.
37. 1 = 1 + Oi = 1 (cosO+ isinG). Using Equation 3 with 7' = 1, n = 8, and 6 = 0, we have
- 1/8 [cos (0+2k1l')
· - 0, 1, 2, ... , 7.
-cos k1l' + .. k1l' , where k+ .. (0+2k1l')J8
4
4
8
~sm
Wk- 1
w0
~sm
1m
= 1(cosO+isinO) = 1, w1 = 1(cosf +isinf) = "7z + 72i,
w 2 = 1 (cos ~ + i sin ~) = i, w 3 = 1 (cos ; + i sin a;.) = -
"7z + 72 i,
w 4 = 1(cos1l'+isinir) = -1, w 5 = 1(cos ; +isin ; ) =
-72 -72i,
3
5
5
w 6 = 1 (cos a;+ isin a;) = ...:i, w 7 = 1(cos 7; + isin 7; ) = 72 - 72i
•
•
0
Re
•
•
39. i = 0 + i = 1 (cos~ + i sin~). Using Equation 3 with r = 1, n = 3, and 6 = %, we have
Wk
1m
' k = 0, 1, 2.
= 11 / 3 [ COS ( J!.2 +32k1l') + i sin ( ~- +q2k1l')] , where
wo = (cos i + i sin i) =
WI =
5
(cos· r,,.
6 + i sin 6") --
4 + ~i
-
v'3
2 + l2 i
•
•
0
-i
0
w2 = ,(cos 6" + i sin 9; ) == - i
41 . Using Euler'sfonnula (6) withy=~. we have ei71'/2 =cos~+ isin ~ = 0 + 1i = i.
© 20 12 Cengnge Lenming. All Rights Reserve~. May not be: scanncJ, copi<'tl, or duplicated, or posted to a publicly accessible website, in whole or in part.
Re
APPENDIX H COMPLEX NUMBERS
43. Using Euler's formula (6) with y=
i· we have ei"/ = cos i + isin i ~ + '!] i.
3
=
47. Taker = 1 and n = 3 in De Moivre'sTheorem to get
(l (cos (} + i sin IJ)] 3 = 13 (cos 39 + i sin 31J)
(cos IJ + isin 0) 3 = cos 31J + i sin30
cos3 (} + 3(cos2 fJ)(i sin 0) + 3(cos IJ)(isln 1J? + (i sin IJ) 3 =cos 38 + i s_in3fJ
cos3 1J + (3 cos2 f) sin fJ)·i - 3 cos IJ sin2 ()- (sin3 fJ)i =cos 31J + i sin 31J
(cos3 1J - 3 sin2 f) cos IJ) + (3 sin f) cos2 f) - sin3 1J)i = cos 30 + i ~in 3(}
Equating real and imaginary parts gives cos 30 = cos~ fJ - 3 sin 2 () cos IJ
and
sin 3B = 3 sin IJ cos 2 (} - sin3
49. F (x) = er:r: = e<n+IJi)x = e""'+l»:i = e""' (cos bx + isin bx) = e'"'' cos bx + i(e'"" sin bx)
e.
=>
F'(x) = (eax cos bx)' + i(e<t:t sinbx)'
= (aea"' cos bx - be""' sin bx) + i(ac""' sin bx + be"~ cos bx)
= a (e""'(cosbx + isinbx)] + b(e""'(- sin bx + icosbx)]
= ae''"' + b(e""'(i2 sinbx + icosbx)]
= aerx + bi(e""' (cos bx + i sin bx )] = aerx + bier% = (a + bi )erx = rerx
® 2012 Cengogc Learning. All Rights Rcs~rvcd. Ma.y not be scanned, copied, or duplicated, or posted 10 n publicly accessible website, in whole or in pln.
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