Student (Print)
Section
Last, First Middle
Student (Sign)
Student ID
Instructor
MATH 152
Exam 3
Spring 2000
Test Form A
Solutions
Part I is multiple choice. There is no partial credit.
Part II is work out. Show all your work. Partial credit will be given.
1-10
/50
11
/10
12
/15
13
/15
14
/10
You may not use a calculator.
TOTAL
1
Part I: Multiple Choice (5 points each)
There is no partial credit. You may not use a calculator.
1. Find the values of x such that the vectors ˜x, "1, 3 ™ and ˜2, "5, x ™ are orthogonal.
a. "1 only
correctchoice
b.
c.
d.
e.
0 only
1 only
0 and 1 only
1 and "1 only
˜x, "1, 3 ™ ˜2, "5, x ™
2x 5 3x 5 5x 0 at x "1
x
3
lim e " 16 " x
xv0
x
HINT: The series for e x may be helpful.
a. 0
b. 1
3!
c. " 1
3!
correctchoice
d. 1
2
3
2. Compute
e. .
2
3
............................................................................ Known Taylor Series
ex 1 x x x C
2!
3!
6
9
3
................................................................................Substitute x v x 3
ex 1 x3 x x C
2!
3!
6
9
1 x3 x x C " 1 " x3
x3
3
2!
3!
lim e " 16 " x lim
1
xv0
xv0
2
x
x6
3. Consider the parametric curve rŸt ˜t, sin t, t 3 ™. Find parametric equations for the line tangent
to the curve at t =.
a. x 1 =t, y "t, z 3= 2 = 3 t
b. x 1 =t, y "1, z 3= 2 = 3 t
c. x = t, y "1 t cos t, z = 3 3t 3
d. x = t, y t cos t, z = 3 3t 3
e. x = t,
rŸt ˜t, sin t, t 3 ™
y "t,
z = 3 3= 2 t
r U Ÿt ˜1, cos t, 3t 2 ™
correctchoice
rŸ= ˜=, 0, = 3 ™
r U Ÿ= ˜1, "1, 3= 2 ™
Tangent line is
rŸ= trU Ÿ= ˜=, 0, = 3 ™ t˜1, "1, 3= 2 ™ ˜= t, "t, = 3 3= 2 t ™
Ÿx, y, z 2
4. Find the Taylor series for fŸx x 2 3 about x 2.
a. 7 4Ÿx " 2 Ÿx " 2 2
b.
c.
d.
e.
correctchoice
7 4Ÿx " 2 2Ÿx " 2 2 4Ÿx " 2 3
7 4Ÿx " 2 Ÿx " 2 2 2 Ÿx " 2 4 C
3
7 4Ÿx " 2 2Ÿx " 2 2 4Ÿx " 2 3 2Ÿx " 2 4 C
7 4Ÿx " 2 2Ÿx " 2 2 2 Ÿx " 2 3 4 Ÿx " 2 4
3
3
fŸx x 2 3 is a quadratic polynomial. So the Taylor series cannot be higher than 2 nd degree.
The answer must be 7 4Ÿx " 2 Ÿx " 2 2 . If you do compute it, fŸ2 7, f U Ÿ2 4 and
f UU Ÿ2 2. All higher derivatives are zero.
5.
The vectors a, b and c b"
a all lie in
the same plane as shown in the diagram.
Which of the following statements is TRUE?
0.
a. a•
b
b. a•
b points into the page.
c.
a • b
c
0.
correctchoice
d. b• a • c points in the direction of "
a.
e. None of These
a • b p 0 because they are not parallel.
a • b points out of the page by the right hand rule, not in.
a • b c 0 because a • b µ c. TRUE
b • a • c points to the right by the right hand rule, not left.
3
6. Find a power series centered at x 0 for the function fŸx of convergence.
!
!
!
!
!
.
a.
Ÿ" 1 n n 3n1
R 1
8
Ÿ" 1 n n 3n1
R8
8 x
n0
.
b.
8 x
x , and determine its radius
1 " 8x 3
n0
.
c.
8 n x 3n1
n!
n0
.
d.
8 n x 3n1
R 1
8
8 n x 3n1
R 1
2
n0
.
e.
n0
1
1"x
R2
!
!
!
.
xn
n0
1
1 " 8x 3
x
1 " 8x 3
|8x | 1
3
for
.
3 n
Ÿ8x n0
.
correctchoice
|x| 1................................................................................Geometric Series
!
.
8 n x 3n
for
|8x 3 | 1.............................................Substitute x v 8x 3
n0
8 n x 3n1
n0
®
for
|x 3 | 1
8
|8x 3 | 1..........................................................................Multiply by x
®
|x| 1
2
®
R 1
2
7. Find the distance from the point Ÿ3, "2, 4 to the center of the sphere
Ÿx
" 1 2 Ÿy 1 2 Ÿz " 2 2 4
a. 2
b. 3
correctchoice
c. 9
d. 61
e. 61
The center of the sphere is Ÿ1, "1, 2 . So the distance is
Ÿ3
" 1 2 Ÿ"2 " "1 2 Ÿ4 " 2 2 4 1 4 3
4
8. Let fŸx sinŸx 2 . Compute f Ÿ14 Ÿ0 , the 14 th derivative of fŸx evaluated at 0.
HINT: Use a series for sinŸx 2 .
1
a. "
14! 7!
b. 7!
14!
c. 14!
7!
d. " 14!
correctchoice
7!
e. "14! 7!
3
5
7
...................................................................Known Taylor Series
sinŸx x " x x " x C
3!
5!
7!
6
10
14
................................................................... Substitute x v x 2
sinŸx 2 x 2 " x x " x C
3!
5!
7!
................... Standard Taylor Series
fŸx fŸ0 f U Ÿ0 x 1 f UU Ÿ0 x 2 C 1 f Ÿ14 Ÿ0 x 14 C
14!
2
1 f Ÿ14 Ÿ0 " 1
Equate coefficients of x 14 :
®
f Ÿ14 Ÿ0 " 14!
7!
7!
14!
9. Find the angle between the vectors u ˜1, 1, 0 ™ and v ˜1, 2, 1 ™.
a. 0°
b. 30°
c. 45°
d. 60°
e. 90°
| |u
11 |v| 2
u v cos 2 ||v|
|u
3
2 6
;
a.
!
!
!
!
!
Ÿ" 1 n0
.
b.
1
3n 1
n0
.
c.
Ÿ" 1 .
d.
Ÿ"1 Ÿ3n
n
n0
.
e.
Ÿ" 1 1"x
;
1/2
0
!
!
3
1
2
3n1
3n1
1 2
1 1 1
C
4 24
7 27
10 2 10
1 "
1 1 "
1
C
7 27
4 24
10 2 10
2
3n"1
" 1 1
2
1
2
3n"1
correctchoice
" 2 " 22 55 " 88 C
2
"2 "
2
2
1 1 " 1 C
2 22
5 25
8 28
x n .............................................................................................................Geometric Series
.
1
3
!
; !
Ÿ" x n0
1x
1
2
0
.
n0
1
1x
n
3n " 1
n0
1
n
3n 1
n0
3n
1
2
n
1/2
1 dx as an infinite series.
1 x3
1 " 13 16 " 19 C
2
2
2
10. Evaluate the integral
.
u v 1 2 3
141 6
3
2 30°
2
dx 3 n
1/2
0
.
.
Ÿ" 1 n 3n
.......................................................................... Substitute
x
n0
n0
Ÿ" 1 x dx n 3n
!
.
n0
Ÿ" 1 n
x 3n1
3n 1
1/2
0
!
.
n0
Ÿ" 1 n
3n 1
1
2
x v "x 3
3n1
5
Part II: Work Out (points indicated below)
Show all your work. Partial credit will be given. You may not use a calculator.
11. (10 points) Consider the planes
P1 :
2x " y z 1
P2 :
x y " 3z 2
a. (2 pts) Fill in the blanks:
A normal to the plane P 1 is N 1 N 1 ˜2, "1, 1 ™
A normal to the plane P 2 is N 2 N 2 ˜1, 1, "3 ™
b. (3 pts) Find a vector parallel to the line of intersection of the two planes.
v N 1 • N 2 i j
k
2 "1
1
1
"3
1
˜2, 7, 3 ™
c. (3 pts) Find a point on the line of intersection of the two planes.
2x " y 1
Set z 0:
A point is Ÿ1, 1, 0 .
xy 2
Add:
3x 3
®
x 1, y 1
d. (2 pts) Find parametric equations for the line of intersection of the two planes.
Ÿx, y, z Ÿ1, 1, 0 t˜2, 7, 3 ™
x 1 2t
y 1 7t
z 3t
12. (15 points) Let fŸx lnx.
a. (10 pts) Find the 3 rd degree Taylor polynomial T 3 for fŸx about x 2.
fŸx lnx
fŸ2 ln 2
f U Ÿx 1x
f U Ÿ2 1
2
f UU Ÿx "12
x
"
UU
f Ÿ2 1
4
f UUU Ÿx 23
x
UUU
f Ÿ2 1
4
T 3 fŸ2 f U Ÿ2 Ÿx " 2 1 f UU Ÿ2 Ÿx " 2 2 1 f UUU Ÿ2 Ÿx " 2 3
3!
2
1
1
1
2
3
ln 2 Ÿx " 2 " Ÿx " 2 Ÿx " 2 2
24
8
b. (5 pts) If this polynomial T 3 is used to approximate fŸx on the interval 1 t x t 3, estimate
the maximum error |R 3 | in this approximation using Taylor’s Inequality.
M |x " 2|n1 where M u f Ÿn1 Ÿx for 1 t x t 3.
|R n Ÿx | n
1 !
Ÿ
Note: n 3 and n 1 4.
On the interval 1 t x t 3,
f Ÿ4 Ÿx "6 is largest at x 1. So M f Ÿ4 Ÿ1 6.
3
x
On the interval 1 t x t 3, |x " 2| is largest at x 1 or 3. So |x " 2| t |3 " 2| 1. Therefore
|R 3 | M |x " 2|4 t 6 1 4 1
4
4!
24
6
13. (15 points) Consider the points
P Ÿ1, 0, "1 ,
Q Ÿ2, 3, 1 and RŸ0, 4, 1 a. (5 pts) Find a vector orthogonal to the plane determined by P, Q and R.
u PQ Q " P ˜1, 3, 2 ™
v PR R " P ˜"1, 4, 2 ™
i j k
u • v 1 3 2 ˜"2, "4, 7 ™
N
"1 4 2
b. (5 pts) Find the area of the triangle with vertices P, Q and R.
• v | 1 4 16 49 1 69
A 1 |u
2
2
2
c. (5 pts) Find the equation of the plane determined by P, Q and R.
N 1 Ÿx " x 0 N 2 Ÿy " y 0 N 3 Ÿz " z 0 0
or
"2Ÿx " 1 " 4Ÿy 7Ÿz 1 0
2x 4y " 7z 9
14. (10 points) Find the radius of convergence and the interval of convergence of the series
!
.
n0
3
n
n
1
Ÿx " 2 .
n1
To find the radius, apply the ratio test:
|a n1 |
|x " 2|n1 3 n n 1
lim
1 |x " 2| 1
L nlim
n
nv. n1
v. |a n |
3
"
2|
|x
3
n2
Convergent on "1 x 5.
Check endpoints:
!
!
.
At x "1:
n0
At x 5:
.
n0
3
3n
n
!
!
n
1
Ÿ" 3 n1
n
1
Ÿ3 n1
.
n0
.
Ÿ" 1 1
n1
|x " 2| 3
®
R3
n
n1
n0
®
is convergent by the alternating series test.
is divergent by the integral test or as a p-series with
p 1 1.
2
So the interval of convergence is "1 t x 5.
7