ANSWk:R.. /(£(1
NAME:
ALPHA NUMBER:
SECTION:
INSTRUCTOR:
CALCULUS n (SM122, SMI22A)
1330-1630 Monday4 May 2009
FINAL EXAMINATION
Page 1 of 10
SHOWALLWORK IN THIS TEST PACKAGE
Tear off the last sheet of this exam. It contains problems27, 28, 29, 30. Do these problemsfirst
WITHOUT YOUR CALCULATOR. When you are finishedwith these 4 problems, return the
sheet to your instructor, take out your calculator, and completethe rest of the exam.
PART ONE: MOLTIPLE CHOICE (50%). The first 20 problemsare multiple-choice. Fillin the
best answer on your Scantronbubble sheet. Write your name, alpha number, instructor and
section on this test and your bubblesheet and bubble in your alpha number. There is no extra
penalty for wrong answerson the multiplechoice. Showall your scratch work on this test.
CALCULA TORS PERMITTED FOR TmS SECTION.
1. Determine fo4/(x)dx for the function I
whose graph on the right consists of 2 line segments.
a) -1
d) 2.5
2.
b) I
e) 4.5
x
€0
- ~ -I
-1.
= cos(x)
Determine the total area bounded between the curve y
over the interval [0, Jr].
~
~
a) -1
c) 1
e) 27r
A
and the x - axis (y
'I
7Yz
~
;;;
r
= 0)
OA-:;Lr-J
4-
0
=. :2. reX)
.
;r;
~
I0
=
r
x
. /rr; \
L Y...)
=.:>...
3. Find the volume of revolution obtained by rotating
about the x - axis the region R bounded by
I
y = x, y = 0, and x = 1.
-
V::;j
a) 1/3
c) 2
e)
b)
0
7r /2
-Ii
,r~J7<
Q
1-J.,
1
- S TiKvv'{0
1[/'0
3
@)
~/~,
I
=Jr~~=%
3
0
ro
4. Find
GU)
d) 2
f I(x)
lit
dx given that
f
t
b) CJ)
c) 0
e) does not exist
- 2t 2
+.
I(x) dx = +1
et
f
f"4 ==t~ [1lt{~I~]
-.:;.t;
~
I'
- j..AA
--r-.
J -J+<;
[
-t -">~
(-t+I)
e~
"
CALCULUS II
FINAL EXAMINATION
4 MAY 2009
Page 2 of 10
5. After a u - substitution, the definiteintegral II2 (3x2 + 1)~x3+'x dx
equalswhich of the following?
a)
0)
c)
rJ;;
I
I~\fz; du~
L
f
f
I
3
10
2
52
~
I-.~X "..
(})
d-
A.-,
10
7
J
du
I 1
--du
2
J;;
following?
x3ex
=('5l~M-f.
v;;: ~
fx3exdx could equal which of the
6. After one integration by parts, the indefinite integral
a)
k
1II:) 2, -"
23..u3/2 du
I
d) J 103..u3/2
2 3
e)
ht).(=
du
/J
1
- - J x2exdx
x3ex -1x2exdV
41
~
1f-I./
CVJ.- e.- vvt-
-
A,if'-
3> '/..
::: y. e- -
2
J x e dx
X
-'.
A==-
=
x2exdx
~
.
~
3
~.
c) x4ex -3f
3 x
d) x' e -
L.
Ace
X~
J
2..1 .
~=
3)(
9
J
r7Vf.
}J::;=e
y:
.
tro&t.
je
of
4
3 'f.
e) none of the above
~""
Use two steps of Euler's method to
approximate y(2) if y' = 2x + y and y(O) = 1 .
:1
(Use step size h == 1.)
()
7.
~
c.:LV
a) 0.1
d) 4
I
c) 2
:2
~
8. The direction field on the right could correspond to
which one of the following differential equations?
a) y' ==x2
b) y' = 1- x
c) y' = x + y
d) y' = x2 + y2
= ,.u... X.+-,£
'f
.2-.:0 )N.
t
(JlzJ':' &.
1>1
/
2{
I
Ci'Y'=y
.
-/
/M-UMj
0+/ =-/
IT!=2-
vI
\
'1\
;/
/
,/
;\
'0
}
\
\
"
>X
NAME:
INSTRUCTOR:
CALCULUS II
FINAL EXAMINATION
ALPHA NUMBER:
SECTION:
4 MAY 2009
}(,
9. The function y = xeX is a solution to
which one of the following differentialequations?
(b)
a) y' + y = eX
c) y' = -4y
e) y' = 16y
10. A chain,
X c:-
Ar=
O,
6
y'- y=0
) y' - y = 0
Page 3 of 10
='>
1-
t<
=:: Ie..- r'i.~
I
~
f
~
d -d ==e--C-,rt e- - xc:-
50 ft long, weighs 2 Ibslft and
4 vJ=-
hangs over the edge of a building 100 ft high.
How much work is done in pulling the rope
to the top of the building?
:d~-t:~
hL
f-.x
a) 10,000 ft-Ibs
Cbf
2,50~
c) 1,000 ft-Ibs
d)
500 ft-lbs
e) none of the above
~-i;:.~
= ~.d =~-X
= (~!b.. XAX it) . J<ft-)
t~o
1;
)(
A w==--~
-=
~~)<
'f.%j~ ~)
~
4 r-~
;:!.:J.)S-()(){t-~
11. A simpleseriesRC circuit consistsof a capacitorwithcapacitanceC = 0.5 farad,a resistor
with resistanceR = 3 ohms, and a constant EMF with E = 7 volts. The initialcharge on the
capacitor is Q = 4 coulombs.
Kirchoff's law leads to which of the following differential equations for the charge Q(t)?
a) 2Q'+3Q=7
3Q' + 2Q =
0f
/(I. + .L, t:1- EE:
b) 3Q'+0.5Q=7
D
d) 4Q'+ 3Q= 7
e) 7Q'+ 2Q = 4
12. Find the best approximatedistancebetween
the two points in the planewhose polar coordinates
(r, B) are (2, JZ"
/6) and (2, JZ"
/ 3) .
(Hint: first change to Cartesiancoordinates.)
/
c.. U(-
L-=q
/
Q -r -:s- Q .::: 7
36/+..2
6'(
7
=9 3
~
~
LIz..J6)= (2 J7f"/~)
(} /d) =-(1,t/lj
0/~) ~)
'fh)
(~Ij) ==(1I3}
a) 0.6
b) 0.8
<CCJ 1. U
d) 1.2
e) 1.4
I
d :: J[-13)~
r{1I'i-/)~
:= 11 xl-if3'/~
:!= I. or
t)
4 MAY 2009
FINAL EXAMINATION
CALCULUS U
13. We learnedthat the area insidea polar region
= J b ~r2 dB.
can be written as A
~
Find the area
°2
inside one loop of the 3-petal curve r = sin(3B) .
{Hint: Use your calculatoror use the trig identity
. 2 z
sm
( )
a)
Jr
c)
TC/6
e)
Jr2
= 1-
~
~~
=<
-1t
~
3
d) TC/12
°
A ~ 3{ r{3e)lJe
cos(2z) .
}
2
bl
TC /
.
'
=
.X
1
.,.
Page 4 of to
1
-L
Lf-
[
+
/
-~((,e)]de
[e - ~~e)] (/3
rfij<
- 0\ - (" -0)] =
Ll
~')
~p-
1
14. The 1nfi rote h annomc senes ISgtven b y 1 + - + - + - + .\.
2
3
4
.
.
'
°
...
1
1
1
2
3
4
The 1nfirote al ternatmg hannomc senes ISgiven by 1 .. - + - .. - +
Which of the followingstatementsis true?
a)
b)
c)
d)
@
Both series converge to the same sum.
Both series converge, but to different sums.
Both series diverge.
The hannonic series converges, but the alternating hannonic series diverges.
The harmonic series diverges, but the alternating harmonic series converges.
15. Find the open intervalof convergencefor
the power series centered at 10,
I
.
n!
n=\
a) (5, 15 )
b) (-5, 25 )
c) (-20,40)
=..
L
1
~
L=o~.Lt.1{"
.:::
{l
+--0.
i
~
J1
~
.A
~
I
/71. +-1
X,
0
~H
16. Find the sum of the geometric series 16 + 12 + 9 +
~e) the seriesdiverges
c) 40
0
/1- -to
- "'--
I aAI
It.-{~ =
'1\..-'>00
a) 32
d) 400
/O--~+I ( ~ L
L= L
/)l-">~
<f)
~
~~:
C1J(x-10)"
~~1~~
-=- L
<:»
0£)
)
...
(
+-a.br
'I
11
~.A::
~..a:::::lf.
/A--ILL ~
.
~
~;t; .0
ih
..!L
I-It
=. --::> .:z./b~1:='7,J
J-~f
.
NAME: INSTRUCTOR:
CALCULUS 0
17.
Two forces
ALPHA NUMBER:
SECTION:
FINAL EXAMINATION
4 MAY 2009
F;
and
F~
t
with magnitudes
4 and 2 Newtons respectively make angles of
30° and 60° respectively with the positive x -axis.
If ft = /5; + F2, the angle F' makes with the
positivex - axisis closestto whichof thefollowing?
---
~
a)
c)
d)
e)
-
35°
~
45°
50°
55°
J(
= -<+i!h(~O.) 1-~
Fl-
'- ~ i-r.>("'Do) ,.;2.
Fr-F; = < f3
;2+ g \
~-
=
F=-
.)
e;:
I
(
[30.»
r
)
Page 5 of 10
::=
<.:J.(3) ;;.>
fi
L~Oo»..:= L. 1)
>
.).+-r/3 >
+1)
.
39. 'f ~
:U/"3 t-- I -;
18. Which of the followingexpressionscouldresult in a vector?
~
c.Q
a)
---=
d) (a. b). c
c) (a. b) xc
~
axb
19. Which of the following is an equation for the plane
going through the point ( 1, 1, 1 ) and perpendicular to
the line x = 3 + 2t, Y= 1 + 3t, z = 5 + 4t?
~2x
e) none of these
'1-:::..
3rd:
~::/~-C
&=S--4t:
+ 3y + 4z = 9:J
b) 3x+ly+5z=9
c) lx+ ly+ lz=9
-"'>
-,
d) 2x + 3y + 4z = 0
~
e) noneof the above
v= L.~ .3)f>
tJ -:=
(
2, $,'i>
'V
~ ~(i--'tp)+b.«rcr)-t-C-(-2--+..,);o
~
=) ~(y.-.)) T-3(J-/) +- "lLc-l) =-'V -~
20. Find the vector projection of b =< 1,2> onto a =< 4,3 >, pro} ab .
.
-
.;)..'I+3(J+- -t: -
y
a)
b)
c)
4
2
9,12
<-->
5 5
(d)~H0
e)
,
--"'I
.
~
Gl
-"'1""-'
't-=
b
ifd..
x
{t.?E ) Cf
J;[ j;J
=.
«
II,-)." £5>\
5') L 1/3?
5"
< 8,6 >
=: ("/;)
:=, ~LS
<,,/,
5"3~ =
) ...t
6- '7
oJ.<
t,3,->
-.".s-
9
CALCULUSn
FINALEXAMINATION
4 MAY 2009
Page 6 of 10
PART TWO. Longer Answers (50%). These are not multiplechoice. Again, SHOW ALL
YOUR WORK ON THESE TEST PAGES. CALCULATORS PERMITTED FOR ALL
PROBLEMS EXCEPT 27,28,29,30.
21. a) Use your calculatorto approximate
J(
answer.)
f
,
I
0
eX dx. (Show three decimal places in your
~A (/. "02-)
J 1-) °/ I
)=
/. 'tb.3
b) Simpson's rule using n subdivisionsis
Ax
8n = 3[f(xo) + 4f(x,) + 2f(X2) + 4f(x3) +... + 2f(xn-z) + 4f(xn-') + f(xJ].
rl ,
Use Simpson's rule with 4 subdivisions,84, to approximate Joe x dx. (Show three decimal
places in your answer.)
£
C
2>if-:::
~
. 2:>-~) .7:) .11~
1-, 'f
~
[ e-
b'
~,
!..
+
I
1/;",
'f e.''" +-,;LC: -f-"T e
]
...1-
1-e--
/.. £/t,1
i-~ t.+
22. We learnedthat hydrostaticforce = (pressure)x(area). An aquarium6 ft long, 3 ft wide, and
4 ft deep is fullof water. Use the fact that water pressure at a depth of d ft is (62.5)(d)lbs/ jtz
to find:
a) The hydrostaticforce on the bottom of the aquarium.
,
'I
Jb ~~)xC6NA-)
{,z5}{1J{!j;X
3,e)f=
1SooIL,
b) The hydrostaticforce on one end (4' x 3') of the aquarium.
4A£=
~F4~"r1~~~
= ~Y(~r=
~
~1
~
(/
L
(p'5';i-Jf {3u)f'
fl(&2.S-Y:)t: 4=; 1600
)
A
.
i-
'.
"
NAME:INSTRUCTOR:
CALCULUSU
-
FINAL EXAMINATION
ALPHA NUMBER:
SECTION: -
4 MAY 2009
Page 7 of to
23. Write out the first three terms of each of the followingseriesand tell whether each series
converges or diverges (you mustjustifyyour answers):
f
a)
~
L
lS-
::;
,101411+]
+ --L-r
~/.3
L
~
1
I a",1 =
/1I...~OO
11\.~
;{~t;:!.~.1
f
~
"=1 3"
b)
~
-
~J--rL ~
~ G1t
1"'+1
~D
==
G""r0/~
/1I.~OO
L
~
~=
/ a~
00
Y3
e>O
~
~
3
1-= .~ <-1-
/ItL~-"">
~)M;t.
LaJI/~
L
L
r)l ~
1
1+-'IrI...
fJ~
I
=- ~
-2
A-'>oO
~
L
~
.J
+1)
~
(3"'V~
~~
~
fr<*
L
/ttz-
/It. -"> oC'
L
::
L=
;::::
"
+.:t- +--L. -+-,. .
1
.:27
.3
-
<=
-r'
(/7\)
L
~
l~)' ~ =L ~~~.~
=g
-1
~
-"7
ex:>
~~~.
~
24. Assume that yet) (lbs) is the amount of radio-active material at time t (hours).
If yet) satisfies the differential equation dy
= -.069 y
, and y( 0) = 10,
dt
solve for yet) and determineapproximatelythe half-life(hours) for this radio-active material.
~ ;
}=-.tJbij-
~z-.oMottd,St~=f-'1Jb?dt9
h I I (-.O"lt+iC\
c -.Ohi:t
~
= eh(jl'= eAe
11
=
~/jl
~
~
~
-oN>?
A e'
='>
<- 069-;{
--. Ofo9t;
-
Ai;:;::- I 0
~
tM 4
'D
r+c
.
~
C1
A
,
.£1(o)=IO~
)',
IO:c;t-ep,Z/O
()
~=I
.
~ f:.
"'IO~1
4J,T4Azj-'-'
/
~
Et~d
~
c)-IDe
;I,
Nt}"..
t;
-,OMit
;t ~;~:~ ==I/O k
(
"
CALCULUS II
4 MAY 2009
FINAL EXAMINATION
Page 8 of 10
=(1+ xr2
25, Find the first 4 non-zero terms of the Maclaurin series for f(x)
Maclaurin series coefficients directly.
by findingthe
ill/...
0 +'f
~(t)~
~ I(j.J
/1/
-'6 I(yJ
V
.
~
3.2/1 let
~
(,,\
)-+ ~
{
-,.3.2
~
~
-2-
1! ~ ~I
b lo)r1J°0-+-1(,1~
I
J...
~
If
=' D ~OJ
,\ -5.1
C
u)=-
,
~ ~)
-2( /+1f'74
~
~
r';", jC'J =.L
~
III
-
/
.3
.iLz. - ~.$-I X- +-
T
' .
)
;;3/-
2.
3
= I -).. 'f.- + 3 'J.. - f
1-
t
,J
I
I'') ='0 (D)~-1.~.2
26. For the triangle shown on the right whose vertices are the points P( 1, 1,0),
and R( 4, 2, 1 ):
+1t!..1
~ 3/ +,.,
~ 'I- + 3>
3.2
III
!$
-
Q( 3,2,2),
-
Q(31).,)..
)
a) Find the vectors PQ, PR and the interiorangle ().
..,
">
PC(::;
e
~
--<- ~
~"(
I I) ..2.
»
r ~ =- < ~ /}
~~~
JfQ/ I rIGI
I '>
1'lJll")~
\
\-1/9~)
--=- .;)5":.;L
'> 0'>- . 41b) Find the area of the triangle !l PQR .
~~
~
L1rQ~:=
~
I
/Ze4/_/1)
=~-.; b+/r2-
~
A
?f< X pr;
A
~j
I
3
11
"-
J
-;I
_.v
l
d-.1.1...
-
" .
::
V /+-1(.1-1J
I
'
I
~
3 I
- "
I
J
2-1
< i ) -1
1
I
t-/'
2-2- i
'>
)
~t{ii::-;C
c) Find an equation for the plane goingthrough
the three points P, Q, and R.
~
l'{{li 0)
~
~!)
=" I(t.-I) - 'liJ-I) .rI@-,,)
ti-rf(j+t:-=-3
-'I; I)== 0
~I
1L-l 2. f
I
ALPHANUMBER:
SECTION:
FINAL EXAMINATION
4 MAY 2009
NAME:
INSTRUCTOR:
CALCULUS II
Page 9 of 10
CALCULATORS NOT PERMITTEDFOR 27,28,29,30.
27. a) Evaluate
f: (4x3 + 8x + 9) dx- .
=
I: == I r4-t- f =='/1
t.J+r'/-r9f
b) Find the area of the region R sketched on the right
bounded by the two parabolas y
2
y=-x
~
- 4x and
~
.
+2x.
~
=
= x2
0
~
-;(>1
1
:
J
-i-lzX
1-;.= -,y..+-2-'/.
"/
~1~'~~O
:L
x
:2.1:.(t -~)=()
\
r(- )(~1-"-)
-( 1--41.)] 4
L~
t..
Z.
z-
'I.--
~
0 rL
3
.
0
J
=
3[ -2-l-l-{. '" 1 04
1)
=
3
-3=-f+3f
-3
3
.3
2.
/a
-;
=
?
'-
[3) + 3(?)
-~~+~7
~
9
==-
28. Evaluate the followingindefiniteintegralsshowingall of your steps:
~~~:
a)
fit~;::: ~(3)()
f[sin(3x)]IO cos(3x)dx10
J
-=-
-b)
f
~'ik
~
~=- 3&nf?cJ4
~
i:rIA-=CP
(},x)
~
1/
II
~
b
.kC (I
---T-L/I
7t -11
(t-l)(t-2)
4"
-];6TI
=
d
t
~(Pt)J
-"
) Lt
- ~Ali-'!-+3hlt-z./
rC.
~(j~;
71:-/1
~
OPt:-
--L
63
Ll
g
==~t-~
6t-t;{-t-z) t-j
=9
7i--i/~
tJ;-f:::z:
:=. fA I (j-04(l-~t I +c-
If-II == 01- f!k-/)
~ ~~3/
i-2Ltt~/:
+-t
!t6t-z.)+f3(!::-t)
7-/1.::;Ai-;)
~~
CALCULUS D
FINAL EXAMINATION
.
f
29. a) Evaluate xe
= ).-{IV5'1.
::;Z
Sx
dx showingall steps.
f lfk
J
.5>"-'1-
==
-
AL==X
Let
.>
Page to of to
M4<-~~~~
166 r;hr: e
!:>-t.
1~ - ~
s
1e
S-
4 MAY 2009
11-
c/M-~14-
~
.!>-J(
~
~
~
-
.s Y.
~£;;""
.5"f
§~
+-C
b) Use separation of variablesto solvefor y explicitlyif dy = cos(!x)
dx
y
d"-71
~
UJ{3X)d1-
Ji"'d = J ~G1-)J'f
1 if = -i ~("K)~
~
3
~
0' =
~
30.
~
{M-\+
3
.
3'::
-==?
11 ~
(J=
Sketch the region R bounded by the curves y
= x2
('51-) +A
'
and y = x, and find the volume of the
solid obtained by rotating the region R about the x - axis. (Use the washer method.)
i
L\v=d1/~
/L
-=-
f.
'fflI2L-h-~)AI
-=-lI( &/_(1:;1:)41-
v=
t Hl l- '/)4
\
0
3
=- 71
=r
{{~
5'
~~
{3 - 5'J
.L
L
:z
)1
/0
)
~
£---~\ =~11
v'l /6 J IS-