NAME:
1\£1
.
ALPHA NUMBER:
SECTION:
INSTRUCTOR:
FINAL EXAMINATION
Page 1 of 10
SHOW ALL WORK IN TIllS TEST PACKAGE
CALCULUSm (SM221,SM221P)
1330-1630Wednesday16Dee 2009
Tear off the last sheet of this exam. It contains problems 27,28,29,30. Do these problems first
WITHOUT YOUR CALCULATOR When you are finishedwith these 4 problems, return the
sheet to your instructor, take out your calculator, and complete the rest of the exam.
PART ONE: MULTIPLE CHOICE (50%). The first 20 problems are multiple-choice. Fill in the
best answer on your Scantron bubble sheet. Write your name, alpha number, instructor and
section on this test and your bubble sheet and bubble in your alpha number. There is no extra
penalty for wrong answers on the multiple choice. Show all your scratch work on this test.
CALCULATORS PERMITTED FOR THIS SECTION.
1. The rectangular box shown on the right has one vertex at the origin
and the opposite vertex at (2,3,1). (The positive x-axis comes out of the
-!-
(:>.,3,,)
page toward you.) The vector AJj in the sketch is:
b) -'11 +3J+k
c) <0,2,1>
e) < 2 -3 0 > A{P/o/,\ B(!-; '3,0)
, ,
a) < 2,3,1>
(J)
d)
'11+3]-k
.
~ AE =<2} 31-I)
2.
,
@
a) 0
d) -9
,
b) 2
e) 9
c) 5
(
<,<11'"
~
~
3{..= 4'/. ~
:::;
0
b) 2../2
d) 6
I
~
~
't::;.q
=>;
~~(f~1
4
@
~I}-7 X)
3C. -7 X+3~::::. 0
3. The area of the parallelogram
on the right is:
'/
a) ../2
c)4
e) 6../2
;I.
.
/ , ,,~).
Find x so that the vectors <6,x,3 >
.
and < 6 -7 x > are perpendicular.
I
~
i
"I
It-
1
D I::;
:;L;;LO
~7Z
)<1/)o»)c<':</~o>J
Ibti
0;
~
01 +(g-~)L
~
4. If f is a scalar field and F isa vector field, which of the following expressions results in a
scalar field?
@
a) divF
A-i'
b) grad(divF)
~i
c) div f
d) curl F
/1L,4,
q,i.
e) none of these
CALCULUSm
5.
C9
FINAL EXAMINATION
16 DEC 2009
Page 2 of 10
=~X2
The contour curves (level curves) in the xy-plane for the function j(x,y)
a) paraboloids
b) cones
c) circles d) righttriangles e) spheres
1/ ~~f
~
6. An equation for the tangent plane to the
surface z = e3x +sin(y) atthe point (0,0,1) is:
a) z = 3x + 2
c) z = 1+ 3x + y
=(
(J2. '
,'Z.
z...
"1-+ff
~
@
+ y2 are:
=~
;!-c..= ,,",-(y:-t..)+~(J-J")
[-to, J"J2;;0)::; {o,o, I~
(\
Jf
~~(J:
I .=. 3c, 'j
b) 3x+y+z=1
d) x - y + Z= 0
771~o)
- W'!1I
At. =- a~ /
e) z = y
.
-~
-
- I
o/fe.oJi- 3 't. ~I d--
JJ-~IO}
-2: =.1-
=--3
!f!?,oJ
7. The maximum directional derivative for
the funcUonf(x,y) = xy' at the point( 2 , 3 ) is:
@
a) 288/5
c) 6
e) 15
b) 1
d) 12~
}
\
Jy..
=
8.
= ~ L ~ i.A~)~
"" I-y'~
r
J
<9
.
(2,3)
/ (9/2.>1
= tfglr/~1
t
\
::!;
1 x2
I
@
a)
we get:
z.
~-1-
I
fo..fYfeY
I
f0 feY
dy dx,
0
L + d_tfi.
41--t\ cy,
dxdy
I
d~
~l
,
c) f feY dxdy
0 0
1 y2
0 0
1..fY
e) f feY dxdy
0 0
~.f-
f
b) f fln(x)dxdy
o..fY
d) f fln(x) dx dy
~
=
f
1=-/
~
J C~4';}
~~Ii
J~o
~
I
I~ ~2.~J>I~/:S
=- ,)7-25"" -=-1S-
When we reverse the order of integration
for the iterated integral
(Z,3)
I
ALPHA NUMBER:
SECTION:
FINAL EXAMINATION
16DEC 2009
NAME:
INSTRUCTOR:
CALCULUSm
i.
3 .J9-x1
9.
f0 f0 sin(x2 + y2)4Y dx
When we convert the iterated integral
.$
ftom rectangular to polar coordinates, we get:
1f/2
a)
3
21f 3
f f sin(r2) dr dB
0
0
b)
1f/2 3
@
c)
e)
f f sin(r2) rdr dB
3
0 0
0
3
J9-( rcos( (J)i
0
0
I I
d)
f
f
0
I
~ "L
\!
£')1-
,id
J J ~(f)~k~)
sin(r2)rdr dB
0
}<
'I,
3
Jf{?-I
1f ~9-(rcos((J»1
f f sin(r2) rdr dB
0
Page 3 of 10
.
e=-
~
(/..:=-0
sin 2 (r)dr dB
10. What shape is the solid of integration Ein the triple integral
11:
211: 4
III
E
1dV =
3
I JJ
p2 sin«(b) dpd(bdB
000
?
r
(~)
y
@
~...
11.
@
1./
r
.y
)/{/
)(.r
~
Suppose that r'(t) ==<2e2t,3t2, cos(t) >, and 1(0) ==<1,1,0>. Find 1(1).
M...'tj=-< e~C!.-,-*3~2. J ~~?»
)
a) <e2,2,sin(l) >
2
b) < 2e + 2,2,0>
PiG\
IL ~
c) < e2 + 1,2,sin(1) >
d) < 8e2 ,3, sin(l) >
e) <8e2,4,-sin(I»
~
~
::;
'-
I t-C-
({
I )
-
/l-(i:) =- Z e.
ZJ
C- '/
3
"'-1: JS
f\-+-I ~
.2.J
I,'
e., J 2.. I ~ltJ
72fJ) =L
=- IN-i-) he
=< I I 0>
J
>
,-1...)
I
. I.J.-\
>
12. Use Green's Theorem to evaluate the line integral
Ic 4ydx+(5x+
@
y)4Y where C is the closed curve boundary of
S
c.
the triangle below, oriented counterclockwise.
:1
a) 0
b) 1
Jc) 2
d) 3
e) 6
3
r4+QI.,~ [f(drJ-ll:\/;f
d ~ ~ Ji)
= fJf&-,'J'
l:.
i~
l-d l'
- JCtl!
)
J;;
rlA-
I(b--f)M= ~{R)
I<-.
~
~)&J(3) -=3
CALCULUSm
13.
FINAL EXAMINATION
The length of the curve r(t) =< t,t2 ,t3 > between the points where t
closest to :
a)
b)
c)
d)
e)
@
16 DEC 2009
1.3
2.7
3.4
3.7
4.0
;f::q
~
.
~
J
Page 4 of 10
= -1
and t = 1 is
I
17L~/r1k;
1::::..
-/
j
.::::.
J 1< IJz-iJ3t~>~
-/
I ,tJ 1+'I~'+-9t1
'
k
~
~dii
~ I
3.7
:p
4'
14.
Use the Divergence Thm
the flux [~F'tiS]
[~ F.tiS = I!Idiv F dV ]
to calculate
if F(x,y,z) =< 5x+ y,z- y,x+ y > and Sis the
;1...
t
boundary surface (6 faces oriented outward) for the box E
shown on the right.
~ F=~'F - i("~3)+t('-aJ+i(..JJ)
a) 0
b) Jr
~
@
:::: .5-/<1-0
Jr2
d) 6
Iff ilV
e) 24
~
=-
4-
<
=-1-
~(~)
)
=- 1- (ILZ<3=-21-
15. The path on the right is the graphofwhat function?
a) z
=cos(x)+sin(y)
for 0 ~ x ~ 6Jr and 0 ~ y ~ 6Jr
b) i(t) =< cos(t), sin( t), t > for 0 ~ t ~ 6Jr
c) i(u, v) =< cos(u),sin(u),v> for 0 ~ u ~ 6Jr and 0 ~ v ~ 1
(l)
.
d) i(t) =< sin(t),t,cos(t) > for 0 ~ t ~ 61f
16.
The function f(s,t)
table on the right.
(])
)(
y
e) none of a - d
a)
b)
c)
d)
e)
-3
3
-2
2
1
is given by the
()
Of
approximately:
os (1,2).is
.
tu,L)~
£(~~;tJL)
=:. 9 ~~ =- 3
,;;L..
I
2
3
0
g 5 J.. -I
I
9 7
3
10 1 t 7
J..
3 JJ. 1/ to 10
NAME:
INSTRUCTOR:
CALCULUSm
17. Let z
::;: x3
ALPHA NUMBER:
SECTION:
FINAL EXAMINATION
16 DEC 2009
Page 5 of 10
+ y2 , where x::;:et and
y is a function of t graphed on the right.
Use the chainruleto approximatedz at t ::;: 0 .
dt
a) 1
(])
1-
j!:::,
b) 4
d) 7
c) 0
e) 12
-1
j\
x
'I
1
~=~!:f,.+~b:
L
Jk
t:
Ie
Jy. J;t;
.:::r.
(3~'
~-c .,., .2~A
tJj::
What statements are true for
-
d::'o'f
y
- curlF=O
- ,
. -- div ~> 0, curl ~- ~
a) divF<O,curIF:t:.O
c)
@
d) divF=O
-+
-+
,
-+
.P
A0
-L
;:;d F ~ .1A.,J.v 1:.
----
curlF:t:.O
'f7-p= ~1-'-!:"+-~t'J1~=O
ch<
cf)'L
O"t:"
~..".
-.. -.. -.. -..
d'i.. ' 0
a;>
aft !J R d6.
~) -
~F=
~
b) divF=O
.
~,,7JJ
F;
y:
0
A..(6-0/
n.
I
/:
e) divF> O:curIF "JJ
0
x
Use the contour map on the right for the function
j(x,y) to solve problems 19 and 20.
+2..}.,AJ
=- 7
2//3'
- dF;.
oF;
~
7f
;?>t>oZ'>
I
F at the point labeled P?
- -
)1t~ = 3./
F-- < F:Y
18. The vectorfield F is shownin the xy-plane
't::-e. ~I
d_~~L?h
ad- ~
and looks the same in all other horizontal planes.
a;t-t~
1:
1:
j
..()
I
'" -J'(0 ~~
~}
.
.
..
.
r
+-1J..o-if) 0'
0
~
8 8
19. Approximate
JJf(x,y)dxdy
by subdividingthe
00
4
region into four equal squares (i.e., n=m=2) and using
midpoints to compute a Riemann Sum.
@
a) 16
j
s"
Q
b) 76
tJA.:-
0
~
c) 100 d) 224
(" X)(4t!
[fr,~J..
e) 304
t(,,z)"I/z,.).
-
6{,,tJ]
(1-)(4\L 3 + S- -I- S-+ ,,]
.:= It [19] - 304-
20. Evaluate the line integral Ie (Vf) .di , where Cis
@
the curve given parametrically by x
=t
2
+I; Y
= 41 -
FrJ.X
2;
where 1::;;t ::;;2, and the level curves of f are shown above.
a) 0
b) I
c) 3
d) 4
e) 5
J(
g
Q
t:=./;
t::;: /
S
C
vt. .u
~~
~@} --!fir)
=
I (J =-1-D) -:1--;.2. ~
-t=z; t =.£r:z..==-,,/
(J==-1tz)-l-=<>
~
=+ 6{,,6)-t(..,>-)~
{.-3
~
3
A-1~1'2.)
,84,')
CALCULUSm
FINAL EXAMINATION
16 DEC 2009
Page 6 of 10
PART TWO. Longer Answers (5()o~). These are not multiplechoice. Again, SHOW ALL
YOUR WORK ON THESE TEST PAGES. CALCULATORS PERMITTED FOR ALL
PROBLEMS EXCEPT 27, 28, 29, 30.
21.
Let P(l,O,ILQ(2,3,4), and R(2,2,2) be three points in space.
a) ..,,-,
Find the angle () betw
~ een the vectors PR andPQ.
.
PI:..=-
<'ll} I>i
nl
TZ.
"PQ
~
/15/
3/
I n.1I pQI ~(f7)
I
1+1.+'3 = ffi r1~/ J 1+r~?' u:&!
-
""
="!>
$
~
~
f(!
)
.
i).i)
Q(~I 3~tf)
-
e.
~ &:!rIIc, =.,21).S'o ...
/((~/:JJ;)..)
.3Slf,uL
b) Find an equation for the plane going through the three points.
--"T
" A 1'
N== pl(.)( PC(=:
.( J tt:
I ~ I
J J 3 3
=:
<31 -.;1.) I>
(
,d;p36-1) -dJ-o) +dc-~=o 6'i.:3/--":1"2
=
({it- Ji.)+b(J-j,)+d~-l..)=
~
c) Findparametricequationsfor the linegoingthrough P and Q.
1-= I +- I ;(:
1::::-{'..t- ~ t
,
- C><$..(i::- <..c;JO ,
{
~~ J. 0 ~h
~
:4-;; 0 ./+:;!JT
~=-'-C::c
+-L-t
~..:::.I
t: )
22. For the functionj(x,y) = x3y+ 12x2-8y,
a)
find the critical points.
J",
l}
=
3
11
b)
:::sy'/.;'2-
~/(!-&
3'1-J- + :2.1'1-=o
== 1'1
-
?~
..
<6
=-0
-=>
/Z4J.."'1-8=0
~
:::
-i
-:= {2J -~
~
~
~~~.
For each critical point, determine whether it is a local maximum,a local minimum,or a saddle
point. Use the second derivative test with D(a,b) = jxx(a,b)jyy(a,b)-
-i'
4r
; fJ =~"
D{-,,_of\= (§1J M){O) - L~'I."7£
t?,..y.J= -/41
=
c,t:r~f j fJJ ~
V.w(a, b)] 2.
c>
<0
~('>J-<fJ
,
MI...-
~
,
f
ALPHA NUMBER:
SECTION: FINAL EXAMINATION
16 DEC 2009
NAME:
INSTRUCTOR:
CALCULUS m
23.
Page 7 of 10
Sketch separate graphs in R3 ,3-d space, represented by each of the following equations:
=6
a) x+2y+3z
b) y
=x
~
1::
~~
;l...
'I
'I
i-.
c) z
=x2
d) Sketch the solid in 3-d space whose
cylindricalcoordinates satisfYthe inequalities:
+ y2
7r:
2~r~4
~
7r:
, -~(}'~-,
.
cc~
~"
2
0~z~2.
(j~
t
t
x
1<
24. A soccer player kicks a ball from the ground toward the center of an empty goal (no
goalkeeper). The ball leaves the ground going 50 ft/sec making an angle of 60 0 with the ground.
If he takes the shot from 60 feet in front of the goal and the top of the goal is 8 ft high, will the
ball go into the goal in the air? (Ignore air resistance.) Show the acceleration of the ball a(t), its
velocity v(t), its position r(t), and solve the problem.
(ji;;= <
~ Vfjj
V~)
= J a.(f;Jck
="
<
5"0
=- L
071:1J:=
~
'7
--'$2
6 I
7l(-fj
f;{)'
riiJ-
~ L... (1. J
~~o,)
t
5O~l(,t)°J)
= J t/~Hb
-/(.
@f:k~
;JSr-
.
=: <e.11
~~>
(j)
2-
= 4..x>-t--H!-I)
-/bt+~~-€:t+ez..>
~ L---- J
'tf!;-)
'-..25", ;2-rJ3"/
+ ;;L'.)cJ3
7f1.0).:= Lv} 0'>::::.. L.C?I
;J.~-t;
=
iJ
t
fL
~6Q
3;)..(H!1..)
J -!)o if.. '7
I;l.-
z- :;...5",-32
/
~y
tl""'\
/! L-'>
~.
L~~j-rJi;
t>
u.~
r
fA) ~
(j(t)
r~1kr
-t=2.'/-.
~ -I&f?~ w,M2.~
=-
~".
~
~-40
-1=&<.1"3-
1/.
't
f
~
Lf.,~-J;L tJ(
f
<J
FINAL EXAMINATION
CALCULUSm
25. Considerthe functionI(x, y) = y -
~
16 DEC 2009
i
.
1-
a) Sketch the level (contour) curves where
I(x,y) = 0; I(x,y) = 1; and I(x,y) = 2.
=r;_J1:=-O ~
,r; -'=-1
-
3
~
=1f+1
~
= ;;~
gb F~d V/(I,1), (gradientof I at (1,1», and
- rr;:=z. ='"
Page 8 of 10
~6((1')
+--L
draw it on yourgraphfrompart a). -~
V A r < I" J'1->
~"r~ 'f- J L> ~ IJ
D
D
-==-
I
I -DQ
c) Find the directional derivative of
I at (1,1) in the direction
h~~ (/,,) = 1'£(~I) 0 ~
26. a) If j(x,y,z)
-~-;.-I '> . L:..fJ
~
~
b)
A.-),
CD
@
I
G)/
At-
i
d/
7J,a ;lqc
'!t
18 Jtc
L.
~
D.
@
'I
/J'I
-. - -'">
F
~ J
==
..,
01-1:
~6.
~
.5
~
~
z.
L(
~
-j~.
;fA ~
~
rO
.
-.
c
~(f )
-~.
~ 7p~
+I{E
-
L::.OJ0, e:>
)
for the vector field F(x,y,z) =<y,x+z,y+
~d
J ~I
cJ,.
o1J
Jl
4 >= ~
~
J-t=
cJ I
i+-~
J
I
2z >.
1-2-~>
.
J(rJ
..cL Lcf'J<v11=
AL-H:
~_~~(Ji.~n:~0
Jfjl")~(1;<+~)
cJa-
1-droi!:+1(~
66,V-) =
~
.4- =' v.:c
~l
-
4/ +3J .
=< 0,0,0 > ).
Find a potential function j(x,y,z)
-
=.
3 '>
Vx VI
~
Vi = J
\7i
'f.
is any scalar field (with continuous second partial derivatives), prove that
curl (grad j) =< 0,0,0 > (i.e.,
v
1
:3
I
- Yz..I L. '>
~ L-.
;2-.
~
(t I I)
L
I
'"i-(J
+
"
.
1ft-:(1+'>"~~[ 1-J +3i: +lflJ] =:r~lo.=)1f4J
~
~(1-'Yc)=1'-:r(ju~~~
~j==
Pr~ =>~~=~~
4
V£::F
NAME: -
ALPHA NUMBER:
SECTION:
FINAL EXAMINATION
16 DEC 2009
INSTRUCTOR:
CALCULUS ill
Page 9 of 10
CALCULATORS NOT PERMITTED FOR 27,28,29,30.
4
27. Use doubleintegralsto find x , the x-coordinate for the center of mass,
for the thin triangular plate with vertices (0,0), (1,0), and (1,1), if the density
of the plate(mass/area)is givenby p(x,y) = y .
Hq
1:[=
.
Jf1-fJA
(f
f( r JA
~
/Tt'l tA4
,)(
~=
'
I
0
1-1-
1,
I
~
~
/'f
-:z-rr =-
:=: JL.
~
~I~
1~ i2
J
0
~
.'
3
28. a) Evaluate the triple integral
:::::
'I
3
fl
.
~
4,.'
fdr.=f{-t
116 " ff.f':d~4--j'{~d4=1
~
3
Jf£f1.]4=f.~4=TI.=f
.
ff(}"<14=
0
.
~
"}..
2
I
JJJ8xyz dzdydx.
L4~/l,~j~4i'{4f(f~~
"1~xl~~:14
~
1- '/.z- /3
1/1.. 4::::r
/0
= 4. 9 3'
::.
~
b) Evaluate a triple integral to find the mass of the right circular cylinder
on the right ifit has density p(x,y,z) = x2 + y2 (mass/vol).
lfS rr/V ~~~~
~~
7.11 3
.
:=
3
~ j
f J S ~
)(
ILrkrAle
e:;:;.oh;° ~=Q
~
f"
t J~IL: f
lA£f)
~7r
~
...
=
r 1- / 3
J
~/o
't&.:::
-
1.j
>rr
J: 3/~
~
+l~£(9
;=
/
;3/e/C7
:lIT
-==/b:J-7I
y
.-
CALCULUSm
.
FINALEXAMINATION
16DEC2~
Page10of 10
29.' a) Evaluatetheline integralIi!'.di , wh~e F(x,y,z) =< z,x,y > and C is the lineiTom
.
(l,I,I)
to (3,4,5)..
(!:(J~r
"/..- I r z.:t
;\: I <-"t-
~
~
--
L
~
f
i 0"
-t
/L(;f) = L...
j+'L
.~
I
.
I
II
~
.
-F.~
.
.
'i= I -I""it-
~
c
#\
-i-?T 1~1t>
J
=-
4> Jj:
f
'\
~"
lJ::
,j
( :2,311 '>
C.
\
=
-'G
J L.~ 1~j'>.
=
~.
b.{
-<
;(,
x~ .JlI ~?
I I.;L(1+1~)+?(/~rt)+-ilr~-t))k:,.
l
')
.
0
~/ I
1 ' 10+-db -k) k =
\.).
0
.
.
~t+/3
t /0
=~
b) Is the value for the line integral in part a) the same for any path from (1,1,1) to (3,4,5)?
~Ian
.
No. T~~
F~ ~
:::::X:
'
.
h
.
.
.
~(f~ -
.
"-rt~
(.
l. ~
Y')( F =-
.
,
:;;-
lr>y. 71(}71)~
Z:::- . 1-
30.
.,. -0
) JI lH.;/
~
+
lI ) +
L.°I ~~ >
.
.' ~
:f
Evaluate a surface integral to find the flux, IIF.dS,
s;
s
.e.;.r-r-L
where F(x,y,z) =< x,y,z > andwhere S is the portionof the
surface z =9-x2 over the rectangular region R = [0,2]x[0,3]
oriented upward. (See the4igure on the right.)
'I
(f~~~S:
-
~ =-1-
\
q
2
>
'/..
7 ~d:S::!L(\L-Sj,-1It IeIt, Ju 4
(r~<f z
~-=q-'I-
1-
()
1S =z '\: j
:~;~ ~
I
A -(J
k
.
JM4-
I
0 - 2."1-.
()
0101
::::r;
<. :;J.]<j
0 J ' /~rlt.
2..
y
.
.9-~
h
ffL-1-'d,/>.<~tJoJ'>~'I
~fSF.;&=
s
=
d
1<.:
f 5~/::;<1_J()1J4
=-17-&~9)(ff~:)4
r
a
0
0
/'(,}+
= ['(31-\-;'7)
0
~
4- = 1-~~7¥-!o
=0 'if","sf = 6;1.
1::2-
'I