-SOL(j1lotJ5
NAME:
ALPHA NUMBER:
SECTION:
INSTRUCTOR:
FINAL EXAMINATION
CALCULUS I (8M121, SM121A, 8M131)
0755-1055 Monday 15 December 2008
Page 1 of 10
SHOW ALL WORK ON THIS TEST PACKAGE
Tear off the last sheet of this exam. It containsproblems27, 28, 29, 30. Do these problems first
WITHOUT YOUR CALCULATOR. Whenyou are finishedwith these 4 problems, retW11the
sheet to your instructor, take out you calculator,and completethe rest of the exam.
PART ONE: MULTIPLECHOICE(50%).Thefirst20 problemsaremultiple-choice.Fillin the
best answer on your Scantron bubble sheet. Writeyour name, alphanumber,instructor and
section on this test and your bubble sheet and bubblein your alphanumber. There is no extra
penalty for \\Tranganswers on the multiplechoice. Showall your scratchwork on this test.
CALCULA TORS ARE PERMITTED FOR THIS MULTIPLE CHOICE SECTION.
1. For the right triangle shown, which trigonometric
AL:J
function applied to e gives a result of 15 ?
2
sm
a)
~
~
d) cot
c) tan
b) cas
2. For the triangle in problem #1, the angle e (in radians)is closestto which of the following?
(!J o.~
b)
3. If loga(x)=2
11:
/6
and loga(y)=3,find
@
a) 0
4. If [(x)
O.SOc)
= X2
+ 1 and
d)
logaex2/y).==
c) 2
.
b) 2
d) 4
r-
1/-3~~I=J
J
i
I
,_:J3
1
.:1
'f
-
i
Q
"'--'
Find the slope of the line through points P and Q
( P has an x coordinate of 1 and Q has an x
coordinate of e.)
1
b) 1
a) -;:;L
e) e-l
J= :J-{;;;'j-3=1
e) 4
e) 5
5. The graph for y = In(x) is sketched on the right.
d) e
=- 0.4"
y
g-I is the inverse
c) 3
2.00
-'dGl.-((J
d) 3
.
1
e)
02(d",(J)
for the function g graphed on the right, find f(g-l (1))..2
a)
11:
/3
--f..
th:- (~)
;:
~~
/Y)\ :::.~:::
A)<
iK(e)
-
e- -/
~/)
=- -!-:.!.-
e./
CALCULUSI
FINALEXAMINATION
15 DEC 29t8
Page2 of 10
X S; 1
6. Find x-+l'
Urn I(x) where I(x)
@
a) . 1
d)
={sm(lr
2, x),
l<x
L.
t-'f. (y" Lfr
::=;
c) 1
J
7T)'=-0
e) none
11:/ 2
= 3 and /'(1) = 2, then the tangent line to the graph of y = lex)
at the point correspondingto x = 1 has a y interceptequal to :
\
(J -,I"':;;' /Y1t(,:.t.)
7. If / is a function with /(1)
~a) - I
b) a
c) 2/3
~-'-3
~
e) 11:/2
8. If / is a function whosegraph is shown on the right,
which of the following couldbe the graphfor /' ?
==
~ 'f-I)
;.
;;2
;c
3'
=;2'/--2-
f.-Z r3 =
,;).'" + I
.~
~x
-1
-1.
Q).--/
b)
right: then the graph of
maxImum
/ has a local
1
b) 2
c) 3
~
~
1
9. If the graph of f' is shown on the
a)
d)
c) .
at what x value?
d) 4
.1.
~
:<
e)
.3
"I
5"""')'x
;
5"
f/>-o
-j;;:
~~O~.
b) 2
(E)
c) 3
11. Find the slope of the line tangent
to the curve given implicitly by
X2Y - y3 = 3 at the point (2,1 ).
a)
b)
~
~
e)
1
-1
none of the above
d) 4
,L~
.
~
e) 5
4;
L/~rif] ;; L3]
~.
/if
;210+
-3i~~o
&~ 3k1.2J it = - .2f}0 d1
~
c4
~
- :4~--I
t ~3 U ~
c~!,)
= -~)Lj
=
;z2-""3{,f
-1 = - f
'i--.3
ALPHA NUMBER:
NAME:
INSTRUCTOR:
CALCULUS I
FINAL EXAMINATION
-
SECTION:
15 nEC 2088
Page3 oflO
Use the graphs of the functions f and g on
the right to solve problems #12 and #13.
1
~
12. If hex) = l(x)1 g(x), find h'(1)-=
a)
d)
-3
6
b) 0
e) d.n.e.
~
,//
=
13. If k(x) = f(g(x)), find k (1). 1J3(/J)
'oYj /
::::.
(3) '1(1)
a -3
b) 1
c) 413
()
0>
e) d.n.e.
=:.
p--}f) ( -3) =:
14. Find the rate of change of the volumeof a
cube with respect to time at the moment
when the edge length (x) of the cube is 2m
and is increasing at a rate of 0.1 m1s .
[U'
'f-
%
v=-
£1/
dvf:
y..3
~3'j..2.i:L
J;t
;-,
a)
(D
0.1 m3 Is
b)
8 m2 1s
e)
0.8 m 1s
1.2 m3jJ)
c)
JJ
0.8 m3 1s
=:;
J 5. The Mean Value Theorem states that for the function
f graphed on the right there is at least one number c
in the interval [a, b] = [-2, 2] where
~
-1
2
c)
a)
-0.20
2.
= e -x
3/
-;l.
-J..
.L
'-
~1
and y = xJ + x2 -1 and determine the
of the point of intersection accurate to two decimal places
cG
c) 115
)( ./ ~,4-')
/I'1\. 1;0-
0
16. Use your calculator to sketch the graphs for y
x - coordinate
/,
~
/----
f'ee) = f(b) - f(a). Based on the graph, which of
b-a
the following numbers is the best extimate for c ?
b)
e)
3 (j
:=0
ciA:;
d) 2.72
e) d.n.e.
CALCULUS I
17. Find
a)
FINAL EXAMINATION
0
18. Oetennine
~fJ
X3
x2sin0)
(I (x)dx
c)
3sin(x)
for the function
1
whose graph on the right consistsof 3 line segments.
xJ
d) -cos(x)
3
'(J
e)
-~cos(x)
3
y
3
2.
1\I, '
~t"3 -r-!S--I,S-~S-
GJ)
b) 3
e) 8
~) 2
d) 6
Page 4 of 10
f(lf~(~Jh)
~ (J: t2 Sin(t)dt).
COS(X2)
15 DEC 2008
-,
-/
-3
19.
If the table on the right gives the velocity
of a car at various times, fmd the
average acceleration (ftls 2) of the car over the
time interval [ 1, 3 ].
a) 5 (1:,) u)
d) 31
c) 26
e) 51
--
O/1K~==-JV~
vh) - VCI)= 31£
3 -I
;L
Ai:
~
/3
r/;
20. Use the table in problem #19 to approximatethe distance traveled (ft) over the time interval
[ 0 , 4 ] by using M 2, the midpoint method using two subintervals.
a) 13
b) 26
@
c) 51
e) 102
~,
/12-= M:!)A-t-t-4(3)~t
Sl;}-) + (? IZz)
rA
31t
"11.
..L
2.-
3
ID +-~;L-
k
[
f
if--=~
=2-
,,2...
~
72-
r
NAME:
ALPHA NUMBER:
INSTRUCTOR:
CALCULUS I
-
SECTION:
FINAL EXAMINATION
1~DEC2008
Page5 of 10
PartTwo. LongerAnswers(50%). Thesearenotmultiplechoice.Again,SHOWALL YOUR
WORKON THESETESTPAGES. CALCULATORSPERMITTED FOR ALL
PROBLEMS EXCEPT 27, 28, 29, 30.
21. The graph of a function
a)
CJ
~
y
1-+1; -/~1-=o
- (:;2.'f.+I'
()~1..:=.L
I
x
Explain why I hasan inverse function /-1.
~ -trt-M-"
(~F-
.--I,:.
0
c) Sketch the graph of I-Ion
d)
is shown on the right (two line segments).
Write the fonnula for y = I(x) as a piece-wisedefinedfunction.
..,
b)
I
Sketch the graph of y
--r:L
ic~N:1;j
011. [-II
/1 . LdeJ:)
the same axes with I.
= - I(x)
.
~:~~
.
-3
f -Lf.)
+2
on the sameaxeswith I.
22. Sketch the graph of a single function (on the axes below) which satisfies all of the following:
a) x~-3
limf(x)=O.
d) f'(-I)
'
= 0;
b) x-tlim
- 2- f(x) = 1.,
c) x-tlim
-2 + f (x) = -1 '.
e ) lim- f(x)
f)
x-t 0
= -00.,
i .~
,
_:--
"1 "
;L
-I
-I
.)..
<,)
I (x) = 2 .,
i) x-t
lim
- 00 I(x)
h) x~oo
lim I(x) = I.'
g) limf(x)
= 3.'
x~l
lim
x~O+
3
= 00
CALCULUSI
FINAL EXAMINATION
15 DEC 2008
Page 6 of 10
23. a) Write the limit definitionfor f'(x).
L ~(i-'l)
A -~l~J
lei; '" A-o
V
b)
Use the limit definitionto show that for f(x)
I
_IClf-J = L
0
~
=,,L.
~~o ji::;2d,+~~~r3-'-
£
=
f'(x) = 2x + 3.
-[ 1~3f]
[(y+-jJ~3{~~~)]
1-0
=X2 + 3x,
}((.;zf
~ -..,0
~)/{
+-"-+~1 = .2(,+3
-f-
)(
24. A plane flying horizontallyat an altitudeof 3 mi and a speed
of 500 milhr passes directlyover a radar station. Findthe rate
at which the distancefrom the plane to the stationis increasing
when the plane is 5 rni from the station.
3~
r:.
.
~rlt Ti 0 Ai
= £t:0F-t-~.
_/- -')
JE k
H::='~
?'
.
rJ;t:
v
/]
~{l+3~
/~ I:L
+
0
£;:
-
1- J
-~
il ,)Ie
elk
::
f,-,
J&
.
~-:;:
I
-C: di:;
=
d/<J'
J.:..
5' ~~
:::::- +00 ~/h
,,--
JOO~
:
(f=-I
~
~
~=-'~)
NAME:-
ALPHANUMBER:
INSTRUCTOR:
CALCULUS I
25.
-
SECTION:
15 DEC 2008
FINAL EXAMINATION
Page 7 of 10
A farmer wants to enclose a rectangulararea of 15,000squarefeet,
and then divide it into 2 pens with fencingparallelto oneof the sides.
What dimensions will minimizethe cost of the fence?
I
\...--..--
I
~
1...
~7
~(j~L?=
~
1J-=
r
IS;(X)Q
.2td}
(
~
d~
?(f)::
IS;ool)
-"-)- +
f(r.) =-
2t
~
-' f
d- 'i -I- '1:;;000
X
<0
.
3
(~,oI
~
0 L. j. ,
'1.== /S-o
?'(p=
;
fj'{)/ohJ ?D ~o
1-=-/S-o
r' g=- loof
26. A can of soda is taken from a refrigerator and placed on a table in a room of temperature
70° F at time t =0. Let f(t) measure the temperature CF) of the soda as a function of
time t (mins).
In all of the following questions, make sure that you give the units.
a) What does f (0)
=40 mean? at:. 1;:,..rt
=0
1/
.t-=/o
b) What does f(10)=60 mean?
p.. \ ~
U-<.
)f
L
/I
-.t:zr f-,o
'h
~
c) What is the average rate of change of the temperature over the time interval [0 , 10] ?
~111
=
(IO)(0) =. ~o-'I()
100
/Q
d) What does f'(10) = 0.5 mean?
7k~~
Mi;:,.(,LL
= ;L"F/
.
e) Use the information in (b) and (d) to a~p;~ri~~e
f(12).
f) Find ~~r;;f(t),
~
-d: t=IO~
= 70" F (~~)
()
~
c
~O,)
/
F ~.
I
l. ~z-) ~~j0°) 1-1 ({oj(Il- -/0
D (,,0 +~5'j(z)
J
= t./
.
F
g) Sketch the graph for y = f(t).
~ ,CF)
70 '
(pO
10
10
tJ
i..).::::. J...2.) j -0 C>
~
~
- ~foOO;x~ =
.;2= if~
x~
~
?~
;J..
t
F)
.
L"I..-
r r.
CALCULUS I
FINALEXAMINATION
15 DEC 2008
PAGE 8 IS BLANK AND CAN BE USED AS SCRATCH PAPER.
Page 8 of 10
NAME:
ALPHA NUMBER:
INSTRUCTOR:
CALCULUSI
SECTION:
FINALEXAMINATION
-
15DEC 2008
PaKe 9 of 10
CALCULATORS NOT PERMITTED FOR 27, 28, 29, 30.
27. a)
Evaluate the following definite integralshowingall of your steps:
7/ 2..
1:, (6t2 +6t+6)dt
=.[J. i'
~
= & {./r-e3~/+ "t.- JJ - [ .l{-J +3
r 3 -f ~ ~ i:1-1
~c. i-f2.+/2.) - ( -2...f-3
..::::
=
- (-5") =-
'/0
-,)
'l-S-
b) A particle moves vertically with acceleration a(t) =3e' + cas(t) + (. Find its velocity vet)
and position set) if v(O)=3 and s(O)=4.
A}'"(f)
~
j
/3e--c+-U={,-tJ
+r)Jt-
~
;t/(!/ =- ~
~
N:{tI
/'
==
-' ...,
0
/,(0)
~
=- 3e-t+-~(+-J.;/., -r
4.«--1:::=J0C
4-0)
L
~
==-
=
if
i
.
==-
~
f-c.
)~
(,,y,
I
~6\
j//,
(!
~
=- ,;2. f-C-'~
~
==-2..
c- ='0
3
zZ -f-o (t-J + ~ +c
I!-L-
+-7~
+~ t-~
4'1:1
t;;.
f/
;-~(d
3 e-~~(t-J
~
~
/0
-f-~
-r
)
3E-- - ~ Ef) +;% t-~
A-CtJ =
28. Find dy for the following functions. (0;' not simplify your answers.)
dx
~
a) Y=(X-2)3cos(5x);
v-;
b)
Y=
7
- 'f '13
.
.:2r:
~ f'- e-.
8
cl.1
~4 -
)
[ k~)
=)
1.; -~
J=
(f l
-
'J
AtC-~/3tl3
~)
==- t" ~(,xJ
J
:)
""
,)
3
3{t 2)w6'XJ +(j-:L~(!:-t)/~-
jjA
,
- e~ )
= xx' j ..L
4
=
cki.
c) y=tan(3x)+tan-1(3x
d) y
~
~
~
e
~f.
/~
)
.;Llce
I
+-
I
..2,1(\.:2-
)
.3
~Kt
-L ~JG
[s-l A{fl +-~f-]
51--~{i) r
(t
t..
17
(-,) rt (.Ilj
CALCULUSI
IS DEC 2008
FINAL EXAMINATION
Pale 10 of 10
29. Use I' and In to graph I(x) = x4 - 2xJ. Labelallrelativemaximumsandminimums
and inflection points,
rr
r;<)::: jt,l-/) -::.'f3(X-~)
)
('
I{~)::: c.}/-~/-=-2/~X-3)
(
]//
~1
I
~I
4
\
~
2.
;1\
0 {t)-=C12':-/2!-=12ft-I)
+- -r-.-r () - - - () ..J1>
il
--
i-t
~i{
1
.
I0
0
1I
- -
0I
0
-+- +- +-
{(,5
f -
t--
e-H
-
,;;L
~...
~
1- -'
2-I
-7
-
f
~
I
- t-
I
/
'\
x
-3
I
"_I
--I
!~
f,
,
'w'~
,
I
~
I
"t
30. Find the following limits. Show all work.
0
~+3
a)
~~ yo + 1
=3-.:=:=:-3
I
'0
otO
b ) lim
x--+oo
xl~3
2
]0
xj
0
~
L
+1
L
&J~31/~/O
,0 =
(,J
1--"?
- C'l
/. , Iv
+ I} t
oQ
~
/1°
lim cos(x)-l
c) x--+0
X
t
6
/~=~
1-" c.o
~I
L
1~O
,},"
6
./0
-~
(f-) .;<'f
j
0
/ ;1'
tJ
'!-~
.
I
-
/
(]A-:; ( .(\
~
"/._0
.
=-
- /21/