Name:
ID#:
Final Examination
V63.0121
Calculus I
December 19, 2008
Please check your section. If you are in section 20, 27, or 34, please also check your
recitation.
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Meets
MW
08:55am–10:45am
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11:00am–12:50pm
MW
02:00pm–03:50pm
MW
06:20pm–08:10pm
TR
08:55am–10:45am
TR
11:00am–12:50pm
TR
02:00pm–03:50pm
TR
T
T
M
M
W
W
TR
T
T
R
R
M
M
MW
T
R
R
11:00am–12:15pm
02:00pm–02:50pm
02:00pm–02:50pm
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09:00am–09:50am
12:30pm–01:20pm
12:30pm–01:20pm
12:30pm–01:45pm
08:00am–08:50am
08:00am–08:50am
08:00am–08:50am
08:00am–08:50am
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09:00am–09:50am
09:30am–10:45am
09:30am–10:20am
09:30am–10:20am
11:00am–11:50am
Location
WAVE 569
SILV 706
SILV 507
SILV 507
SILV 509
WAVE 369
TISC UC57/TISC UC58
SILV 504
MEYR 102/SILV 507
SILV 705
SILV 507
SILV 711
SILV 706
SILV 507
SILV 705
SILV 509
SILV 706
SILV 507
WAVE 369/SILV 520
SILV 714
WAVE 566B
TISC UC53
SILV 804
SILV 514
WAVE 566B
TISC UC55
SILV 714
194M 206
194M 202
194M 207
48CS 114
WAVE 431
SILV 810
5 WP 101
145F 210
SILV 803
WAVE 668
Instructor
HAMEIRI, ELIEZER
JIANG, NING
CHUMAKOVA, LYUBOV
LEINGANG, MATTHEW
JANG, JUHI
KRAHMER, FELIX
REN, WEIQING
DIAZ-ALBAN, JOSÉ
FREIRE, JULIANA
AUFFINGER, ANTONIO
LEINGANG, MATTHEW
LALIBERTÉ, FRÉDÉRIC
MOLINO, VAN
MONSOUR, PAUL D
JOHNS, JOSEPH A
RYAN, JEFFREY A
LIN, FANG-HUA
STUCCHIO, CHRIS
YU, YONG
KOSYGIN, DENIS
KIM, SUNGWOOK
SHMIDHEISER, HANS
LI, SEAN
SHMIDHEISER, HANS
LI, SEAN
KIM, SUNGWOOK
CHEN, YU
CHEN, XI (ROGER)
CORWIN, IVAN
WONG, TAK KWONG (DANNY)
CORWIN, IVAN
CHEN, XI (ROGER)
WONG, TAK KWONG (DANNY)
GERMAIN, PIERRE
LUSHI, ENKELEIDA
LUSHI, ENKELEIDA
SILJANDER, JUHA
version 1.0α-1, 2009-01-14 08:46
2 Section
Rules
you may put your ID number on the
top of each page you write on.
• This is a 110-minute exam.
• No calculators or formula sheets are
allowed.
Hints
• Unless otherwise stated, show all of
your work. Full credit may not be
given for an answer alone.
• Read the entire exam to scan for obvious typos or questions you might
have.
• You may use the backs of the pages or
the extra pages for scratch work. Do
not unstaple or remove pages as they can
be lost in the grading process.
• Budget your time so that you don’t
run out.
• Problems may stretch across several
pages.
• Please do not put your name on any
page besides the first page. If you like,
• Relax and do well!
Problem
Number
Possible
Points
1
10
2
10
3
10
4
15
5
12
6
12
7
10
8
10
9
5
10
6
Total
100
Points
Earned
“Santa’s Lap” image owned by Flickr user yksin and used under the Creative Commons license
version 1.0α-1, 2009-01-14 08:46
Good luck!
1
V63.0121
Final Examination
December 19, 2008
1
1. (10 Points) Compute the following derivatives. Please leave your answers unsimplified.
(i)
d
(ln ( x sin x + 1))
dx
arctan
x+1
ex + 1
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d
(ii)
dx
–1–
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2
V63.0121
Final Examination
December 19, 2008
2
2. (10 Points) Find the y-intercept of the line that is tangent to the ellipse 4x2 + 9y2 = 900
at the point (12, 6). Put your answer in the box.
Hint. Implicit differentiation may help here.
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y-intercept:
–2–
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3
V63.0121
Final Examination
December 19, 2008
3
3. (10 Points) Evaluate the following limits. Put your answers in the boxes. Show your work.
(i) (5 points) lim
x →1
x4 − 1
x2 − 1
2
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(ii) (5 points) lim (cos x )1/x
x →0
–3–
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4
V63.0121
Final Examination
December 19, 2008
4
4. (15 Points) Let f ( x ) = x4 + 4x3 − 2. Explain your answers on each of these parts.
(i) (4 points) The derivative of f is f 0 ( x ) = 4x3 + 12x2 . On which intervals is f increasing? decreasing? Find all critical points and classify them as local maxima, local
minima, or neither.
–4–
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(ii) (4 points) The second derivative of f is f 00 ( x ) = 12x2 + 24x. On which intervals is f
concave up? concave down? Find all inflection points.
4
V63.0121
Final Examination
4
December 19, 2008
(iii) (4 points) Find the absolute maximum and minimum values of f on [−4, 1].
(iv) (3 points) Sketch the graph of f . Label all the critical points and inflection points.
y
−3
−2
−1
1
–5–
2
3
4
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−4
x
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5
V63.0121
Final Examination
5
December 19, 2008
5. (12 Points) A department store is fencing off part of the store for children to meet and
be photographed with Santa Claus. They have decided to fence off a rectangular region of
fixed area 800 ft2 . There will be two 6 ft openings in the fencing, one on each side, to let the
kids in and out. Find the dimensions that will minimize the length of fencing used.
store wall
6 ft gap
6 ft gap
fencing
10 ft gap
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Dimensions:
–6–
ft ×
ft
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6
V63.0121
Final Examination
6
December 19, 2008
6. (12 Points) A cannonball is shot into the air. Its velocity is given as a function f (t) m/s,
where t measured in seconds since 1:00PM. We know that f (t) takes the following values:
t
f (t)
0
10.0
7.5
6.46
15
5.00
22.5
3.88
30
2.93
37.5
2.09
(i) (2 points) What quantities do the integrals I =
Z 60
pute? What is the difference between the two?
0
45
1.34
52.5
0.65
60
0
f (t) dt and J =
Z 30
0
f (t) dt com-
For the two parts below, let Ln be the Riemann sum for I using n subintervals and left
endpoints, Rn be the Riemann sum for I using n subintervals and right endpoints, and Mn
be the Riemann sum for I using n subintervals and midpoints.
(ii) (2 points) Write out the terms in M4 .You may leave your answer unsimplified.
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(iii) (2 points) Assume that f (t) is decreasing for all t ≥ 0. Without computing, put these
in order from least to greatest: L2 , R4 , R2 , I, L4 . Put your answers in the boxes. No
justification is necessary.
Hint. A picture might help your thinking here.
<
<
<
<
10 √
(iv) (6 points). It turns out f (t) = 10 − √
t. Compute I exactly.
60
–7–
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7
V63.0121
Final Examination
December 19, 2008
7
7. (10 Points) Find the following indefinite integrals. Your answer should be the most general
antiderivative.
(i) (5 points)
Z
√
12x
2x2 + 5
dx
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(ii) (5 points)
Z
2x sin(3x ) dx
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8
V63.0121
Final Examination
8
December 19, 2008
8. (10 Points) Let
F (y) =
Z y
0
2
e−s ds
F (y)
y
(i) (5 points) In each of these, select > if the quantity in column A is greater, < if the
quantity in column B is greater, = if the two quantities are the same, and ? if it is
impossible to determine which is greater. No justification is necessary. No partial credit
will be given. Please fill in the circle completely.
Hint. It is mathematically impossible to compute F (y) exactly by antidifferentiation,
so please do not try. That is not the point of this problem.
B
Your answer
1)
F (0)
0
2)
F (1)
F (2)
3)
F (−1)
F (−2)
4)
F 0 (0)
0
5)
F 0 (1)
F 0 (−1)
> < =
?
> < =
?
> < =
?
> < =
?
> < =
?
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A
(ii) (5 points) Suppose y(t) = 9 sin(πt) and let g(t) = F (y(t)). In other words,
g(t) =
Find g0
1
2
Z 9 sin πt
0
2
e−s ds
. Put your answer in the box.
–9–
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9
V63.0121
Final Examination
December 19, 2008
9
9. (5 Points) Evaluate the following. No justification is necessary for this problem.
In the first three, express your answer as an integer or a fraction.
(i) log3 (27)
(ii) log4
1
2
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(iii) ln (1)
In the next two, express your answer as an angle in radians.
Note. Remember that arcsin is the inverse of sin, sometimes also written as sin−1 . But this
1
is not the same as
.
sin
(iv) arcsin 12
(v) arctan(1)
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/5
10
V63.0121
Final Examination
10
December 19, 2008
10. (6 Points) Determine whether the following statements are true (i.e. true in general) or
false (i.e. not true in all cases). As long as there is one example where the statement does
not hold, it is considered false. Please fill in the circle completely. No justification is necessary.
No partial credit will be given.
Z b
Z b
Z b
(i) If f and g are continuous on [ a, b], then
f ( x ) g( x ) dx =
f ( x ) dx
g( x ) dx
a
a
a
T F
(ii) If f is differentiable at a, then f is continuous at a.
arcsin( x )
arccos( x )
(iv) If lim f ( x ) = 0 and lim g( x ) = 0, then lim [ f ( x )/g( x )] does not exist.
x →5
(v) lim
x →1
x →5
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(iii) If −1 < x < 1, then arctan( x ) =
T F
T F
T F
x →5
lim ( x − 3)
x−3
x →1
=
x2 + 2x − 4
lim ( x2 + 2x − 4)
T F
x →1
(vi) If f is continuous on [ a, b] and differentiable on ( a, b), then there is a point c in ( a, b)
f (b) − f ( a)
T F
with f 0 (c) =
.
b−a
–11–
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