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Math 41: CalculusFirst Exam — October 16, 2007
Name :
Section Leader: David Xiannan Jason Tracy Ziyu(Circle one)
Fernandez-Duque Li Lo Nance Zhang
Section Time: 11:00 1:15(Circle one)
• This is a closed-book, closed-notes exam. No calculators or
other electronic devices will bepermitted. You have 2 hours.
• In order to receive full credit, please show all of your work
and justify your answer. You donot need to simplify your answers
unless specifically instructed to do so.
• If you need extra room, use the back sides of each page. If
you must use extra paper, makesure to write your name on it and
attach it to this exam. Do not unstaple or detach pagesfrom this
exam.
• Please sign the following:
“On my honor, I have neither given nor received any aid on
thisexamination. I have furthermore abided by all other aspects
ofthe honor code with respect to this examination.”
Signature:
The following boxes are strictly for grading purposes. Please do
not mark.
1 8 6 6
2 20 7 8
3 5 8 16
4 12 9 7
5 8 10 10
Total 100
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 2 of
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1. (8 points) Depicted is a graph of the function f .
3 3
3. (20 Points) Depicted is the graph of a function f .
1
1-1
-1
Draw the graphs of the following functions, labeling the
endpoints of the domainas well as the maximum and minimum
points.
(i) g(x) = f(x + 1)
3
Sketch the graphs of the following functions. Label clearly the
coordinates of the graphs’endpoints as well as both the “maximum”
point and “minimum” point. (You don’t have tokeep the same length
scale as above, and no additional justification is necessary.)
(a) g(x) = 3f(5x)
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 3 of
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For easy reference, here again is the graph of the original
function f :
3 3
3. (20 Points) Depicted is the graph of a function f .
1
1-1
-1
Draw the graphs of the following functions, labeling the
endpoints of the domainas well as the maximum and minimum
points.
(i) g(x) = f(x + 1)
3
(b) h(x) = f(−x− 2)− 1
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 4 of
14
2. (20 points) Find each of the following limits, with
justification. If there is an infinite limit,then explain whether
it is ∞ or −∞.
(a) limh→0
h
(2 + h)2 − (2− h)2
(b) limx→4−
3− xx2 − 2x− 8
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 5 of
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(c) limx→∞
sin
(ln
(1
x
))
(d) limx→0+
e−1/x sin
(2π
x
)
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 6 of
14
3. (5 points) Show that limx→2
(7− 2x) = 3 by finding a δ > 0 such that
|(7− 2x)− 3| < � whenever 0 < |x− 2| < δ.
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 7 of
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4. (12 points) Let f(x) =5ex
ex − 2.
(a) Find the domain of f .
(b) Find the equations of all vertical asymptotes of f , or
explain why none exist. As jus-tification for each asymptote x = a,
calculate both the one-sided limits lim
x→a+f(x) and
limx→a−
f(x), showing your reasoning.
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 8 of
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(c) Find the equations of all horizontal asymptotes of f , or
explain why none exist. Justifyeach asymptote with a limit
computation.
(d) It is a fact that f is a one-to-one function. Find an
expression for the function f−1(x),the inverse of f .
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 9 of
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5. (8 points) Let f(x) =1√
x + 2. Find a formula for f ′(x) using the limit definition of
the
derivative. Show the steps of your computation.
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 10 of
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6. (6 points) The graph of the function g is given below, as
well as the graphs of the function’sfirst and second derivatives,
g′ and g′′, respectively. Indicate which graph belongs to
whichfunction, and give your reasoning.
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 11 of
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7. (8 points) Let h(x) = x4 − 4x3 + 2x2 + x.
(a) Find h′(x) using any method.
(b) Show that there is a number a between 0 and 1, satisfying
the property that the tangentline to the graph of h at the point
(a, h(a)) is horizontal.
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 12 of
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8. (16 points) Differentiate, using any method you like. You do
not need to simplify your answers.
(a) f(x) =x2 +
√x
5− x
2 +√
x3√
x
(b) h(t) = −4t3et
(c) g(x) =x2 − 1x4 + x
(d) P (t) = e2 − 3 cos t + t sin t
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 13 of
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9. (7 points) Candidates in a Presidential primary believe their
support in a certain small stateis affected by the number of
different advertisements they make and broadcast on local TV.Let V
(x) be the number of voters (in thousands) that end up voting for a
candidate who usesx ads during the campaign. The following table
gives values of V (x) for certain x.
x 180 190 200 210 220V (x) 16 20 28 34 37
(a) Estimate the value of V ′(200). What are its units?
(b) What is the practical meaning of the quantity V ′(200)? Give
a brief one- or two-sentenceexplanation that is understandable to
someone who is not familiar with calculus.
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Math 41, Autumn 2007 First Exam — October 16, 2007 Page 14 of
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10. (10 points) Sketch the graph of a function f with all of the
following properties. Be sure tolabel the scales on your axes.
• The domain of f is all real numbers, and f is continuous
everywhere• f is an odd function• f has a local maximum at (−2, 2)•
f is decreasing for |x| < 2• f is increasing for 2 < |x| <
3• f ′(x) = −3 if |x| > 3• lim
x→0f ′(x) = −∞
• f is concave down for −3 < x < 0• f has an inflection
point at (0, 0)
