SSEA Math 41 Track
Final Exam
September 1st, 2011
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• This is the final exam for the Math 41 track at SSEA. Answer as many problems as possible
to the best of your ability; do not worry if you are not able to answer all of the problems.
Partial credit is available. No calculators, notes, or other electronic devices are permitted.
• As with all math tests at Stanford, you are required to show your work in order to receive
credit. In particular, you should not do computations in your head; instead, write them out
on the test paper. You should also justify all conclusions that you make - do not be afraid to
explain yourself by writing a sentence or two. The goal is to make your thought process as
clear as possible. However, you are not required to simplify your answers.
• Please sign below to indicate your acceptance of the following statement:
“On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination.”
Problem
Total Points
1
12
2
16
3
12
4
12
5
12
6
10
7
8
8
16
Total
100
Score
SSEA Math 41 Track
1
Final Exam
September 1st, 2011
(12 points)
Compute the following limits if they exist, or write DNE if they do not exist.
Write ∞ or −∞ to indicate an infinite limit.
(a)
t2 − 9
.
t→3 t2 + t − 12
lim
(b)
lim−
x→1
x3 + 2x − 1
.
x−1
SSEA Math 41 Track
Final Exam
(c)
lim cos y.
y→∞
(d)
(x2 + 1)2 (x − 3)
.
x→∞
(x − 5)6
lim
September 1st, 2011
SSEA Math 41 Track
2
Final Exam
September 1st, 2011
(16 points) Compute the first derivative of each of the following functions: (problems worth
3/3/3/3/4 points)
(a)
f (t) = t2 sin t.
(b)
f (x) = cos(3e2x ).
(c)
Z
x
g(x) =
ln(t + 4) dt.
−3
SSEA Math 41 Track
Final Exam
September 1st, 2011
(d)
f (x) =
arctan x
.
2x + 1
(e) Compute f 0 (x) directly from the limit definition of the derivative:
f (x) = x2 + 3.
SSEA Math 41 Track
3
Final Exam
September 1st, 2011
(12 points)
Sketch the graph of a function f (x) which satisfies each of the following
conditions (your answer to this problem should be a single graph satisfying all twelve
conditions). You will get one point for each correct condition.
• limx→−∞ f (x) = −1, limx→∞ f (x) = 1, limx→−1− f (x) = 1, limx→−1+ = −∞;
limx→0 f (x) = −1, f (0) = −2;
• f (x) is differentiable everywhere except for x = −1, x = 0, and x = 3, and is continuous
everywhere except for x = −1 and x = 0;
• If x < 3 and f 0 (x) is defined, f 0 (x) ≥ 0, but f 0 (x) = 0 only at x = 2;
• f (x) has an inflection point at (−2, 0) and a local maximum at (3, 2).
SSEA Math 41 Track
4
Final Exam
4
September 1st, 2011
2
(12 points)
Sketch the graph of f (x) = x − 2x + 2. You should find the intervals of
increase and decrease, find the intervals of concavity, and label all local and absolute maxima
and minima as well as all inflection points.
SSEA Math 41 Track
5
Final Exam
September 1st, 2011
(12 points)
(a) (6 points) Find the equation of the tangent line to f (x) = ex cos x at x = 0, and then
use linear approximation to estimate e0.1 cos(0.1).
(b) (6 points) Find the absolute maximum and minimum values of f (x) = ex cos x on the
interval [0, π/2]. Hint: it may help to remember that tan(π/4) = 1.
SSEA Math 41 Track
6
Final Exam
September 1st, 2011
(10 points)
A comet is making a near approach to Earth. The comet travels along the
graph of the function y = 5 − 2x, where Earth is located at the origin of our coordinate
system and x and y are measured in millions of kilometers. (Of course, comets do not travel
in straight lines, but it’s a reasonable approximation when a comet is in a small region of the
solar system.)
(a) (4 points) Draw a diagram and write an expression for the distance from the comet to
Earth. Your answer should only depend on x, the x-coordinate of the comet.
(b) (6 points) At what point is the comet closest to Earth, and how far away is the comet
at that point?
SSEA Math 41 Track
7
(8 points)
Final Exam
September 1st, 2011
Consider the integral
Z
8
√
1 + x dx.
3
(a) Write an approximation to this integral, using the Left Endpoint Rule with 5 subintervals.
(b) Do the same with the Right Endpoint Rule and 10 subintervals.
(c) Do the same with the Midpoint Rule and 5 subintervals.
(d) Are your answers to a) and b) overestimates or underestimates? Explain.
(e) (Extra Credit - 2 points) Is your answer to c) an overestimate or underestimate? Explain.
SSEA Math 41 Track
8
(16 points)
Final Exam
Evaluate the following integrals:
(a)
Z
1
(x3 − 2x2 + x − 5) dx.
0
(b)
Z
0
π 1/3
x2 sin(x3 ) dx.
September 1st, 2011
SSEA Math 41 Track
Final Exam
(c)
Z
x cos x dx.
(d) If
(sin x)9 (cos x)2001
F (x) =
,
1+x
evaluate
Z
0
π/4
F 0 (x) dx.
September 1st, 2011