SSEA Math 41 Track
Final Exam
August 30, 2012
Name:
• 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, and 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.”
Signature:
Problem
Total Points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
Total
100
Score
SSEA Math 41 Track
1
Final Exam
August 30, 2012
Compute the following limits if they exist, or write DNE if they do not exist. Write
∞ or −∞ to indicate an infinite limit.
(a)
t3
lim
.
t→2− t − 2
(b)
lim
x→∞
3ex − e−x
.
ex + e−x
(c)
lim x2 2 + sin(1/x) .
x→0
SSEA Math 41 Track
2
Final Exam
August 30, 2012
Compute the derivative of each of the following functions:
(a)
f (x) = ex cos x .
(b)
f (x) = sin2 (ln(x)).
(c)
Z
x
g(x) =
sin
−8
√
t4 + 1 dt.
SSEA Math 41 Track
3
Final Exam
2
August 30, 2012
(a) Compute the derivative of f (x) = 3x + x using the definition of the derivative
(involving limits).
(b) Find the absolute maximum and minimum values of
f (x) =
on the interval [−4, 4].
x2 − 4
x2 + 4
SSEA Math 41 Track
4
Final Exam
August 30, 2012
(a) Sketch the graph of a function f (x) which satisfies the following conditions:
•
•
•
•
•
•
•
•
f (0) = 0,
f 0 (−2) = f 0 (1) = f 0 (9) = 0,
limx→∞ f (x) = 0,
limx→6 f (x) = −∞,
f 0 (x) < 0 on (−∞, −2), (1, 6), and (9, ∞),
f 0 (x) > 0 on (−2, 1) and (6, 9),
f 00 (x) > 0 on (−∞, 0) and (12, ∞),
f 00 (x) < 0 on (0, 6) and (6, 12).
SSEA Math 41 Track
Final Exam
3
2
August 30, 2012
(b) Sketch the graph of f (x) = 2x + 3x − 36x. You should find the intervals of
increase and decrease, find the intervals of concavity, label all local and absolute
maxima and minima, and label all inflection points.
SSEA Math 41 Track
Final Exam
August 30, 2012
√
5 (a) Let f (x) = 3 + x. Find the linear approximation L(x) to f (x) at x = 1. This
is also the equation of the tangent line to f (x) at x = 1.
(b) Use the linear approximation from part (a) to estimate
√
3.99.
SSEA Math 41 Track
6
Final Exam
August 30, 2012
A rectangular storage container with an open top is to have a volume of 10 m3 . The
length of the base is twice the width of the base. Material for the base of the container
costs $10 per square meter. Material for each of the four sides costs $6 per square
meter. What is the cost of materials for the cheapest such container? You do not need
to simplify the expression you obtain.
SSEA Math 41 Track
7
Final Exam
August 30, 2012
Consider the integral
Z
9
ln(x) dx.
1
(a) Write an approximation to this integral, using left endpoints and 4 subintervals.
You do not need to simplify. Is this an overestimate or underestimate?
(b) Write an approximation to this integral, using right endpoints and 4 subintervals.
You do not need to simplify. Is this an overestimate or underestimate?
(c) Write an approximation to this integral, using midpoints and 4 subintervals.
SSEA Math 41 Track
8
Final Exam
Evaluate the following integrals:
(a)
Z
π
(2x3 + 5 + sin x) dx.
0
(b)
Z
x3/2 ln x dx.
August 30, 2012
SSEA Math 41 Track
Final Exam
(c)
Z
1
(x2 + 1)(x3 + 3x)4 dx.
0
(d) If
F (x) = x3/2 ln x,
evaluate
Z
1
3
F 0 (x) dx.
August 30, 2012
SSEA Math 41 Track
9
Final Exam
August 30, 2012
For the following true and false questions, you do not need to explain your answer at
all. Just write “True” or “False”.
(a) True or false: If a function f (x) is differentiable at all real numbers x, then it is
also continuous at all real numbers x.
(b) True or false: If function f (x) is not defined at x = a, then limx→a f (x) does not
exist.
(c) True or false: If a function f (x) is continuous on [2, 5] then it attains an absolute
maximum on [2, 5].
(d) True or false: If a function f (x) has an antiderivative function, then it has many
different antiderivative functions.
(e) True or false: Suppose f 00 is continuous near c. If f 0 (c) = 0 and f 00 (c) < 0, then
f has a local minimum at c.
SSEA Math 41 Track
Final Exam
August 30, 2012
10 Two athletes run a 10 mile race. It’s a tie! They both finish in exactly one hour. In
part (b), we will prove that at some time during the race the two athletes have the
exact same speed.
(a) What is the name of the theorem you want to use for this proof? What is the
statement of this theorem, as best as you can remember?
(b) Prove that at some time during the race the two athletes have the exact same
speed. For notational purposes, you may want to let g(t) be the position in miles
of the first athlete at time t and let h(t) be the position in miles of the second
athlete at time t. You may want to let 0 ≤ t ≤ 1 be measured in hours. If so,
then g(0) = h(0) = 0 miles and g(1) = h(1) = 10 miles.