Math 41: Calculus

Final Exam — December 10, 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 be

permitted. You have 3 hours.
• In order to receive full credit, please show all of your work and justify your answer. You do
not 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, make
sure to write your name on it and attach it to this exam. Do not unstaple or detach pages
from this exam.
• Please sign the following:
“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:

The following boxes are strictly for grading purposes. Please do not mark.

1 10 8 5

2 8 9 13

3 15 10 10

4 8 11 22

5 8 12 8

6 10 13 17

7 8 14 8

Total 150
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 2 of 20

1. (10 points) Find each of the following limits, with justification. If there is an infinite limit,
then explain whether it is ∞ or −∞.
 
3π
(a) lim − x − tan x
x→ 3π
2
2

√
(b) lim ( t2 − 5t − t)
t→∞
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 3 of 20

2. (8 points) Suppose c is a constant, and let g(x) be the function

(
2
x
+ c2 x 0 < x < 1
g(x) =
cx + 2c x ≥ 1.

(a) Determine all values of c for which g(x) is continuous for all x > 0. Explain your
reasoning.

(b) Determine all values of c for which g 0 (x) is continuous for all x > 0. Explain your
reasoning.
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 4 of 20

3. (15 points) Differentiate, using any method you choose. You do not have to simplify your
answers.

(a) h(t) = (t − ln 2t + cos 3t)5

√
3x 5 x2 − 1
(b) f (x) =
(x + 7)10

2
et
Z
(c) g(z) = dt
z2 sin t + 2
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 5 of 20

4. (8 points) You are told that a conical funnel has a height of twice its radius, and you wish
to calculate its volume. If you measure the radius of the funnel to be 10 cm with a possible
error of 0.05 cm, use linear approximations (or differentials) to estimate the maximum error
you would obtain in the volume calculation.
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 6 of 20

5. (8 points) The height of a cylinder is decreasing at a rate of 5 cm/min. If the volume of the
cylinder is to be kept constant, at what rate must the radius be increasing when the height is
50 cm and the radius is 80 cm?
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 7 of 20

6. (10 points) It’s 9:30 p.m. Your snowmobile is out of gas and you are 3 miles due south of a
major east-west highway. The nearest service station on the highway is 4 miles east of your
position; it closes at midnight. You can walk a mile in 15 minutes on icy roads, but each mile
takes 30 minutes on snowy fields.
4 mi

3 mi

In order to quickly get to the station, you consider the ways in which you could first walk to
some point on the highway and then continue due east the rest of the way.

(a) Determine, with justification, the route that gets you to the station in the least time.

(b) Can you make it to the service station before it closes? Explain.
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 8 of 20

7. (8 points) An airplane takes off at 1:00 p.m. on a 2000 mile flight and arrives at its destination
at 5:00 p.m. Use reasoning from calculus to explain why there were at least two times during
the journey when the speed of the plane was 400 miles per hour. (You may assume anything
you like about the continuity or differentiability of the plane’s position function or one of its
derivatives, but state clearly what conditions you are using.)
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 9 of 20
Z
8. (5 points) Verify by differentiation that sec x dx = ln | sec x + tan x| + C.
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 10 of 20

9. (13 points)

(a) Let f (x) = 1 + 2x2 . Let R be the region in the xy-plane bounded by the curve y = f (x)
and the lines y = 0, x = 0, and x = 3. Find the area of R by evaluating the limit of a
Riemann sum that uses the Right Endpoint Rule. (That is, do not use the Fundamental
Theorem of Calculus.)
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 11 of 20

(b) Express the limit

n  
X π π πi
lim cos +
n→∞
i=1
n 2 n
as a definite integral, and evaluate the integral using the Evaluation Theorem.
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 12 of 20

10. (10 points) Starting at time t = 0 hours, water leaks out of a tank at the rate r(t), measured
in gallons per hour. A table of some values for r(t) is given below.

t 0 0.5 1.0 1.5 2.0 2.5 3.0

r(t) 0 6 11 15 18 20 21
R2
(a) What is the meaning of the quantity 1 r(t)dt? Express your answer in terms relevant to
this situation, and make it understandable to someone who does not know any calculus;
be sure to use any units that are appropriate.

R3
(b) Use the Midpoint Rule with n = 3 to estimate 0 r(t)dt; give your answer as an expression
in terms of numbers alone, but you do not have to simplify it.

(c) Name
R3 another approximation method, including a value of n, that can be used to estimate
0
r(t)dt, and write the numerical expression that corresponds to this method and this
n.
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 13 of 20

11. (22 points) Evaluate each of the following integrals, showing all reasoning.
Z  
1 2 1
(a) √ − sec x + dx
1 − x2 3x

Z 2
x − x3 − sin x dx


(b)
−2
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 14 of 20
Z
x2 (1 + x3 )9 − (1 + x3 )5 + cos(1 + x3 ) dx


(c)

Z e2
ln x
(d) dx
e x3
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 15 of 20

√ √
Z
(e) x cos x dx
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 16 of 20

12. (8 points) The figure below shows the graph of a function f that has continuous first, second,
and third derivatives. The dashed lines are tangent to the graph of y = f (x) at (1, 1) and
(5, 1).

Based on what is shown, determine whether the following integrals are positive, negative, zero,
or if there is not enough information to tell; give brief explanations.
Z 5
(a) f (x) dx
1

Z 5
(b) f 0 (x) dx
1

Z 5
(c) f 00 (x) dx
1

Z 5
(d) f 000 (x) dx
1
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 17 of 20
Z x
Z x
13. (17 points) Let
6. Let F (x) = f (t) dt,
g(x) f=(t) dt, where f is where f iswhose
the function the function whose
graph is given below.graph is given
Note that below.
the graph of f Note
−3 −4
is made up of straight lines and a semicircle. Also note that −3 is the lower limit of integration in the
that the definition
graph ofof fF . is made up of straight lines and a semicircle.

f HxL

x
-5 -3 -1 1 3 5

-2

(a) Identify the x-values of all critical points of F in the interval (−5, 5).
(a) Find each of the following values. If a value is not defined, explain why not.
(i) g(3)

g 0 (3)
(ii) (b) On what interval(s) in (−5, 5) is F decreasing? Justify your answer.

(iii) g 00 (3)

(c) At what x-values in the interval (−5, 5), if any, does f have a local maximum? Justify your
(b) Identify the critical numbers of g in the interval (−5, 5).
answer.

(d) At what x-values in the interval (−5, 5), if any, does f have a local minimum? Justify your answer.
(c) For each critical number that you found in part (b), determine if it is a point where g
has a local maximum, local minimum, or neither. Give reasons for your answer(s).

12
 Z x
6. Let F (x) = f (t) dt, where f is the function whose graph is given below. Note that the graph of f
Math 41,
−3 Autumn 2007 Final Exam — December 10, 2007 Page 18 of 20
is made up of straight lines and a semicircle. Also note that −3 is the lower limit of integration in the
definition of F .
For easy reference, here again are the graph of f and the definition of g:
f HxL

Z x
2
g(x) = f (t) dt
−4

x
-5 -3 -1 1 3 5

-2

(a) Identify the

(d)x-values
On what of allportion(s)
critical pointsofofthe
F ininterval
the interval (−5,5)
(−5, 5).is the graph of g concave up? concave down?

(b) On what interval(s) in (−5, 5) is F decreasing? Justify your answer.

(c) At what x-values in the interval (−5, 5), if any, does f have a local maximum? Justify your
answer.

(e) Find a formula that gives the value of g(x) for any x such that 3 ≤ x ≤ 5.

(d) At what x-values in the interval (−5, 5), if any, does f have a local minimum? Justify your answer.

12
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 19 of 20

14. (8 points) Which of the following shaded areas A,B,C,D are equal? Give reasons. (Hint: you
don’t need to compute all four areas.)

A B

C D
Math 41, Autumn 2007 Final Exam — December 10, 2007 Page 20 of 20

Formulas for Reference

Geometric Formulas:

4
Vsphere = πr3 , SAsphere = 4πr2 , Acircle = πr2
3

1
Vcylinder = πr2 h, Vcone = πr2 h
3

Summation Formulas:

n
X n(n + 1)
i=
i=1
2

n
X n(n + 1)(2n + 1)
i2 =
i=1
6

n  2
X
3 n(n + 1)
i =
i=1
2