Math 41: Calculus Final Exam — December 10, 2007Math 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

Math 41: CalculusFinal 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 bepermitted. You have 3 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 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 −∞.

(a) limx→ 3π

2

−

(x− 3π

2

)tan x

(b) limt→∞

(√

t2 − 5t − t)


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

g(x) =

{2x

+ c2x 0 < x < 1

cx + 2c x ≥ 1.

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

(b) Determine all values of c for which g′(x) is continuous for all x > 0. Explain yourreasoning.


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

(a) h(t) = (t− ln 2t + cos 3t)5

(b) f(x) =3x 5√

x2 − 1

(x + 7)10

(c) g(z) =

∫ 2

z2

et

sin t + 2dt


4. (8 points) You are told that a conical funnel has a height of twice its radius, and you wishto calculate its volume. If you measure the radius of the funnel to be 10 cm with a possibleerror of 0.05 cm, use linear approximations (or differentials) to estimate the maximum erroryou would obtain in the volume calculation.


5. (8 points) The height of a cylinder is decreasing at a rate of 5 cm/min. If the volume of thecylinder is to be kept constant, at what rate must the radius be increasing when the height is50 cm and the radius is 80 cm?


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


7. (8 points) An airplane takes off at 1:00 p.m. on a 2000 mile flight and arrives at its destinationat 5:00 p.m. Use reasoning from calculus to explain why there were at least two times duringthe journey when the speed of the plane was 400 miles per hour. (You may assume anythingyou like about the continuity or differentiability of the plane’s position function or one of itsderivatives, but state clearly what conditions you are using.)


8. (5 points) Verify by differentiation that

∫sec x dx = ln | sec x + tan x|+ C.


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 aRiemann sum that uses the Right Endpoint Rule. (That is, do not use the FundamentalTheorem of Calculus.)


(b) Express the limit

limn→∞

n∑i=1

π

ncos

(π

2+

πi

n

)as a definite integral, and evaluate the integral using the Evaluation Theorem.


10. (10 points) Starting at time t = 0 hours, water leaks out of a tank at the rate r(t), measuredin 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 3.0r(t) 0 6 11 15 18 20 21

(a) What is the meaning of the quantity∫ 2

1r(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.

(b) Use the Midpoint Rule with n = 3 to estimate∫ 3

0r(t)dt; give your answer as an expression

in terms of numbers alone, but you do not have to simplify it.

(c) Name another approximation method, including a value of n, that can be used to estimate∫ 3

0r(t)dt, and write the numerical expression that corresponds to this method and this

n.


11. (22 points) Evaluate each of the following integrals, showing all reasoning.

(a)

∫ (1√

1− x2− sec2 x +

1

3x

)dx

(b)

∫ 2

−2

(x− x3 − sin x

)dx


(c)

∫x2

((1 + x3)9 − (1 + x3)5 + cos(1 + x3)

)dx

(d)

∫ e2

e

ln x

x3dx


(e)

∫ √x cos

√x dx


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.

(a)

∫ 5

1

f(x) dx

(b)

∫ 5

1

f ′(x) dx

(c)

∫ 5

1

f ′′(x) dx

(d)

∫ 5

1

f ′′′(x) dx


13. (17 points) Let g(x) =

∫ x

−4

f(t) dt, where f is the function whose graph is given below. Note

that the graph of f is made up of straight lines and a semicircle.

6. Let F (x) =

∫ x

−3

f(t) dt, where f is the function whose graph is given below. Note that the graph of f

is made up of straight lines and a semicircle. Also note that −3 is the lower limit of integration in thedefinition of F .

-5 -3 -1 1 3 5x

-2

2

4

f HxL

(a) Identify the x-values of all critical points of F in the interval (−5, 5).

(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 youranswer.

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

12

(a) Find each of the following values. If a value is not defined, explain why not.

(i) g(3)

(ii) g′(3)

(iii) g′′(3)

(b) Identify the critical numbers of g in the interval (−5, 5).

(c) For each critical number that you found in part (b), determine if it is a point where ghas a local maximum, local minimum, or neither. Give reasons for your answer(s).


For easy reference, here again are the graph of f and the definition of g:

6. Let F (x) =

∫ x

−3

f(t) dt, where f is the function whose graph is given below. Note that the graph of f

is made up of straight lines and a semicircle. Also note that −3 is the lower limit of integration in thedefinition of F .

-5 -3 -1 1 3 5x

-2

2

4

f HxL

(a) Identify the x-values of all critical points of F in the interval (−5, 5).

(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 youranswer.

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

12

g(x) =

∫ x

−4

f(t) dt

(d) On what portion(s) of the interval (−5, 5) is the graph of g concave up? concave down?

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


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

A B

C D


Formulas for Reference

Geometric Formulas:

Vsphere =4

3πr3, SAsphere = 4πr2, Acircle = πr2

Vcylinder = πr2h, Vcone =1

3πr2h

Summation Formulas:

n∑i=1

i =n(n + 1)

2

n∑i=1

i2 =n(n + 1)(2n + 1)

6

n∑i=1

i3 =

(n(n + 1)

2

)2

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Math 41: Calculus Final Exam — December 10, 2007Math 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