Math 115 — Practice for Exam 3

Generated April 18, 2014

Name:

Instructor: Section Number:

1. This exam has 10 questions. Note that the problems are not of equal difficulty, so you may want toskip over and return to a problem on which you are stuck.

2. Do not separate the pages of the exam. If any pages do become separated, write your name on themand point them out to your instructor when you hand in the exam.

3. Please read the instructions for each individual exercise carefully. One of the skills being tested onthis exam is your ability to interpret questions, so instructors will not answer questions about examproblems during the exam.

4. Show an appropriate amount of work (including appropriate explanation) for each exercise so that thegraders can see not only the answer but also how you obtained it. Include units in your answers whereappropriate.

5. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad).However, you must show work for any calculation which we have learned how to do in this course. Youare also allowed two sides of a 3′′ × 5′′ note card.

6. If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of thegraph, and to write out the entries of the table that you use.

7. You must use the methods learned in this course to solve all problems.

Semester Exam Problem Name Points Score

Fall 2009 3 5 icicle 13

Winter 2002 3 6 Hankytown3 10

Winter 2002 3 4 naked mile 9

Winter 2005 3 9 Simpsons 12

Winter 2004 3 2 music trading 4

Fall 2007 3 4 12

Winter 2009 3 6 Turtle vs Bunny 14

Winter 2002 3 2 rectangle 7

Fall 2001 3 12 BarBQ Ice Cream 12

Fall 2011 3 1 FarmVille 14

Total 107

Recommended time (based on points): 128 minutes

Math 115 / Final (December 17, 2009) page 6

5. [13 points] A cone-shaped icicle is dripping from above the en-trance to Dennison Hall. The icicle is melting at a rate of 1.2cm3 per hour. At 10:00 a.m., the icicle was 25 cm long and hada 2 cm radius at its widest point. Assume that the icicle keepsthe same proportions as it melts. [Note: the volume of a cone isV = 1

3πr2h.]

a. [5 points] Determine the rate at which the length of theicicle is changing at 10:00 a.m.

b. [4 points] At what rate is the radius of the icicle changing at 10:00 a.m.?

c. [4 points] Let V (t) and r(t) denote the volume and radius, respectively, of the icicle t

hours after 10:00 a.m. Assume that the icicle continued to melt from t = 0 (10:00 a.m.)to t = M . Circle all of the statements below that must be true if “After the icicle begandripping at 10:00 a.m., it took exactly M hours for the icicle to melt completely.” [Circlethe entire expression, and be certain that your circled answers are VERY clear!!]

i.

∫ M

0

V′(t) dt >

∫ M/2

0

V′(t) dt ii.

∫ M

0

V′(t) dt = 0

iii.

∫ M

0

V′(t) dt = −V (0) iv.

∫2

0

r(t) dt = 0

v.

∫ M

0

r(t) dt = −2 vi.

∫ M

0

r′(t) dt = −2

vii.

∫0

2

V′(r) dr = M viii.

∫ M

0

h(t) dt = 0

University of Michigan Department of Mathematics Fall, 2009 Math 115 Exam 3 Problem 5 (icicle)

0 2 4 6 8 10 12

−60

−40

−20

0

20

40

60

Rate of change of Valentine supply

Graph of r(t)

University of Michigan Department of Mathematics Winter, 2002 Math 115 Exam 3 Problem 6 (Hankytown3)

University of Michigan Department of Mathematics Winter, 2002 Math 115 Exam 3 Problem 4 (naked mile)

9

9. (5+2+2+3 points) The three happy wizards leave the fair and go home to watch the Simpsons. Inthis episode, Homer needs to deliver Lisa’s homework to her at school, and he must do so before PrincipalSkinner arrives. Suppose Homer starts from the Simpson home in his car and travels with velocity givenby the figure below. Suppose that Principal Skinner passes the Simpson home on his bicycle 2 minutesafter Homer has left, following him to the school. Principal Skinner is able to sail through all the trafficand travels with constant velocity 10 miles per hour.

t (minutes)

v (miles/hr)

15

20

1 2 3 4 5

5

6 7 8 9

10

10

(a) How far does Homer travel during the 10 minutes shown in the graph?.

(b) What is the average of Homer’s velocity during the 10 minute drive?

(c) At what time, t > 0, is Homer the greatest distance ahead of Principal Skinner?

(d) Does Principal Skinner overtake Homer, and if so, when? Explain.

University of Michigan Department of Mathematics Winter, 2005 Math 115 Exam 3 Problem 9 (Simpsons)

3

2. (4 points) As an avid online music trader, your rate of transfer of mp3’s is given by m(t)measured in songs/hour where t = 0 corresponds to 5 pm. Explain the meaning of the quantity∫

5

0

m(t) dt.

3. (8 points) Suppose

∫4

−3

f(x)dx = 10,

∫4

0

f(x)dx = 2, and that f is an odd function. For each

of the following integrals fill in the answer in the space provided.

(a)

∫4

−3

6f(x) dx =

(b)

∫0

−3

f(x) dx =

(c)

∫0

−4

f(x) dx =

(d)

∫−3

−4

f(x) dx =

University of Michigan Department of Mathematics Winter, 2004 Math 115 Exam 3 Problem 2 (music trading)

4

4. (12 points) Suppose that f , g and h are all continuous and differentiable functions such that:

• f is an odd function

•

∫ 3

0

f(t)dt = 3

• g(t) = t2 + 2

• h(t) = g′(t − 1)

Evaluate the following, where possible. If evaluation is not possible, simply state “insufficientinformation.”

(a)

∫a+3

a+3

f(t) dt

(b)

∫ 10

−10

f(t) dt

(c) The average value of g on the interval [−2, 2]

(d)

∫ 0

−3

f(t) dt

(e)

∫ 1

−1

h(t) dt

University of Michigan Department of Mathematics Fall, 2007 Math 115 Exam 3 Problem 4

7

6. The Awkward Turtle is competing in a race! Unfortunately his archnemesis, the Playful Bunny,is also in the running. The two employ very different approaches: the Awkward Turtle takes thefirst minute to accelerate to a slow and steady pace which he maintains through the remainder ofthe race, while the Playful Bunny spends the first minute accelerating to faster and faster speedsuntil she’s exhausted and has to stop and rest for a minute - and then she repeats this processuntil the race is over. The graph below shows their speeds (in meters per minute), t minutes intothe race. (Assume that the pattern shown continues for the duration of the race.)

t, min

v, m / min

1 2 3 4 5 6

6

18Bunny’s speedTurtle’s speed

(a) (6 points) What is the Awkward Turtle’s average speed over the first two minutes of therace? What is the Playful Bunny’s?

(b) (3 points) The Playful Bunny immediately gets ahead of the Awkward Turtle at the start ofthe race. How many minutes into the race does the Awkward Turtle catch up to the PlayfulBunny for the first time? Justify your answer.

(c) (5 points) If the race is 60 meters total, who wins? Justify your answer.

University of Michigan Department of Mathematics Winter, 2009 Math 115 Exam 3 Problem 6 (Turtle vs Bunny)

University of Michigan Department of Mathematics Winter, 2002 Math 115 Exam 3 Problem 2 (rectangle)

University of Michigan Department of Mathematics Fall, 2001 Math 115 Exam 3 Problem 12 (BarBQ Ice Cream)

Math 115 / Final (December 15, 2011) page 2

1. [14 points] You are online playing the Facebook-based game, FarmVille, and you receive landwith 5 stalks of corn on it. You decide that you would like to model the corn population onthis patch of land using your calculus skills, so you recall that a good model for populationgrowth is the logistic model

P (t) =L

1 +Ae−ktL > 0, A > 0, k > 0.

a. [5 points] Using the limit definition of the derivative, write an explicit expressionfor the derivative of the function P (t) at t = 1. Do not evaluate this expression.

b. [5 points] Using the definition of the logistic model above, compute the following in termsof L, k, and A, showing your work or providing an explanation for each part:

i. [1 points] limt→∞

P (t)

ii. [1 points] limt→−∞

P (t)

iii. [1 points] P (0)

iv. [2 points] P ′(0)

c. [4 points] Your farmland satisfies the following conditions:

P (0) = 5, P ′(0) = 1, limt→∞

P (t) = 100.

Based on your answers in part (b), compute the correct values of L, k, and A for thelogistic equation modeling corn population on your land.

L = A = k =

University of Michigan Department of Mathematics Fall, 2011 Math 115 Exam 3 Problem 1 (FarmVille)