math 116 — practice for exam 2 - university of michigantghyde/practiceexam2sols.pdf · math 116...

Math 116 — Practice for Exam 2

Generated November 7, 2017

Name: SOLUTIONS

Instructor: Section Number:

1. This exam has 12 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

Winter 2015 3 10 ladybugs2 12

Winter 2015 2 11 spaceship 6

Winter 2015 2 9 10

Fall 2014 2 7 8

Fall 2013 3 4 9

Fall 2013 3 10 movie tickets 13

Winter 2013 3 3 boat 10

Winter 2017 3 2 disc tower 7

Winter 2017 3 3 9

Winter 2014 3 8 12

Winter 2013 3 8 8

Fall 2012 3 5 9

Total 113

Recommended time (based on points): 128 minutes

Math 116 / Final (April 23, 2015) page 10

10. [12 points] Vic is watching the ladybugs run around in his garden. His garden is in the shapeof the outer loop of a cardioid with polar equation r = 2 + 3 sin θ where r is measured inmeters and θ is measured in radians. The outline of the garden is pictured below for yourreference. At a time t minutes after he begins watching, Apple, his favorite red ladybug, is atthe xy-coordinate (sin2 t, cos2 t), and Emerald, his prized green ladybug, is at the xy-coordinate(− cos(2t), sin(2t) + 1.5). Vic watches the ladybugs for 2π minutes.

x

y

-2 2

5

Using the information above, circle the correct answer for each part below. There is only onecorrect answer for each part. You do not need to show your work.

a. [3 points] Which of the following integrals gives the area of the garden?

Solution:

A)1

2

∫π

0(2 + 3 sin θ)2dθ

B)1

2

∫π+arcsin( 2

3)

arcsin( 23)

(2 + 3 sin θ)2dθ

C)1

2

∫π+arcsin( 2

3)

− arcsin( 23)

(2 + 3 sin θ)2dθ

D)1

2

∫ 2π

0(2 + 3 sin θ)2dθ

E)1

2

∫ 4π−arcsin( 23)

2π−arcsin( 23)(2 + 3 sin θ)2dθ

b. [3 points] Which of the following is not true about Apple while Vic is watching?

Solution:

A) Apple runs through the point (12 ,12) more than once.

B) Apple crosses the path made by Emerald exactly 4 times.

C) Apple’s speed is zero at least once.

D) Apple does not leave the garden.

E) Apple is moving faster than Emerald for some of the time.

University of Michigan Department of Mathematics Winter, 2015 Math 116 Exam 3 Problem 10 (ladybugs2) Solution


c. [3 points] How far does Emerald run while Vic is watching?

Solution:

A) π meters

B) 2π meters

C) 4π meters

D)√

8π meters

E) 8π meters

d. [3 points] After the 2π minutes, Vic stops watching. Apple runs from the point (x, y) =(0, 1) in the positive y-direction with speed of g(T ) = 5Te−T , T seconds after Vic stopswatching. Which of the following is true?

Solution:

A) Apple leaves the garden eventually, but never runs further than 5 meters total.

B) Apple’s speed is always increasing after Vic stops watching.

C) If given enough time, Apple would eventually be more than 1000 meters fromthe garden.

D) Apple is still in the garden 5 minutes after Vic stops watching.

E) Apple changes direction, eventually.

University of Michigan Department of Mathematics Winter, 2015 Math 116 Exam 3 Problem 10 (ladybugs2) Solution

Math 116 / Exam 2 (March 23, 2015) page 14

11. [6 points] Jane is trying to make a spaceship in the shape of the region enclosed by the polarcurve

r =2 cos(2θ)

cos θ

where r is measured in meters and θ is measured in radians. The region is graphed below.

x

y

1 2

-1

1

a. [4 points] Find values of θ that trace out the boundary of the region exactly once.

Solution: There are many options here, we give two. If you take −π

4≤ θ ≤

π

4, the curve

will be traced exactly once. You can also take 0 ≤ θ ≤π

4and

7π

4≤ θ ≤ 2π.

b. [2 points] Write an expression involving integrals that give the area of the region. Do notevaluate your integral.

Solution: The area of the region is given by

1

2

∫ π/4

−π/4

4 cos2(2θ)

cos2(θ)dθ

University of Michigan Department of Mathematics Winter, 2015 Math 116 Exam 2 Problem 11 (spaceship) Solution

Math 116 / Exam 2 (March 23, 2015) page 11

9. [10 points] Determine if the following integrals converge or diverge. If the integral converges,circle the word “converges” and give the exact value (i.e. no decimal approximations). Ifthe integral diverges, circle “diverges”. In either case, you must show all your work and

indicate any theorems you use. Any direct evaluation of integrals must be done without

using a calculator.

a. [5 points]

∫1

0

ln(x) dx

CONVERGES DIVERGES

Solution: We have

∫1

0

ln(x)dx = lima→0+

∫1

a

ln(x)dx

= lima→0+

x ln(x)− x|1a

= lima→0+

−1− a ln(a) + a.

L’Hospital’s Rule tells us that lima→0+

a ln(a) = 0. Thus,

∫1

0

ln(x)dx = −1

b. [5 points]

∫∞

2

x+ sinx

x2 − xdx

CONVERGES DIVERGES

Solution: We have

sin(x) ≥ −1,

x+ sin(x) ≥ x− 1,

x+ sin(x)

x2 − x≥

x− 1

x2 − x=

1

x.

We also know that

∫∞

2

1

xdx diverges, as it is an improper integral of the form

∫∞

a

1

xpdx

for p ≤ 1. Thus, by the comparison test we have that

∫∞

2

x+ sinx

x2 − xdx diverges.

University of Michigan Department of Mathematics Winter, 2015 Math 116 Exam 2 Problem 9 Solution

Math 116 / Exam 2 (November 12, 2014) page 10

7. [8 points] Suppose that f(x) is a differentiable function, defined for x > 0, which satisfies theinequalities 0 ≤ f(x) ≤ 1

xfor x > 0. Determine whether the following statements are always,

sometimes or never true by circling the appropriate answer. No justification is necessary.

a. [2 points]

∫∞

1f(x)dx converges.

Always Sometimes Never

b. [2 points]

∫∞

1(f(x))2dx converges.


c. [2 points]

∫ 1

0f(x)dx converges.


d. [2 points]

∫∞

1ef(x)dx converges.


University of Michigan Department of Mathematics Fall, 2014 Math 116 Exam 2 Problem 7 Solution

Math 116 / Final (December 17, 2013) page 7

4. [9 points]Determine if each of the following sequences is increasing, decreasing or neither, and whetherit converges or diverges. If the sequence converges, identify the limit. Circle all your answers.No justification is required.

a. [3 points] an =

∫n3

1

1

(x2 + 1)1

5

dx.

Converges to DIVERGES.

INCREASING Decreasing Neither.

Solution: (Not required)

Since

∫∞

1

1

(x2 + 1)1

5

dx ≈

∫∞

1

1

x2

5

dx which diverges by p-test (with p = 2

5≤ 1). Hence,

limn→∞

an = limn→∞

∫n3

1

1

(x2 + 1)1

5

dx =

∫∞

1

1

(x2 + 1)1

5

dx = ∞. The sequence is increasing

since1

(x2 + 1)1

5

> 0.

b. [3 points] bn =n∑

k=0

(−1)k

(2k + 1)!.

CONVERGES TO sin 1 Diverges.

Increasing Decreasing NEITHER.

Solution: (Not required)

Since sinx =

∞∑k=0

(−1)kx2k+1

(2k + 1)!, then lim

n→∞

bn =

∞∑k=0

(−1)k(1)2k+1

(2k + 1)!= sin 1. Since the series

is alternating, the sequence is neither increasing or decreasing.

c. [3 points] cn = cos (an) , where 0 < a < 1.

CONVERGES TO 1 Diverges.

INCREASING Decreasing Neither.

Solution: (Not required)Since lim

n→∞

an = 0, then limn→∞

cos an = cos 0 = 1.



10. [13 points] The blockbuster action movie Mildred’s Adventures with Calculus! was just re-leased. During the first week after the premiere, 2.5 million people went to see it. The studiohas conducted a study to gauge the impact of the film on audiences, and found that: the

number of tickets sold in a given week is 60% of the number of tickets sold the previous week.Assume that this process repeats every week.

a. [5 points] Let pk be the number of movie tickets, in millions, sold during the kth weekafter the premiere of the movie. Determine p2, p3 and a formula for pk.

Solution:

p1 = 2.5

p2 = 2.5(0.6)

p3 = 2.5(0.6)2

pk = 2.5(0.6)k−1.

b. [6 points] A movie ticket costs $8. Let Tn be the total amount of money earned inticket sales, in millions of dollars, during the first n weeks the movie has been exhibited.Determine T3 and a closed formula for Tn. Show all your work.

Solution:

T1 = 8(2.5)

T2 = 8(2.5 + 2.5(0.6))

T3 = 8(2.5 + 2.5(0.6) + 2.5(0.6)2)

Tn = 8(2.5 + 2.5(0.6) + 2.5(0.6)2 + · · ·+ 2.5(0.6)n−1)

Tn = 8(2.5)1− (0.6)n

1− .6= 50(1− (0.6)n)

c. [2 points] Determine the value of limn→∞

Tn.

Solution: limn→∞

Tn = limn→∞

50(1− (0.6)n) = 50



3. [10 points] A boat’s initial value is $100, 000; it loses 15% of its value each year. Theboat’s maintenance cost is $500 the first year and increases by 10% annually.In the following questions, your formulas should not be recursive.

a. [2 points] Let Bn be the value of the boat n years after it was purchased. Find B1

and B2.

Solution:

B1 = $100, 000(0.85).B2 = $100, 000(0.85)2.

b. [3 points] Find a formula for Bn.

Solution: Bn = 100, 000(.85)n

c. [2 points] Let Mn be the total amount of money spent on the maintenance of theboat during the first n years. Find M2 and M3.

Solution:

M2 = 500(1 + 1.1)M3 = 500(1 + 1.1 + (1.1)2)

d. [3 points] Find a closed form formula for Mn.

Solution: Mn = 500(1− (1.1)n)

1− 1.1

University of Michigan Department of Mathematics Winter, 2013 Math 116 Exam 3 Problem 3 (boat) Solution

University of Michigan Department of Mathematics Winter, 2017 Math 116 Exam 3 Problem 2 (disc tower) Solution


8. [12 points] Suppose an and bn are sequences of positive numbers with the following properties.

•

∞∑

n=1

an converges.

•

∞∑

n=1

bn diverges.

• 0 < bn ≤ M for some positive number M .

For each of the following questions, circle the correct answer. No justification is necessary.

a. [2 points] Does the series∞∑

n=1

anbn converge?

Converge Diverge Cannot determine

b. [2 points] Does the series∞∑

n=1

(−1)nbn converge?


c. [2 points] Does the series

∞∑

n=1

√

bn converge?


d. [2 points] Does the series∞∑

n=1

sin(an) converge?


e. [2 points] Does the series∞∑

n=1

(an + bn)2 converge?


f. [2 points] Does the series

∞∑

n=1

e−bn converge?




8. [8 points] Consider the power series

∞∑

n=1

1

2n√n(x− 5)n.

In the following questions, you need to support your answers by stating and properlyjustifying the use of the test(s) or facts you used to prove the convergence or divergenceof the series. Show all your work.

a. [2 points] Does the series converge or diverge at x = 3?

Solution: At x = 3, the series is∞∑

n=1

(−1)n√n

, which converges by the alternating

series test, since 1/√n is decreasing and converges to 0.

b. [2 points] What does your answer from part (a) imply about the radius of conver-gence of the series?

Solution: Because it converges at x = 3, we know that the radius of convergenceR ≥ 2.

c. [4 points] Find the interval of convergence of the power series.

Solution: Using the ratio test, we have

limn→∞

1

2n+1√n+1

|x− 5|n+1

1

2n√n|x− 5|n =

1

2|x− 5| = L,

so the radius of convergence is 2. Now we have to check the endpoints. We know

from part (a) that it converges at x = 3. For x = 7, we get∞∑

n=1

1√n, which diverges.

Thus, the interval of convergence is 3 ≤ x < 7.



5. [9 points] Consider the following power series:

∞∑

n=1

1

n5n(x− 4)n+1

a. [4 points] Find the radius of convergence of the power series. Show all your work.

Solution:

limn→∞

1(n+1)5n+1

1n5n

|x− 4|n+2

|x− 4|n+1=

1

5|x− 4| lim

n→∞

n

n+ 1=

1

5|x− 4|.

1

5|x− 4| < 1 ⇔ |x− 4| < 5,

so R = 5.

b. [5 points] For which values of x does the series converge absolutely? For whichvalues of x does it converge conditionally?

Solution: Converges absolutely inside radius: (−1, 9).Left endpoint: x = −1,

∞∑

n=1

1

n5n(−5)n+1 =

∞∑

n=1

(−1)n+1

n,

converges conditionally (alternating harmonic series).Right endpoint: x = 9,

∞∑

n=1

1

n5n5n+1 = 5

∞∑

n=1

1

n,

diverges. So, converges conditionally for x = −1, absolutely for −1 < x < 9.


math 116 — practice for exam 2 - university of michigantghyde/practiceexam2sols.pdf · math 116...

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