math 115 — practice for exam 2 · 2. do not separate the pages of the exam. if any pages do...

Math 115 — Practice for Exam 2

Generated November 10, 2019

Name:

Instructor: Section Number:

1. This exam has 16 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 2018 3 1 unicycle 10

Fall 2006 2 7 Poiseuille’s Law 12

Fall 2008 2 6 portfolio 16

Fall 2017 2 6 6

Winter 2019 2 6 10

Fall 2014 2 2 11

Winter 2017 2 4 10

Winter 2018 2 5 15

Fall 2016 2 3 7

Winter 2007 2 7 Octopus 14

Winter 2007 3 4 octopus2 12

Winter 2009 2 4 turtle river 16

Fall 2015 2 6 dog park 11

Fall 2015 2 4 10

Winter 2015 3 10 5

Fall 2018 2 6 4

Total 169

Recommended time (based on points): 160 minutes

Math 115 / Final (Friday, December 14, 2018) page 2

1. [10 points] Brianna is riding her unicycle on William Street. As she rides, she passes the AnnArbor District Library. The function u(t) represents Brianna’s location (in meters west of thelibrary) when she has been riding her unicycle for t seconds. The table below shows some values ofu′(t), the derivative of u(t).

t 0 2 5 10 15 18 20 23 25 30

u′(t) 0 1 2 2.5 1.5 0 -1 -1.5 -2 -3Note the following:

i) u(23) = 2.

ii) u′(t) is continuous.

iii) u′(t) satisfies:

• u′(t) is increasing on (0, 10).

• u′(t) is decreasing on (10, 30).

a. [2 points] Circle any of the following intervals on which u(t) could be invertible.

[3, 8] [2, 15] [5, 20] [10, 25] none of these

b. [3 points] u(t) is invertible on the interval [20, 30]. Let f(t) be the inverse of u(t) on thatinterval. Calculate f ′(2) and include units.

Answer:

c. [2 points] Find the value of limx→23

u(x)− u(23)

x− 23. If the limit does not exist, write DNE. If it

cannot be determined based on the information given, write NI.

Answer:

d. [1 point] Estimate the value of u′′(24).

Answer:

e. [2 points] Which of the following values of t could be inflection points of u(t)?

5 10 17 18 23 none of these

University of Michigan Department of Mathematics Fall, 2018 Math 115 Exam 3 Problem 1 (unicycle)

8

7. (12 points) The flux F , in millilitres per second, measures how fast blood flows along a bloodvessel. Poiseuille’s Law states that the flux is proportional to the fourth power of the radius, R,of the blood vessel, measured in millimeters. In other words F = kR

4 for some positive constantk.

(a) Find a linear approximation for F as a function of R near R = 0.5. (Leave your answer interms of k).

(b) A partially clogged artery can be expanded by an operation called an angioplasty, whichwidens the artery to increase the flow of blood. If the initial radius of the artery was 0.5mm,use your approximation from part (a) to approximate the flux when the radius is increasedby 0.1mm.

(c) Is the answer found in part (b) an under- or over-approximation? Justify your answer.

University of Michigan Department of Mathematics Fall, 2006 Math 115 Exam 2 Problem 7 (Poiseuille’s Law)

6

6. In Modern Portfolio Theory, a client’s portfolio is structured in a way that balances risk andreturn. For a certain type of portfolio, the risk, x, and return, y, are related by the equationx − 0.45(y − 2)2 = 2.2. This curve is shown in the graph below. The point P represents aparticular portfolio of this type with a risk of 3.8 units. The tangent line, l, through point P isalso shown.

x, Risk

(Standard Deviations)

y, Return (% per year)

b

x − 0.45(y − 2)2 = 2.2

l

P

(a) (5 points) Using implicit differentiation, find dy/dx, and the coordinates of the point(s)where the slope is undefined.

(b) (8 points) The y-intercept of the tangent line for a given portfolio is called the Risk Free Rateof Return. Use your answer from (a) to find the Risk Free Rate of Return for this portfolio.

(c) (3 points) Now, estimate the return of an optimal portfolio having a risk of 4 units by usingyour information from part (b). Would this be an overestimate or an underestimate? Why?

University of Michigan Department of Mathematics Fall, 2008 Math 115 Exam 2 Problem 6 (portfolio)

Math 115 / Exam 2 (November 13, 2017) page 7

6. [6 points] Let L(x) be the linear approximation and Q(x) be the quadratic approximation to thefunction d(x) near x = 1. Suppose that d′(x), d′′(x) and d′′′(x) are defined for all real numbers.Let Q(x) = 7(x− 1)2 − 8(x− 1) + 3. Find the exact value of the following quantities. If there is notenough information to answer the question, write “NI”.

d(0) = d′(1) = d′′(1) =

L′(2) = Q′′′(1) = d′′′(1) =

7. [5 points] Sketch graphs of functions f(x) and g(x) satisfying the conditions below, or circle no

such function exists. You do not need to explain your answer.

A function f(x) defined on the interval(0, 4) that satisfies:

i) f ′(x) > 0 for all x 6= 2.

ii) x = 2 is a global minimum.

−1 1 2 3 4 5

−1

1

2

3

4

5

x

y = f(x)

or

no such function exists

A continuous function g(x) defined on theinterval (0, 4) that satisfies:

i) limx→2−

g′(x) = ∞.

ii) limx→2+

g′(x) = 0.

−1 1 2 3 4 5

−1

1

2

3

4

5

x

y = g(x)

or

no such function exists

University of Michigan Department of Mathematics Fall, 2017 Math 115 Exam 2 Problem 6

Math 115 / Exam 2 (March 26, 2019) page 8

6. [10 points]

a. [4 points] Below is a table of values for a differentiable function g(x). Also shown are somevalues of g′(x), which is an increasing function and also differentiable.

x 3 8 10

g(x) 10 1 0

g′(x) −4 0.6 2

i. [2 points] Write a formula for L(x), the linear approximation of g(x) at x = 3.

Answer: L(x) =

ii. [1 point] Use your formula for L(x) to estimate g(3.2).

Answer: g(3.2) ≈

iii.[1 point] Is your estimate of g(3.2) an overestimate or an underestimate? Circle youranswer.

Overestimate Underestimate Cannot be determined

b. [2 points] The quadratic approximation of g(x) at x = 10 is

Q(x) = 2(x− 10) + 2(x− 10)2.

Find g′′(10).

Answer: g′′(10) =

c. [4 points] Let h(x) = (g(x))3. The linear approximation of h(x) at x = 6 is

K(x) = 8 + 3(x− 6).

Find g(6) and g′(6).

Answer: g(6) = Answer: g′(6) =

University of Michigan Department of Mathematics Winter, 2019 Math 115 Exam 2 Problem 6

Math 115 / Exam 2 (Nov 11, 2014) page 3

2. [11 points]

Shown to the right is the graph of a functionf(x).

1234567

−1−2−3−4−5

1 2 3 4 5 6 7−1

bc

bc

b

b

b

b

b

x

y

y = f(x)

Note that you are not required to show your work on this problem. However, limited partialcredit may be awarded based on work shown.

Find each of the following values. If the value does not exist, write does not exist.

a. [3 points] Let h(x) = f(3x+ 1). Find h′(1).

Answer: h′(1) =

b. [3 points] Let k(x) = ef′(x). Find k′(6).

Answer: k′(6) =

c. [2 points] Find (f−1)′(0).

Answer: (f−1)′(0) =

d. [3 points] Let j(x) =f(2x+ 1)

x+ 1. Find j′(1).

Answer: j′(1) =


Math 115 / Exam 2 (March 22, 2017) do not write your name on this exam page 5

4. [10 points] A portion of the graph of the function w(x) is shown below.

For each of the parts below, find the value of thegiven quantity. If there is not enough informationprovided to find the value, write not enough

info. If the value does not exist, write does not

exist. You are not required to show your workon this problem. However, limited partial creditmay be awarded based on work shown. All youranswers must be in exact form.

−4 −3 −2 −1 1 2 3 4

−5

−4

−3

−2

−1

1

2

3

y = w(x)

x

y

a. [2 points] Let k(x) = w−1(x). Find k′(−1.5).

Answer: k′(−1.5) =

b. [2 points] Let h(u) = ln(3w(u)). Find h′(1).

Answer: h′(1) =

c. [2 points] Let n(x) =w(x)

1− x2. Find n′(−2).

Answer: n′(−2) =

d. [2 points] Let s(x) be the exponential function s(x) = 4w(x). Find s′(2).

Answer: s′(2) =

e. [2 points] Let p(x) = x · w−1(x). Find p′(−1).

Answer: p′(−1) =


Math 115 / Exam 2 (March 20, 2018.) do not write your name on this exam page 10

5. [15 points] The graph of the function f(x) with domain −4 ≤ x ≤ 8 is shown below.

The function f(x) satisifies:

• f(x) = 1.5x1

3

for −1 < x < 1,

• f(x) = 4 + sin⇣π

4(x− 3)

⌘

for 3 ≤ x < 5 and 5 < x ≤ 8.

−4 −3 −2 −1 1 2 3 4 5 6 7 8

−2

−1

1

2

3

4

5

x

y = f(x)

a. [2 points] Estimate the x-coordinate(s) of all the local minimum(s) of f(x) in−4 < x < 8. Write “None” if f(x) does not have any local minimums.

Answer: x =

b. [3 points] Find the value(s) of b in −4 < b < 8 for which the limit limh→0

f(b+ h)− f(b)

hdoes not exist. Write “None” if there are no such values of b.

Answer: b =

c. [4 points] Estimate the x-coordinate(s) of all critical points of f(x) in −4 < x < 8. Write“None” if f(x) does not have any critical points.

Answer: x =

d. [3 points] On which of the following intervals is the conclusion of the Mean ValueTheorem true? Circle your answer(s).

[−4, 0] [0, 5] [1, 3] [3, 7] None

e. [3 points] On which of the following intervals are the hypotheses of the Mean ValueTheorem true? Circle your answer(s).

[−3,−1] [−2, 2] [0, 2] [3, 5] None


Math 115 / Exam 2 (November 14, 2016) do not write your name on this exam page 4

3. [7 points] Consider the curve D defined by the equation

x2y(1− y) = 9.

Note that the curve D satisfiesdy

dx=

2xy(y − 1)

x2(1− 2y).

a. [4 points] Exactly one of the following points (x, y) lies on the curve D.Circle that one point.

(0.9, 10) (1,−8) (3, 9) (9, 3) (10, 0.9)

Then find an equation for the tangent line to the curve D at the point you chose.

Answer: y =

b. [3 points] Find all points on the curve D where the slope of the curve is undefined. Giveyour answers as ordered pairs. Write none if there are no such points.

Answer: (x, y) =


8

7. (14 points) No matter what is done with the other exhibits, the octopus tank at the zoo mustbe rebuilt. (The current tank has safety issues, and there are fears that the giant octopus mightescape!) The new tank will be 10 feet high and box-shaped. It will have a front made out of glass.The back, floor, and two sides will be made out of concrete, and there will be no top. The tankmust contain at least 1000 cubic feet of water. If concrete walls cost $2 per square foot and glasscosts $10 per square foot, use calculus to find the dimensions and cost of the least-expensive newtank. [Be sure to show all work.]

GIANT OCTOPUS (Enteroctopus)2

Dimensions:

Minimum Cost:

2See http://www.cephbase.utmb.edu/Tcp/pdf/anderson-wood.pdf. (They really DO escape....)

University of Michigan Department of Mathematics Winter, 2007 Math 115 Exam 2 Problem 7 (Octopus)

5

4. (12 points) The zoo has decided to make the new octopus tank spectacular. It will be cylindricalwith a round base and top. The sides will be made of Plexiglas which costs $65.00 per squaremeter, and the materials for the top and bottom of the tank cost $50.00 per square meter. If thetank must hold 45 cubic meters of water, what dimensions will minimize the cost, and what is theminimum cost?

r

h

radius

height

cost

University of Michigan Department of Mathematics Winter, 2007 Math 115 Exam 3 Problem 4 (octopus2)

5

4. (16 points) The Awkward Turtle is going to a dinner party! Unfortunately, he’s running quitelate, so he wants to take the quickest route. The Awkward Turtle lives in a grassy plain (his homeis labeled H in the figure below), where his walking speed is a slow but steady 3 meters perhour. The party is taking place southeast of his home, on the bank of a river (denoted by P inthe figure). The river flows south at a constant rate of 5 meters per hour, and once he gets to theriver, the Awkward Turtle can jump in and float the rest of the way to the party on his back. Atypical path the Awkward Turtle might take from his house to the party is indicated in the figurebelow by a dashed line.

What is the shortest amount of time the entire trip (from home to dinner party) can take? [Recall

that rate × time = distance.]

25 m

15 mT

he

Riv

erH

P

3 m/hr

5 m/hr

Minimal time =

University of Michigan Department of Mathematics Winter, 2009 Math 115 Exam 2 Problem 4 (turtle river)


6. [11 points]

The engineer Elur Niahc has been commissioned tobuild a park for the citizens of Srebmun Foyoj. Thepark will consist of a square attached to a rectangulardog park (as shown in the diagram on the right).The fencing for the dog park (bold, dashed line) costs$4 per linear meter, and the fencing for the three re-maining sides of the square portion of the park (bold,solid line) costs $6 per linear meter.

w

ℓ

w

a. [5 points] Assume that Elur spends $2400 on fencing. The resulting park will have widthw meters, and the length of the dog park will be ℓ meters, as shown in the diagramabove. Find a formula for ℓ in terms of w.

Answer: ℓ =

b. [3 points] Let A(w) be the total area (in square meters) of the resulting park (includingthe dog park) if the width is w meters and Elur spends $2400 on fencing. Find a formulafor the function A(w). The variable ℓ should not appear in your answer.(Note: This is the function that Elur would use to find the value of w maximizing thearea of the park, but you should not do the optimization in this case.)

Answer: A(w) =

c. [3 points] In the context of this problem, what is the domain of A(w)?

Answer:University of Michigan Department of Mathematics Fall, 2015 Math 115 Exam 2 Problem 6 (dog park)


4. [10 points] A function f(x) is defined and differentiable on the interval 0 < x < 10. Inaddition, f(x) and f ′(x) satisfy all of the following properties:

• f ′(x) is continuous on the interval 0 < x < 10.

• f ′(1) = 2.

• f ′(x) is differentiable on the interval 1 < x < 5.

• f(x) is concave up on the interval 3 < x < 5.

• f(x) has a local minimum at x = 4.

• f(x) is decreasing on the interval 6 < x < 8.

• f(x) has an inflection point at x = 7.

• f ′(x) is not differentiable at x = 9.

On the axes provided below, sketch a possible graph of f ′(x) (the derivative of f(x)) on theinterval 0 < x < 10.Make sure your sketch is large and unambiguous.

Graph of y = f ′(x)

−4

−3

−2

−1

1

2

3

4

x

y

1 2 3 4 5 6 7 8 9 10


Math 115 / Final (April 23, 2015) page 10

10. [5 points] Shown on the axes below are the graphs of y = f(x), y = f ′(x), and y = f ′′(x).

x

y

I

II

III

Determine which graph is which and circle theone correct response below.

• f(x): I, f ′(x): II, and f ′′(x): III

• f(x): I, f ′(x): III, and f ′′(x): II

• f(x): II, f ′(x): I, and f ′′(x): III

• f(x): II, f ′(x): III, and f ′′(x): I

• f(x): III, f ′(x): I, and f ′′(x): II

• f(x): III, f ′(x): II, and f ′′(x): I

11. [4 points] Suppose w and r are continuous functions on (−∞,∞), W (x) is an invertibleantiderivative of w(x), and R(x) is an antiderivative of r(x).Circle all of the statements I-VI below that must be true.If none of the statements must be true, circle none of these.

I. W (x) +R(x) + 2 is an antiderivative of w(x) + r(x).

II. W (x) +R(x) is an antiderivative of w(x) + r(x) + 2.

III. cos(W (x)) is an antiderivative of sin(w(x)).

IV. eW (x) is an antiderivative of w(x)ew(x).

V. eR(x) is an antiderivative of r(x)eR(x).

VI. If w is never zero, then W−1(R(x)) is an antiderivative ofr(x)

w(W−1(R(x)).

VII. none of these



6. [4 points] The graph of the function f(x) is shown below. Note that f(x) has a vertical tangentline at x = 5.

1 2 3 4 5 6 7 8 9 10 11 12−1

1

2

3

4

5

x

y = f(x)

a. [2 points] On which of the following intervals does the function f(x) satisfy the hypotheses ofthe Mean Value Theorem? Circle the correct answer(s).

[0,2] [1,3] [2,4] [3,5] none of

these

b. [2 points] On the interval [8, 12] the hypotheses of the Mean Value Theorem are true for thefunction f(x). What does the conclusion of this theorem say in this interval?Answer:

7. [5 points]Yi is constructing a cardboard box. The base of the box willbe a square of width w inches. The height of the box will beh inches. Yi will use gray cardboard for the sides of the boxand brown cardboard for the bottom (the box does not have atop). Gray cardboard costs $0.05 per square inch, while browncardboard costs $0.03 per square inch. Yi wants to spend $20on the cardboard for his box.

Write a formula for h in terms of w. ww

h

Answer: h =


math 115 — practice for exam 2 · 2. do not separate the pages of the exam. if any pages do...

Documents