Math 115 — Practice for Exam 2mconger/dhsp/115packet2.pdfMath 115 / Exam 2 (March 22, 2017) do not write your name on this exam page 6 5. [12 points] Let f(x) = x(x−4)4/5e−x

Math 115 — Practice for Exam 2

Generated November 12, 2017

Name:

Instructor: Section Number:

1. This exam has 18 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 2017 2 5 12

Winter 2014 3 3 potential energy 11

Fall 2009 2 4 UM mug 12

Winter 2017 2 6 corn snake 9

Fall 2009 2 6 gravity 14

Fall 2016 2 11 6

Winter 2012 2 6 16

Winter 2011 2 4 antihistamine 15

Fall 2016 2 6 14

Fall 2009 2 7 Kampyle of Eudoxus 14

Winter 2014 2 11 5

Winter 2014 2 4 ship 8

Fall 2014 2 9 caffeine 10

Fall 2014 2 4 garden 12

Winter 2007 3 4 octopus2 12

Fall 2002 3 10 trough 10

Winter 2016 2 6 11

Winter 2013 2 6 9

Total 200

Recommended time (based on points): 190 minutes

Math 115 / Exam 2 (March 22, 2017) do not write your name on this exam page 6

5. [12 points] Let

f(x) = x(x− 4)4/5e−x and f ′(x) =(5− x)(5x− 4)e−x

5 5√x− 4

.

Note that the domain of f(x) is (−∞,∞).

a. [6 points] Find all values of x at which f(x) has a local extremum. Use calculus to findand justify your answers, and be sure to show enough evidence to demonstrate that youhave found all local extrema. For each answer blank below, write none if appropriate.

Answer:

Local max(es) at x = Local min(s) at x =

b. [6 points] Find the values of x for which f(x) attains a global maximum and globalminimum. Use calculus to find and justify your answers, and be sure to show enoughevidence to demonstrate that you have found all global extrema. Write none ifappropriate.

Answer:

Global max(es) at x = Global min(s) at x =

University of Michigan Department of Mathematics Winter, 2017 Math 115 Exam 2 Problem 5

Math 115 / Final (April 28, 2014) page 4

3. [11 points] For positive constants a and b, the potential energy of a particle is given by

U(x) = a

(

5b2

x2−

3b

x

)

.

Assume that the domain of U(x) is the interval (0,∞).

a. [2 points] Find the asymptotes of U(x). If there are none of a particular type, write none.

Answer: Vertical asymptote(s): Horizontal asymptote(s):

b. [6 points] Find the x-coordinates of all local maxima and minima of U(x) in the domain(0,∞). If there are none of a particular type, write none. You must use calculus to findand justify your answers. Be sure to provide enough evidence to justify your answers fully.

Answer: Local max(es) at x = Local min(s) at x =

c. [3 points] Suppose U(x) has an inflection point at (6,−14). Find the values of a and b.Show your work, but you do not need to verify that this point is an inflection point.

Answer: a = and b =University of Michigan Department of Mathematics Winter, 2014 Math 115 Exam 3 Problem 3 (potential energy)

Math 115 / Exam 2 (November 24, 2009) page 5

4. [12 points] In preparation for the holidays, a local bookstore is planning to sell mugs of avariety of shapes. Suppose that the amount of liquid in a “UM” mug if filled to a depth of h

cm is L(h) = Uh(3M2− 3Mh + h2) cm3 for U,M > 0.

a. [4 points] Find and classify any critical points of L on the interval (0, 5M).

b. [2 points] Determine any points of inflection of L on the interval (0, 5M).

c. [6 points] Suppose you are pouring coffee into a “UM” mug at a rate of 15 cm3 per second.At what rate is the depth of the coffee in the mug changing when the coffee reaches adepth of 4 cm in the mug?

University of Michigan Department of Mathematics Fall, 2009 Math 115 Exam 2 Problem 4 (UM mug)

Math 115 / Exam 2 (March 22, 2017) do not write your name on this exam page 7

6. [9 points] A group of biology students is studying the length L of a newborn corn snake (incm) as a function of its weight w (in grams). That is, L = G(w). A table of values of G(w) isshown below.

w 5 10 15 20 25

G(w) 24.5 31.6 38.7 44.7 50

G′(w) 2.23 1.58 1.30 1.12 1.05

Assume that G′(w) is a differentiable and decreasing function for 0 < w < 25.

a. [2 points] Find a formula for H(w), the tangent line approximation of G(w) near w = 20.

Answer: H(w) =

b. [1 point] Use the tangent line approximation of G(w) near w = 20 to approximate thelength of a corn snake that weighs 22 grams.

Answer:

c. [2 points] Is your answer in part (b) an overestimate or an underestimate? Circle youranswer and write a sentence to justify it.

Circle one: Overestimate Underestimate cannot be determined

Justification:

d. [4 points] In their study of the growth of corn snakes, they found the results of a recentarticle that states that the average weight w of a corn snake (in grams) t weeks afterbeing born is given by w = 1

5t2. Let S(t) = G

(

1

5t2)

be the length of a corn snake t weeksafter being born. Find a formula for P (t), the tangent line approximation of S(t) neart = 5.

Answer: P (t) =



6. [14 points] The force F due to gravity on a body at height h above the surface of the earth isgiven by

F (h) =mgR2

(R + h)2

where m is the mass of the body, g is the acceleration due to gravity at sea level (g < 0), andR is the radius of the earth.

a. [3 points] Compute F ′(h).

b. [3 points] Compute F ′′(h).

c. [5 points] Find the best linear approximation to F at h = 0.

d. [3 points] Does your approximation from part (c) give an overestimate or an underestimateof F? Why?

University of Michigan Department of Mathematics Fall, 2009 Math 115 Exam 2 Problem 6 (gravity)

Math 115 / Exam 2 (November 14, 2016) do not write your name on this exam page 11

10. [4 points] Let a and b be constants. Consider the curve C defined by the equation

cos(ax) + by ln(x) = 3 + y3.

Find a formula for dydx

in terms of x and y. The constants a and b may appear in your answer.To earn credit for this problem, you must compute this by hand and show every step of yourwork clearly.

Answer:dy

dx=

11. [6 points] Let h(x) = xx. For this problem, it may be helpful to know the following formulas:

h′(x) = xx (ln(x) + 1) and h′′(x) = xx(

1

x+ (ln(x) + 1)2

)

.

a. [2 points] Write a formula for p(x), the local linearization of h(x) near x = 1.

Answer: p(x) =

b. [4 points] Write a formula for u(x), the quadratic approximation of h(x) at x = 1.(Recall that a formula for the quadratic approximation Q(x) of a function f(x) at x = a

is Q(x) = f(a) + f ′(a)(x− a) + f ′′(a)2 (x− a)2.)

Answer: u(x) =

University of Michigan Department of Mathematics Fall, 2016 Math 115 Exam 2 Problem 11

Math 115 / Exam 2 (March 20, 2012) page 7

6. [16 points] Consider the piecewise linear function f(x) graphed below:

8

10

f(x)

x

(10,−6)

(4, 6)

For each function g(x), find the value of g′(3):

a. [4 points] g(x) = sin(

[f(x)]3)

b. [4 points] g(x) =f(x2)

x

c. [4 points] g(x) = ln(f(x)) + f(2)

d. [4 points] g(x) = f−1(x)



4. [15 points] A model for the amount of an antihistamine in the bloodstream after a patienttakes a dose of the drug gives the amount, a, as a function of time, t, to be a(t) = A(e−t

−e−kt).In this equation, A is a measure of the dose of antihistamine given to the patient, and k isa transfer rate between the gastrointestinal tract and the bloodstream. A and k are positiveconstants, and for pharmaceuticals like antihistamine, k > 1.

a. [5 points] Find the location t = Tm of the non-zero critical point of a(t).

b. [3 points] Explain why t = Tm is a global maximum of a(t) by referring to the expressionfor a(t) or a′(t).

c. [4 points] The function a(t) has a single inflection point. Find the location t = TI of thisinflection point. You do not need to prove that this is an inflection point.

d. [3 points] Using your expression for Tm from (a), find the rate at which Tm changes as kchanges.

University of Michigan Department of Mathematics Winter, 2011 Math 115 Exam 2 Problem 4 (antihistamine)

Math 115 / Exam 2 (November 14, 2016) do not write your name on this exam page 7

6. [14 points] The entire graph of a function g(x) is shown below. Note that the graph of g(x)has a horizontal tangent line at x = 1 and a sharp corner at x = 4.

1 2 3 4 5

1

2

3

4

5y = g(x)

x

y

For each of the questions below, circle all of the available correct answers.(Circle none of these if none of the available choices are correct.)

a. [2 points] At which of the following values of x does g(x) appear to have a critical point?

x = 1 x = 2 x = 3 x = 4 none of these

b. [2 points] At which of the following values of x does g(x) attain a local maximum?

x = 1 x = 2 x = 3 x = 4 none of these

c. [6 points] Let L(x) be the local linearization of g(x) near x = 3. Circle all of thestatements that are true.

L(3) > g(3)

L(3) = g(3)

L(3) < g(3)

L′(3) > g′(3)

L′(3) = g′(3)

L′(3) < g′(3)

L(2.5) > g(2.5)

L(2.5) = g(2.5)

L(2.5) < g(2.5)

L′(2.5) > g′(2.5)

L′(2.5) = g′(2.5)

L′(2.5) < g′(2.5)

L(0) > g(0)

L(0) = g(0)

L(0) < g(0)

L(5) > g(5)

L(5) = g(5)

L(5) < g(5)

none of these

d. [2 points] On which of the following intervals does g(x) satisfy the hypotheses of theMean Value Theorem?

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

e. [2 points] On which of the following intervals does g(x) satisfy the conclusion of theMean Value Theorem?

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

University of Michigan Department of Mathematics Fall, 2016 Math 115 Exam 2 Problem 6


7. [14 points] The Kampyle of Eudoxus is a family of curves that was studied by the Greekmathematician and astronomer Eudoxus of Cnidus in relation to the classical problem ofdoubling the cube. This family of curves is given by

a2x4 = b4(x2 + y2).

where a and b are nonzero constants and (x, y) 6= (0, 0)—i.e.. the origin is not included.

a. [5 points] Finddy

dxfor the curve a2x4 = b4(x2 + y2).

b. [5 points] Find the coordinates of all points on the curve a2x4 = b4(x2 + y2) at which thetangent line is vertical, or show that there are no such points.

c. [4 points] Show that when a = 1 and b = 2 there are no points on the curve at which thetangent line is horizontal.

University of Michigan Department of Mathematics Fall, 2009 Math 115 Exam 2 Problem 7 (Kampyle of Eudoxus)


11. [5 points] A curve C gives y as an implicit function of x. The curve C passes through the point(1, 2) and satisfies

dy

dx=

y2 − 2xy + 4y − 5

4(y − x).

a. [1 point] One of the values below is the slope of the curve C at the point (1, 2). Circlethat one value.

Answer: The slope at (1, 2) is1

4

1

3

1

2

5

8

2

3

3

4

4

5

b. [4 points] One of the following graphs is the graph of the curve C.Which of the graphs I-VI is it? To receive any credit on this question, you must circleyour answer next to the word “Answer” below.

1

2

3

−1

−2

−3

1 2 3−1−2−3

x

y

I

1

2

3

−1

−2

−3

1 2 3−1−2−3

x

y

II

1

2

3

−1

−2

−3

1 2 3−1−2−3

x

y

III

1

2

3

−1

−2

−3

1 2 3−1−2−3

x

y

IV

1

2

3

−1

−2

−3

1 2 3−1−2−3

x

y

V

1

2

3

−1

−2

−3

1 2 3−1−2−3

x

y

VI

Remember: To receive any credit on this question, you must circle your answer next to theword “Answer” below.

Answer: I II III IV V VI



4. [8 points] A ship’s captain is standing on the deck while sailing through stormy seas. Therough waters toss the ship about, causing it to rise and fall in a sinusoidal pattern. Supposethat t seconds into the storm, the height of the captain, in feet above sea level, is given by thefunction

h(t) = 15 cos (kt) + c

where k and c are nonzero constants.

a. [3 points] Find a formula for v(t), the vertical velocity of the captain, in feet per second,as a function of t. The constants k and c may appear in your answer.

Answer: v(t) =

b. [2 points] Find a formula for v′(t). The constants k and c may appear in your answer.

Answer: v′(t) =

c. [3 points] What is the maximum vertical acceleration experienced by the captain? Theconstants k and c may appear in your answer. You do not need to justify your answer orshow work. Remember to include units.

Answer: Max vertical acceleration:

University of Michigan Department of Mathematics Winter, 2014 Math 115 Exam 2 Problem 4 (ship)

Math 115 / Exam 2 (Nov 11, 2014) page 10

9. [10 points] Our friend Oren, the Math 115 student, wants to minimize how long it will takehim to complete his upcoming web homework assignment. Before starting the assignment, hebuys a cup of tea containing 55 milligrams of caffeine.Let H(x) be the number of minutes it will take Oren to complete tonight’s assignment if heconsumes x milligrams of caffeine. For 10 ≤ x ≤ 55

H(x) =1

120x2−

4

3x+ 20 ln(x) .

Instead of immediately starting the assignment, he solves a calculus problem to determine howmuch caffeine he should consume.

a. [8 points] Find all the values of x at which H(x) attains global extrema on the interval10 ≤ x ≤ 55. Use calculus to find your answers, and be sure to show enough evidencethat the points you find are indeed global extrema.

(For each answer blank below, write none in the answer blank if appropriate.)

Answer: global min(s) at x =

Answer: global max(es) at x =

b. [2 points] Assuming Oren consumes at least 10 milligrams and at most 55 milligrams ofcaffeine, what is the shortest amount of time it could take for him to finish his assignment?Remember to include units.

Answer:

University of Michigan Department of Mathematics Fall, 2014 Math 115 Exam 2 Problem 9 (caffeine)

Math 115 / Exam 2 (Nov 11, 2014) page 5

4. [12 points] Researchers are constructing a rectangular garden adjacent to their building. Thegarden will be bounded by the building on one side and by a fence on the other three sides.(See diagram below.) The fencing will cost them $5 per linear foot. In addition, they will alsoneed topsoil to cover the entire area of the garden. The topsoil will cost $4 per square foot ofthe garden’s area.Assume the building is wider than any garden the researchers could afford to build.

building

h feet (fencing)

w feet (fencing)

h feet (fencing) garden

a. [5 points] Suppose the garden is w feet wide and extends h feet from the building, as shownin the diagram above. Assume it costs the researchers a total of $250 for the fencing andtopsoil to construct this garden. Find a formula for w in terms of h.

Answer: w =

b. [3 points] Let A(h) be the total area (in square feet) of the garden if it costs $250 andextends h feet from the building, as shown above. Find a formula for the function A(h).The variable w should not appear in your answer.

(Note that A(h) is the function one would use to find the value of h maximizing the area.You should not do the optimization in this case.)

Answer: A(h) =

c. [4 points] In the context of this problem, what is the domain of A(h)?

Answer:

University of Michigan Department of Mathematics Fall, 2014 Math 115 Exam 2 Problem 4 (garden)

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)

°O±ÔFE�Õ ³j°O±X�.�¢��º�q�C·�b��G¤q�¢�S«¢�Á��]�v�q��¹��q�S�%ø�«¢�S¤À�]Á��G�»©.�%�[�]§S�%¹��²�q��|©��]�G�%�q��¥�v�� Î ¯Q�O�O�v¹��§S��]��§¿°O±1¯=�O�O�x�²�¢�S«S¬sg��c�q�¨§S��0�]¯��q��t�G¤q�¢��«¢�|�¥¤À�c�]��q� Î ¯=�O�O�K¹��¨§S��©� ª°O±1¯=�O�O�x�²�¢�S«1�]��§|�¥¤q�r�G�9©.��#��]®��§�G�[�q��¥�g�q�S�O X�[�¥Å]�9�]�ª�]��«¢�²�vÙ�¹��q�X�q�S�nµ]�O¤q�q��®O�]� ¬

Trough Cross Section

x

10 feet

2 feet 2 feet

Ö �.×cØ ��¥�t��q�S�9�¥¤q��?Á.��G�O¤À�[�g�]¯Ù~Á#�]¯�|®�¤q�¢�q�t�G��®��q�²�¢��]¯�q�S�¶�G¤q�¢��«¢��.�O¤q�.��§��¨®O��¥¤x�²�q��¢�S«Æ�q��§��AÝØ ��¥��K�q�S�¶µ]�¢��¦�v�]¯3�q�S�v�G¤q�¢��«¢�#Ý o �S��¹y ]�¢��¤�¹*�]¤qÅ#¬

ÖGÞ ×�Ø ��¥�g�]�S«¢�²�nÙ�¹��#«¢�²µ]�n�q��v�G¤q�¢�S«¢�»�]¯��¥¤q«]��q�xµ]�¢��¦�]Á��]��§ª¹��¥�g��K�q��¥��µ]�¢��¦�AÝ�e�?��]��X�S�A¹ ]�¢�ª¯Q�¢��§X ]�¢��¤��]��G¹*�O¤�Á��]�²�¢��«¦¹��²�q�»�]�� X�À�S��.�]¤q�q��S«1�Oµ?��§S��®��Æ³Q ]�¢��[�� »��G�v ]�¢��¤�®O�]��®O��¥�G�]¤�·D¬

University of Michigan Department of Mathematics Fall, 2002 Math 115 Exam 3 Problem 10 (trough)


6. [11 points] On the axes provided below, sketch the graph of a single function y = g(x)satisfying all of the following:

• g(x) is defined for all x in the interval −6 < x < 6.

• g(x) has at least 5 critical points in the interval −6 < x < 6.

• The global maximum value of g(x) on the interval −5 ≤ x ≤ −3 is 4, and this occurs atx = −4.

• g(x) is not continuous at x = −2.

• g′(x) (the derivative of g) has a local maximum at x = 0.

• g(x) is continuous but not differentiable at x = 1.

• g′′(x) ≥ 0 for all x in the interval 2 < x < 4.

• g(x) has at least one local minimum on the interval 4 < x < 6 but does not have aglobal minimum on the interval 4 < x < 6.

• g(x) has an inflection point at x = 5.

Make sure your sketch is large and unambiguous.

Graph of y = g(x)

−5

−4

−3

−2

−1

1

2

3

4

5

−6 −5 −4 −3 −2 −1 1 2 3 4 5 6

x

y



6. [9 points] In each of the following problems, draw a graph of a function with all of the indicatedproperties. If there is no such function, then write “NO SUCH FUNCTION EXISTS”. Youdo not need to write any explanations. No partial credit will be given on each part of thisproblem.

a. [3 points] A continuous function f(x), whose domain is all real numbers, with the followingfour properties:(i.) f(x) attains a local minimum somewhere.(ii.) f(x) attains a local maximum somewhere.(iii.) f(x) does not attain a global minimum.(iv.) f(x) does not attain a global maximum.

b. [3 points] A continuous function g(x), whose domain is the closed interval [0, 1], with thefollowing two properties:(i.) g(x) does not attain a global maximum on the interval [0, 1](ii.) g(x) attains a global minimum on the interval [0, 1].

c. [3 points] A differentiable function j(x) with the following two properties:(i.) The linear approximation to j(x) at x = 3 gives an overestimate when used toapproximate j(2).(ii.) The linear approximation to j(x) at x = 3 gives an underestimate when used toapproximate j(4).


Documents

Math 115 — Practice for Exam 2mconger/dhsp/115packet2.pdfMath 115 / Exam 2 (March 22, 2017) do not write your name on this exam page 6 5. [12 points] Let f(x) = x(x−4)4/5e−x