Math 2144 Exam 3 Spring 2019 NAME:

INSTRUCTIONS: This exam is a closed book exam. You may not use your text, home-work, or other aids except for a 3 × 5-inch notecard. You may use an allowable calculator,TI-83 or TI-84 to• perform operations on real numbers,

• evaluate functions at specific values, and

• look at graphs and/or tables.A TI-89, TI-Nspire, or a calculator with a computer algebra system, any technology withwireless or Internet capability (i.e. laptops, tablets, smart phones or watches), a QWERTYkeyboard, or a camera are not allowed. Unless otherwise stated, you must show all ofyour work including all steps needed to solve each problem and explain your reasoning inorder to earn full credit. This means that correct answers using incorrect reasoningmay not receive any credit. This exam assesses your understanding of material coveredup to and including Section 5.7 of Rogawski and Adams.

Turn off all noise-making devices and all devices with an internet connection and put themaway. Put away all headphones, earbuds, etc.

This exam consists of 7 problems on 10 pages. Make sure all problems and pages are present.

The exam is worth 63 points in total.

You have 60 minutes to work starting from the signal to begin. Good luck!

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Math 2144 Exam 3

Exam 3 Grade byProblem Number

No. Out of Pts.

1 10

2 6

3 10

4 9

5 16

6 8

7 4

Total 63

Current Course Grade by Category

Category Out of Current

Exam 1 100%

Exam 2 100%

Exam 3 100%

WebAssign 100%

Classwork/Quiz/HW 100%

Overall 14 Week Grade 100%

2

Math 2144 Exam 3

1. (2 points each) Answer the following multiple choice questions by circling your answer.No justification or explanation is required.

(i) If g′(t) represents a child’s rate of growth in pounds per year, which of the followingexpressions represents the increase in the child’s weight (in pounds) between years2 and 5?

(a)

∫ 5

2

g′(t) dt

(b) g′(5)− g′(2)

(c)

∫ 2

5

g′(t) dt

(d)g′(5)− g′(2)

5− 2

(e)d

dt(g′(t))

∣∣∣∣52

(ii) The function y = g(t) represents the relationship between the rate of change inthe value of investment stocks (in dollars per month) and the number of monthst elapsed since the stocks were purchased. Which of the following sums approxi-mates the change in the value of the stocks over the interval of time from 4 to 7months after the stocks were purchased?

(a)7∑

k=4

g(k)

(b)7∑

k=4

g(t) ·∆t

(c)6∑

k=1

g(4 + 0.5k) · 0.5

(d)3∑

k=0

g(4 + k) ·∆t

(e)3∑

k=0

g(4 + k)

3

Math 2144 Exam 3

(iii) The value of the definite integral

∫ π

0

cos(2x) dx is equal to the value of which of

the following expressions?

(a)

∫ π

0

cos(u) du

(b)1

2

∫ 2π

0

cos(u) du

(c)

∫ 2π

0

2 cos(u) du

(d) 2

∫ π

0

cos(u) du

(e)1

2

∫ π

0

cos(u) du

(iv) How many values of k in the interval[−π

2, π]

satisfy the equation∫ k

0

sin(4x) dx = 0? (The graph of f(x) = sin(4x) is given below.)

(a) 4

(b) 7

(c) 0

(d) 3

(e) 1

(v) Suppose

∫ 9

−2f(x)dx = 12 and

∫ 9

3

f(x)dx = 4. What is

∫ 3

−2f(x)dx?

(a) 16

(b) −8

(c) 3

(d) 8

(e) −16

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Math 2144 Exam 3

2. (6 points) Suppose you want to determine the maximum area of an isosceles trianglethat has a perimeter of 10 inches (see the image below). Write the function that youwould need to differentiate (in terms of a single variable) to solve this optimizationproblem.

(Note: Do not solve the optimization problem; simply write the function that you wouldneed to differentiate to solve the problem.)

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Math 2144 Exam 3

3. (10 points) A box with a square base is pictured below.

(a) (2 points) Write a formula for the volume of the box.

(b) (2 points) Write a formula for the surface area of the box.

(c) (2 points) Given that we have 48 ft2 of material to use to make the box, expressthe volume of the box as a function of one variable (use parts (a) and (b)).

(d) (4 points) Find the dimensions of the box of maximum volume whose surface areais 48 ft2.

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Math 2144 Exam 3

4. (3 points each) The table below gives the rate r(t) at which a mayoral candidate isgaining votes t days after she announced her candidacy. Assume that the rate at whichthe candidate is gaining votes is increasing for the first 32 days since she announcedher candidacy. Also assume that at the moment she announced her candidacy, she had520 supporters.

t 0 4 8 12 16 20 24 28 32r(t) 75 82 96 107 152 210 287 360 449

(a) Use the right endpoint approximation with 4 equal-width subintervals, R4, toapproximate the total number of supporters the mayoral candidate has after 32days.

(b) Is your approximation in part (a) an overestimate or an underestimate of the totalnumber of supporters the mayoral candidate has after 32 days? Explain yourresponse.

(c) Write an integral that represents the exact total number of supporters the mayoralcandidate has after 32 days. Do not evaluate this integral. (Note: You donot need to construct a function definition for r(t). Simply use “r(t)”.)

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Math 2144 Exam 3

5. (4 points each) Evaluate each of the following. Show all of your work and give exactanswers. If the answer is a number, do not round.

(a)

∫ (18x5 − 10x4 − 28

x

)dx =

(b)

∫ 3

1

dt

t2=

(c)

∫θ sin

(θ2)dθ

(d)d

dx

∫ x2

1

(t5 − 9t3

)dt =

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Math 2144 Exam 3

6. (8 points) The graph of the function y = f(t) is given below. Note that the graph of fis a semicircle from t = −6 to t = 0. Define A(x) by

A(x) =

∫ x

−6f(t) dt.

(a) (4 points) Evaluate A(−8) and A(0).

(b) (2 points) For what value(s) of x on the interval [−8, 8] does A(x) have a localmaximum?

(c) (2 points) Find A′(2). (Note that this is the same as ddxA(x)

∣∣x=2

.)

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Math 2144 Exam 3

7. (4 points) The graph of the function g is given below.

Each of the expressions below represents a numerical value. Identify which of the fol-lowing expressions represents the largest value and which represents the smallest value.To receive credit, you must convey your rationale for your selections.

(i) g′(1)

(ii)

∫ 5

1

g(x)dx

(iii)10∑k=1

g(1 + 0.4k) · 0.4

(iv) g′′(1)

Smallest:

Largest:

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Math 2144 Exam 3

BASIC FORMULAS

d

dxxn = nxn−1

d

dxex = ex

d

dxln(x) =

1

x

d

dxloga(x) =

1

x ln(a)

d

dxax = ax ln(a)

d

dxsin(x) = cos(x)

d

dxcos(x) = − sin(x)

d

dxtan(x) = sec2(x)

d

dxcot(x) = − csc2(x)

d

dxsec(x) = sec(x) tan(x)

d

dxcsc(x) = − csc(x) cot(x)

d

dxsin−1(x) =

1√1− x2

d

dxcos−1(x) = − 1√

1− x2d

dxtan−1(x) =

1

1 + x2

d

dxcot−1(x) = − 1

1 + x2∫un du =

un+1

n+ 1+ C, n 6= −1∫

du

u= ln |u|+ C

∫eu du = eu + C∫au du =

au

ln a+ C∫

sin(u) du = − cos(u) + C∫cos(u) du = sin(u) + C∫sec2(u) du = tan(u) + C∫csc2(u) du = − cot(u) + C∫sec(u) tan(u) du = sec(u) + C∫csc(u) cot(u) du = − csc(u) + C∫tan(u) du = ln | secu|+ C∫cot(u) du = ln | sinu|+ C∫sec(u) du = ln | secu+ tanu|+ C∫csc(u) du = ln | cscu+ cotu|+ C∫

du√a2 − u2

= sin−1(ua

)+ C∫

du

a2 + u2=

1

atan−1

(ua

)+ C∫

f (u(x)) · u′(x) dx =

∫f(u) du

Ln = ∆xn∑k=1

f (a+ (k − 1)∆x)

Rn = ∆xn∑k=1

f (a+ k∆x)

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