Name:

Section #:

Math 165: Final Exam — Part 1

Spring 2017

This part of the exam has 5 problems for a total of 40 possible points. Each problem is

worth 8 points.

You may NOT use a calculator on this section. You must show all work, but you need not

simplify your answers unless instructed to do so. This part of the exam will be collected

after 50 minutes.

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:

Total Points: /40

Question 1. Given that

Z 3

�1

f(t) dt = 10, find the value of

Z 2

0

(f(2x� 1)� x) dx.

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Question 2. Find, in simplest form, the value of

Ze

2

1

p5 + 2 ln x

3x

dx.

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Question 3. Evaluate

lim

x!0+(1 + sin(2x))

1x

.

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Question 4. The function g(t) = t

3 � 7t + 5 is one-to-one and g(�3) = �1. Find the

linearization to the inverse function y = g

�1(t) at t = �1.

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Question 5.

a. (4 points.) For what values of x is it true that

cos

�1(cos x) = x ?

b. (4 points.) Simplify

sin(tan

�14).

Your answer should involve no trigonometric and no inverse trigonometric functions.

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Name:

Section #:

Math 165: Final Exam — Part 2

Spring 2017

This part of the exam has 5 problems. Each problem is worth 12 points.

Answer each question completely. Show all work. No credit is allowed for mere answers withno work shown. Show the steps of calculations. State the reasons that justify conclusions.

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:

Total Points: /60

Question 1. A function F is defined for all real numbers x by

F (x) =

Zx

1

e

t(t+ 1)(t� 1)(t� 3)2 dt.

Find, with complete justification, the x-coordinates of all critical points for F and tell, withjustification, whether each corresponds to a maximum, a minimum, or neither.

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Question 2. A large decorative window is to be constructed in the form of a glass rectanglesurmounted by a half disk made of a di↵erent glass, as in the figure below. The perimeter(distance around the outside edge) of the completed window is to be 100 feet. Find thedimensions of the window of maximal area, that is, for this window find the radius of thehalf disk and the height of the rectangular portion.

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Question 3.

a. In the figure below, the shaded region is bounded below by the parabola y = x

2 andabove by the line y = b. Find the area of this region. (6 points.)

b. If the shaded area is growing at a rate of 324 square feet per hour, at what rate is b

changing when the area of the shaded region is 972 square feet? (6 points.)

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Question 4. The function f is defined on the interval (0, 10). On this interval f is positive,decreasing, and concave down. Define a new function h by

h(x) = tan�1(f(x))� 6x2, for 0 < x < 1.

a. Determine, if possible, whether h is increasing or decreasing on (0, 10). If it is possibleto make this determination, then state which is the case, with justification. If youcannot make the determination (i.e. you need more information) state this and tellwhat additional information you would need. (6 points.)

b. Determine, if possible, whether h is concave up or concave down on (0, 10). If it ispossible to make this determination, then state which is the case, with justification. Ifyou cannot make the determination (i.e. you need more information) state this andtell what additional information you would need. (6 points.)

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Question 5. Invested money in an account grows at a rate proportional to the money inthe account. You invest $10,000 and are pleased to see that two years later your investmenthas grown to $12,500. How many years (from your initial investment date) will it take yourinitial investment to double? Assume that the interest rate on your investment is constant.

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