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MATH115Summer2022.pdf
MATH 115 Precalculus Summer, 2022
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MATH 115 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator. You must complete the exam individually. Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.
There are 28 problems. Problems #1–6 are Multiple Choice. Problems #7–17 are Short Answer. (Work not required to be shown) Problems #18–28 are Short Answer with work required to be shown.
MULTIPLE CHOICE
1. Solve | 6 – 5x | ≤ 14 and write interval notation for the solution set.
A. ��−∞,−8/5 �∪ �4�,∞ �
B. ��−∞,−8/5 �
C. �4�,∞ �
D. �−8/5, 4
2. Which of the following polynomials has a graph which exhibits the end behavior of downward to the left and upward to the right?
A. f (x) = –5x 6 + 2x
+ 6
B. f (x) = 3x 4 + 6x
3 – x
C. f (x) = 4x 3 + 3x
3 – x
D. f (x) = –6x 5 – 5x
3 – 1
3. Express as a single logarithm: log x − 8 log z + log 1
A. log �� − 8� + 1
B. log � ����
C. log ����� � �
D. log � ������
vvvv
MATH 115 Precalculus Summer, 2022
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4. Determine the interval(s) on which the function is increasing.
A. (–4.5, –1) and (2.5, ∞)
B. (–∞, –3) and (1, ∞)
C. (–3, 1)
D. (–2, 2)
5. Which of the functions corresponds to the graph?
A. ( ) 3xf x e−= +
B. ( ) 3xf x e= − +
C. ( ) 4xf x e= +
D. ( ) 4xf x e−= +
MATH 115 Precalculus Summer, 2022
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6. Which of the functions corresponds to the graph?
A. f (x) = 2 – sin x
B. f (x) = 2 + cos x
C. f (x) = 3 – cos x
D. f (x) = 2(1 – cos x)
SHORT ANSWER:
7. Points (–6, 5) and (2, 1) are endpoints of the diameter of a circle.
(a) What is the exact length of the diameter? (Simplify as much as possible)
(b) What is the center of the circle?
(c) What is the equation of the circle?
8. Find the value of the logarithm: log� � � ��.
9. Ted, a resident of Metropolis, pays Metropolis an annual tax of $85 plus 1.3% of his annual income. If Ted paid $1,086 in tax, what was Ted’s income?
MATH 115 Precalculus Summer, 2022
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10. A can of soda at 80° F. is placed in a refrigerator that maintains a constant temperature of 37° F. The temperature T of the soda t minutes after it is placed in the refrigerator is given by
T(t) = 37 + 43 e – 0.058 t
Find the temperature of the soda 10 minutes after it is placed in the refrigerator. (Round to the nearest tenth of a degree.)
11. Given the function ��� = 13� −8, find a formula for the inverse function.
12. (a) State the reference angle associated with 304°.
(b) Convert 304° to radians. Leave the answer in terms of π.
13. Given y = 5 sin(6x – π), state the
(a) period
(b) phase shift
14. Solve the trigonometric equation (cos x - 1/2)(2 sin x − 1) = 0 in the interval [0, 360°).
15. (a) Find the exact value of cos$� �sin'(� �
(b) Find the exact value of sin$� �sin'() �
16. For the parabola given by (y + 3)2 = 8 (x – 6), find the following:
(a) direction parabola opens (to the left, right, up, or down)
(b) vertex
(c) focus
MATH 115 Precalculus Summer, 2022
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17. Let ( ) 3
f x x
3x − 4 =
− .
(a) State the domain.
(b) State the vertical asymptote(s).
(c) State the horizontal asymptote.
(d) Which of the following represents the graph of 3 2
( ) 2
x f x
x
− =
− ?
GRAPH A. GRAPH B.
GRAPH C. GRAPH D
MATH 115 Precalculus Summer, 2022
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SHORT ANSWER, with work required to be shown, as indicated.
18. Find the equation for a line which passes through the points (–3, 11) and (1, 7). Write the equation in slope-intercept form. Show work.
19. Find the exact solutions and simplify as much as possible: 32x 2 + 13 = 40x. Show work.
20. Let f (x) = 7x2 – 4 and g(x) = x – 3.
(a) Find the composite function ( f o g )( x) and simplify. Show work. (b) Find ( f o g )(−2) . Show work.
21. A projectile is launched from a platform 15 feet high with an initial velocity of 96 feet per second,
The height h of the projectile at t seconds after launch is given by h = –16t 2 + 96t + 15 feet.
(a) How many seconds after launch does the projectile attain maximum height? Show work.
(b) What is the maximum height? Show work.
22. Solve � � * � � + +
�� �, $ )* = 0. Show work.
23. Suppose that sin θ = 12/13 and that θ is a Quadrant II angle.
(a) Find the exact value of cos θ. Show work.
(b) Find the exact value of sin 2θ. Show work.
24. Prove the identity 1 − (sin x − cos x) 2 = sin(2x)
MATH 115 Precalculus Summer, 2022
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25. From a point 65 feet from the base of a redwood tree, the angle of elevation to the top of the tree is 49.3°. Find the height of the tree to the nearest foot. Show work. (sketch is not to scale)
26. For the triangle ABC, we are given that A = 48°, B = 62°, and c = 35.0.
Find the length of side b, rounded to the nearest tenth. Show work.
27. Let u = ⟨4, -5⟩ and v = ⟨10, 8⟩.
(a) Calculate the dot product u • v. Show work.
(b) Determine the angle between u and v. Round the result to the nearest degree. Show work.
28. An ellipse has the equation �� $ � , .) +
�/ $ * , �0 = 1
(a) Is the major axis horizontal or vertical?
(b) Find the exact values of the foci of the ellipse. Show work.
