percale exam
MATH 115 Precalculus Fall, 2018, V3.1
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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 | 4 – 3x | 1 and write interval notation for the solution set. 1. _______
A. (−∞, 5
3 ] ∪ [1, ∞)
B. (−∞, 1] ∪ [ 5
3 , ∞)
C. [1, ∞)
D. [1, 5
3 ]
2. Which of the following polynomials has a graph which exhibits the end behavior of
downward to the left and downward to the right? 2. _____
A. f (x) = 5x4 – 3x2 – x
B. f (x) = 3x5 + x2 – x
C. f (x) = –7x3 – x3 – 5
D. f (x) = –6x6 + x3 + 8
3. Express as a single logarithm: 3 log 2 + log x – log (4x) 3. ______
A. log (8𝑥)
log (4𝑥)
B. log (2)
C. log (4x)
D. log (6 – 3x)
MATH 115 Precalculus Fall, 2018, V3.1
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4. Determine the interval(s) on which the function is increasing. 4. ______
A. ( )1.25, 3.25−
B. ( ), 1/ 3− − and (3, 4.75)
C. ( ),1− and ( )4,
D. ( )1/ 3, 3− and ( )4.75,
5. Which of the functions corresponds to the graph? 5. ______
A. ( ) 2xf x e=
B. ( ) 1 xf x e−= +
C. ( ) 1 xf x e= +
D. ( ) 21 xf x e= −
MATH 115 Precalculus Fall, 2018, V3.1
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6. Which of the functions corresponds to the graph? 6. ______
A. f (x) = 2(1 – cos x)
B. f (x) = 2 + cos x
C. f (x) = 2 – sin x
D. f (x) = 3 – cos x
SHORT ANSWER:
7. Points (–7, 2) and (–5, 4) are endpoints of the diameter of a circle.
(a) What is the exact length of the diameter? (Simplify as much as possible) Answer: ________
(b) What is the center of the circle? Answer: ____________
(c) What is the equation of the circle? Answer: ___________________________
8. Find the value of the logarithm: log3 ( 1
27 ). Answer: ____________
9. A salesperson earns a base salary of $1,210 per month and a commission of 4.7% on the
amount of sales. If the salesperson has a paycheck of $2,873.80 for one month, what was the
amount of sales for the month?
Answer: _____________
MATH 115 Precalculus Fall, 2018, V3.1
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10.
A can of soda at 83 F. is placed in a refrigerator that maintains a constant temperature of 35 F.
After t minutes, the temperature T of the soda is given by T(t) = 35 + 48e – 0.058 t .
Find the temperature of the soda 15 minutes after it is placed in the refrigerator. (Round to the nearest tenth of a degree.)
Answer: _________
11. Given the function 𝑓(𝑥) = 8 − 1
2 𝑥, find a formula for the inverse function.
Answer: __________________
12. (a) State the reference angle associated with 240°. Answer: ________
(b) Convert 240° to radians. Leave the answer in terms of . Answer: ________
13. Given y = 5 sin(6x – ), state the
(a) period Answer: ________
(b) phase shift Answer: ________
14. Solve the trigonometric equation (√2 cos(𝑥) − 1)(2 sin(𝑥) + 1) = 0 in the interval [0, 360°). Find the exact values of the solutions in degrees.
Answer: _________________
15. (a) Find the exact value of arccos (sin 𝜋
2 ) Answer: ________
(b) Find the exact value of arcsin (sin 3𝜋
4 ) Answer: ________
16. For the parabola given by (x – 2)2 = –8(y + 3), find the following:
(a) direction parabola opens (to the left, right, up, or down) Answer: ___________
(b) vertex Answer: ___________
(c) focus Answer: ___________
MATH 115 Precalculus Fall, 2018, V3.1
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17. Let 2
( ) 5
x f x
x =
− .
(a) State the domain. Answer: _________________
(b) State the horizontal asymptote. Answer: _________________
(c) State the vertical asymptote(s). Answer: _________________
(d) Which of the following represents the graph of 2
( ) 5
x f x
x =
− ? Answer: ______________
GRAPH A. GRAPH B.
GRAPH C. GRAPH D
MATH 115 Precalculus Fall, 2018, V3.1
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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 (–1, 9) and (–3, 1). Write the
equation in slope-intercept form. Show work.
19. Find the exact solutions and simplify as much as possible: 2x2 + 15 = 14x. Show work.
20. Let f (x) = 3x2 − 15 and g(x) = x − 5.
(a) Find the composite function ))(( xgf o and simplify. Show work.
(b) Find (𝑓 ∘ 𝑔)(2). Show work.
21. A projectile is launched from a platform 18 feet high with an initial velocity of 104 feet per second,
The height h of the projectile at t seconds after launch is given by h = –16t2 + 104t + 18 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: 𝑥 + 10
𝑥 + 2 +
32
𝑥2 − 4 = 0. Show work.
23. Suppose that sin = 4/5 and that is a Quadrant II angle.
(a) Find the exact value of cos . Show work.
(b) Find the exact value of sin(2 ). Show work.
24. Prove the identity sec(x) (sin(x) + cos(x))2 = sec(x) + 2 sin(x)
MATH 115 Precalculus Fall, 2018, V3.1
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25. From a point 60 feet from the base of a redwood tree, the angle of elevation to the top of the
tree is 47.3°. 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 = 43, B = 55, and c = 28.0.
Find the length of side b, rounded to the nearest tenth. Show work.
27. Let �⃗⃗� = 3, − 6 and �⃗⃗� = 8, 4.
(b) Calculate the dot product �⃗⃗� ∙ �⃗⃗� . Show work.
(c) Determine the angle between �⃗⃗� and �⃗⃗� . Round the result to the nearest degree. Show work.
28. An ellipse has the equation (𝑥 − 2)2
16 +
(𝑦 + 1)2
36 = 1
(a) Is the major axis horizontal or vertical?
(b) Find the exact values of the foci of the ellipse. Show work.