percale exam

MATH 115 Precalculus Fall, 2018, V3.1

Page 1 of 7

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

Page 2 of 7

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

Page 3 of 7

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

Page 4 of 7

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

Page 5 of 7

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

Page 6 of 7

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

Page 7 of 7

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.