MATH 115 Precalculus Spring, 2016, V4.3 MATH 115

MATH 115 Precalculus Spring, 2016, V4.3 MATH 115

MATH 115 Precalculus Spring, 2016, V4.3 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, and express the answer in interval notation: | 7 – 3x | ≤ 11.

A.

B.

C.

D. 1. ______ (–∞, –4/3]

(–∞, –4/3] ∪ [6, ∞)

[–4/3, 6]

(–∞, 6] ∪ [–4/3, ∞) 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.

B.

C.

D. f (x) = –3x6 + 6x3 – x

f (x) = –4x5 – 5x3 – 1

f (x) = 7x4 + 2x + 6

f (x) = 6x3 + 3x2 – x vvvv 3. Express as an equivalent expression: 9 log x – log (y + 3) + log 1

A. 9x +1 log y+3 B. log ( 9 x − y − 2 ) C. log x9

log ( y + 3) D. x9 log y+3 3. ______ Page 1 of 7 MATH 115 Precalculus

4. Determine the interval(s) on which the function is increasing.

A. (–3, 1) B. (–3.6, 0) and (6.7, ∞) C. (–∞, –2) and (4, ∞) D. (–2, 4) 5. Which of the functions corresponds to the graph? Spring, 2016, V4.3

4. ______ 5. ______ A. f ( x ) = e x − 2

B. f ( x ) = e− x − 1

C. f ( x ) = e− x + 1

D. f ( x ) = −e x Page 2 of 7 MATH 115 Precalculus Spring, 2016, V4.3 6. Which of the functions corresponds to the graph? 6. ______ A. f (x) = cos(2x) + 2

B. f (x) = sin x + 3

C. f (x) = 3 – sin x

D. f (x) = 2 cos x + 1 SHORT ANSWER:

7. Points (–5, 8) and (–9, 2) 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: ___________________________

ଵ 8. Find the value of the logarithm: log ଻ ቀ ቁ.

ସଽ Answer: ____________ 9. Mimi, a resident of Metropolis, pays Metropolis an annual tax of $35 plus 1.9% of her annual

income. If Mimi paid $1,127.50 in tax, what was Mimi’s income?

Answer: _____________ Page 3 of 7 MATH 115 Precalculus Spring, 2016, V4.3 10. Water initially at 190° F. is left in a room of temperature 70° F to cool.

– 0.096 t

After t minutes, the temperature T of the water is given by T(t) = 70 + 120e

Find the temperature of the water 10 minutes after it is left to cool. (Round to the nearest degree.) Answer: ________

1 11. Given the function ݂ሺ‫ݔ‬ሻ = 9 − 7 ‫ݔ‬, find a formula for the inverse function.

Answer: __________________ 12. (a) State the reference angle associated with 210°. Answer: ________ (b) Convert 210° to radians. Leave the answer in terms of π. Answer: ________ 13. Given y = 6 sin(8x – π), state the

(a) period Answer: ________ (b) phase shift Answer: ________ (cos x)(2cos x + 1) = 0 in the interval [0, 360°). 14. Solve the trigonometric equation Answer: _________________ 15. Find the exact value of arcsin ቀsin ଷగ (b) Find the exact value of arccos ቀsin ଷగ (a) ସ ଶ ቁ Answer: ________ ቁ Answer: ________ 2 16. For the parabola given by (y + 9) = 8(x – 4), find the following:

(a) direction parabola opens (to the left, right, up, or down) Answer: ___________ (b) vertex Answer: ___________ (c) focus Answer: ___________ Page 4 of 7 MATH 115 Precalculus

17. Let f ( x) = Spring, 2016, V4.3 2 x − 18

.

x−5 (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 f ( x) = 2 x − 18

?

x−5 GRAPH A. GRAPH B. GRAPH C. GRAPH D.



Answer: ______________ Page 5 of 7 MATH 115 Precalculus Spring, 2016, V4.3 SHORT ANSWER, with work required to be shown, as indicated. 18. Find the equation for a line which passes through the points (6, 3) and (2, – 9). Write the

equation in slope-intercept form. Show work. 2 19. Find the exact solutions and simplify as much as possible: 24x + 1 = 12x. Show work. 2 20. Let f (x) = 6x + 7 and g(x) = x – 4.

(a) Find the composite function ( f o g )( x) and simplify. Show work.

(b) Find ( f o g )(−1) . Show work. 21. A projectile is launched from a platform 14 feet high with an initial velocity of 80 feet per second,

2

The height h of the projectile at t seconds after launch is given by h = –16t + 80t + 14 feet.

(a) How many seconds after launch does the projectile attain maximum height?

(b) What is the maximum height? 22. Solve: Show work. Show work. x − 10

42

+ 2

= 0 . Show work.

x − 7 x − 49 23. Suppose that cos θ = 5/13 and that θ is a Quadrant IV angle.

(a) Find the exact value of sin θ. Show work. (b) Find the exact value of sin 2θ. Show work. 2 24. Prove the identity (sin x + cos x) − 1 = sin(2x) Page 6 of 7 MATH 115 Precalculus Spring, 2016, V4.3 25. From a point 48 feet from the base of a redwood tree, the angle of elevation to the top of the

tree is 56.3°. 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 = 46°, B = 64°, and c = 35.0. Find the length of side a rounded to the nearest tenth. Show work. ሬԦ = 〈2, –6〉 and ࢜

ሬԦ = 〈15, 5〉.

27. Let ࢛

(b) ሬԦ ∙ ࢜

ሬԦ. Show work.

Calculate the dot product ࢛ (c) ሬԦ and ࢜

ሬԦ. Round the result to the nearest degree. Show work.

Determine the angle between ࢛ 28. An ellipse has the equation ሺ௫ ା ଵሻమ

଼ଵ + ሺ௬ି ଺ሻమ

ଷ଺ =1 (a) Is the major axis horizontal or vertical?

(b) Find the exact values of the foci of the ellipse. Show work.

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