MATH 115 FINAL EXAMINATION

Fall, 2016, V4.3

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:  | 3 – 8x | ≤ 21.

 

A.    (–∞, −9/4] ∪ [3, ∞)

B.     (–∞, –9/4]

C.     [–9/4, 3]

D.    [–9/4, ∞)

 

 

1. ______

2.  Which of the following polynomials has a graph which exhibits the end behavior of

 

downward to the left and downward to the right?

A.     f (x) = –6x⁵– x³– x²– 5

B.     f (x) = 8x³+ 6x²– x + 3

 

2. _______

 

C.

D.

 

f (x) = 5x⁶+ 7x²+ x

f (x) = –3x⁴ + 9x²– x – 1

 

 

3. Express as an equivalent expression:   8 log y  + log 1 – log (x – 3)

 

 

3. ________

A.



log 

y

 

 

8





B.

   y +  

C.

log

  x

(

)

 

-13D.

 

-13log  8y + 4 − x

 

MATH 115 Precalculus

 

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

 

 

A.     (− 4 / 3, 1)

 

B.     (−1,1) and (3,∞)

 

Fall, 2016, V4.3

 

4. ______

 

 

C.     (

 

 

)

 

 

11/ 3, ∞)

 

C.   f ( x) = −e^x

D.     (  )     x    1

-40f   x  = e−   +

5. _____

Page 2 of 7

 

MATH 115 Precalculus

 

6. Which of the functions corresponds to the graph?

 

 

 

 

 A.  f (x) = cos(2x) + 2

 

B.  f (x) = 3 – sin x

 

C.   f (x) = sin x + 3

 

 D.   f (x) = 2 cos x + 1

 

 

SHORT ANSWER:

 

7. Points (–7, 2) and (5, 6)  are endpoints of the diameter of a circle.

 

Fall, 2016, V4.3

 

6. ______

 

 

(a) What is the exact length of the diameter? (Simplify as much as possible)     Answer:  ________

 

 

 (b) What is the center of the circle?

 

(c) What is the equation of the circle?

 

 

Answer:  ____________

 

Answer:  ___________________________

 

 

 

8. Find the value of the logarithm: log

 Answer:  ____________

9.  Bill, a resident of Metropolis, pays Metropolis an annual tax of $55 plus 1.8% of his annual

income. If Bill paid $1,441 in tax, what was Bill’s income?

Answer:  _____________

 

 

 

Page 3 of 7

 

MATH 115 Precalculus

 

Fall, 2016, V4.3

 

 

10. A can of soda at 71° F. is placed in a refrigerator that maintains a constant temperature of 39°

F.  The temperature T of the soda t minutes after it is placed in the refrigerator is given by

 

T(t) = 39 + 32 e – 0.058 t

 

Find the temperature of the soda 20 minutes after it is placed in the refrigerator. (Round to the

nearest tenth of a degree.)

 Answer:  ________

 

 

 

 

11. Given the function

 

 

 

= 3 − 18   , find a formula for the inverse function.

  Answer:  __________________

12.   (a) State the reference angle associated with 300°.

 

 (b) Convert 300° to radians. Leave the answer in terms of ð.

 

13. Given y = 9 sin(8x – ð), state the

 

 (a)  period

 

(b)  phase shift

 

 Answer:  ________

 

 Answer:  ________

 Answer:  ________

 Answer:  ________

14. Solve the trigonometric equation   (cos x)(2cos x + 1)  = 0  in the interval [0, 360°).

 

    Answer:  _________________

 

15.   (a)    Find the exact value of  arccos  sin

 

(b)   Find the exact value of  arcsin  tan

16. For the parabola given by  (y + 5)²= 8(x – 2), find the following:

 

(a)  direction parabola opens (to the left, right, up, or down)

(b)  vertex

 

(c)  focus

 

Answer:  ________

Answer:  ________

 Answer:  ___________

Answer:  ___________

 

Answer:  ___________

Page 4 of 7

 

3x + 1

Fall, 2016, V4.3

 

17. Let

 

f (x) =

 

x − 2

   (a) State the domain.

(b) State the vertical asymptote(s).

(c) State the horizontal asymptote.

(d) Which of the following represents the graph of

f  x

(  )

=

3x + 1

x − 2

 

Answer:  _________________

Answer:  _________________

Answer:  _________________

Answer:  ______________

GRAPH A.

GRAPH C.

GRAPH B.

GRAPH D.

MATH 115 Precalculus

SHORT ANSWER, with work required to be shown, as indicated.

 

Fall, 2016, V4.3

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

 

(b) Find  (

 

f  o g   −   . Show work.

 

21. A projectile is launched from a platform 15 feet high with an initial velocity of 48 feet per second.

The height h of the projectile at t seconds after launch is given by  h = –16t² + 48t + 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:

x − 6

x − 4

+

16

x²−16

 

= 0 .  Show work.

23. Suppose that cos è  = 5/13 and that è  is a Quadrant IV angle.

(a) Find the exact value of sin è.

(b) Find the exact value of sin(2è ).

Show work.

 

Show work.

 

24.  Prove the identity   (sin x + cos x)²−  sin(2x) = 1

MATH 115 Precalculus

 

Fall, 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 52.3°. Find the height of the tree to the nearest tenth of a 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.

27. Let    = 〈10, –5〉 and     = 〈2, 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

+

= 1

(a) Is the major axis horizontal or vertical?

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