# Math 115 final exam fall, 2016

**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 – 8*x* | ≤ 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*) = –6*x*^{5}– *x*^{3}– *x*^{2}– 5

B. *f* (*x*) = 8*x*^{3}+ 6*x*^{2}– *x* + 3

2. _______

C.

D.

*f* (*x*) = 5*x*^{6}+ 7*x*^{2}+ *x*

*f* (*x*) = –3*x*^{4} + 9*x*^{2}– *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 8*y* + 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

-40*f* *x* = *e*− +

5. _____

Page **2** of **7**

MATH 115 Precalculus

6. Which of the functions corresponds to the graph?

A. *f* (*x*) = cos(2*x*) + 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(8*x* – 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**

3*x* + 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*

( )

=

3*x* + 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* = –16*t*^{2} + 48*t* + 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

*x*^{2}−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(2*x*) = 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.**