Math 111, Final Exam215 pts, 3 hours

Name:April 27, 2013 Page 1 of 3

No notes, books, electronic devices, or outside materials of any kind.

Read each problem carefully. Unless otherwise indicated, supporting work will be requiredon every problem; one-word answers, or answers which simply restate the question, willreceive no credit.

Give all answers in exact form (i.e., “ready-to-calculate form”). You are not required torationalize denominators but should simplify answers down to rational numbers or integerswhenever possible. You are expected to know the values of all trigonometric functions atmultiples of !/4 and of !/6.

Please circle or box your answers if an answer box is not provided on the problem.

1 (5 pts). Solve the inequality. Write the solution set in interval form. |6! 5x| " 10

2 (4 pts). Find the domain of f(x) =

!

x! 3

x+ 6. Write your answer in interval notation.

3 (6 pts). Use factorization to simplify as much as possible:x5 + 2x4 + x3

x4 ! 2x2 + 1

4 (4 pts). Find the equation of the line through the point (2,3) and perpendicular tox! 4y + 1 = 0. Give your answer in slope-intercept form.

5 (6 pts). Graph each of the given functions. Include the coordinates of at least tworeference points on your graph.

a. f(x) = x2/3 b. g(x) = !x2/3 + 26a (4 pts). Complete the square to write f(x) = 2x2 + 4x! 3 in standard form.

6b (6 pts). Find the coordinates of the vertex and of all intercepts of the graph of f(x).(Make sure you indicate which is which in your answers.)

111 Final Exam 4/27/2013 No calculators p. 2 of 3

7 (8 pts). Give a piecewise definition ofthe function whose graph is given at right.

x

y

-4

-3

-2

-1

1

2

3

4

1 2 3 4 5-5 -4 -3 -2 -1

-4

-3

-2

-1

1

2

3

4

1 2 3 4 5-5 -4 -3 -2 -1

y=f(x)8 (7 pts). Find the di!erence quotientf(x+ h)! f(x)

hif f(x) =

#2x+ 1.

Simplify your answer until it is definedat h = 0.

9 (8 pts). Sneezium is a radioactivesubstance. If 12 mg of Sneezium de-cays to 10 mg in 15 hours, what is itshalf-life?

10. Solve for x.

a (6 pts). log6(x+ 4) + log6(x! 1) = 1

b (6 pts). 4x = 53!2x

11. Evaluate the expression. Express your answer as a rational number (possibly aninteger) in lowest terms.

a (2 pts). log21

32b (4 pts). ln

"

3#e4#e

#

12 (5 pts). Write as a single logarithm. ! ln(2x! 1) + 2 ln(3x+ 1)! 3 ln(4x+ 1)

13. Let f(x) =1

x! 3and g(x) =

x

x! 3. Find the following functions and their domains.

Simplify your answers. Express domains in interval notation.

a (6 pts).

$

f

g

%

(x)

b (7 pts). (f $ g)(x)14 (8 pts). Determine the polynomial of degree 4 whosegraph is shown at right. Express your answer in factoredform.

x

y

-16

-12

-8

-4

4

8

12

16

20

1 2 3 4-4 -3 -2 -1

(1,12)15 (8 pts). Find all zeros of the func-tion p(x) = 2x4!5x3+5x2!20x!12.(Hint: x = !2i is a zero.)

16 (5 pts). Find the equations of allhorizontal and slant asymptotes of the

graph of y =x3 ! 3x2

x2 + 4x+ 3. (If they

don’t exist, write none.)

17 (4 pts). Ship A is 3 km due east from Charleston, and ship B is 2 km southeast fromCharleston. How far apart are the ships?(Here “southeast” refers to the direction exactly halfway between south and east.)

111 Final Exam 4/27/2013 No calculators p. 3 of 3

18 (15 pts). Find the following and sketch an accurate graph of w(x) =3(x+ 2)(x+ 1)

(x! 1)(x+ 2).

a. Domain of w(x) (in interval notation).

b. Range of w(x) (in interval notation).

c. Coordinates of x-intercepts.

d. Coordinates of y-intercepts.

e. Equations of all vertical asymptotes.

f. Equations of all horizontal or slant asymptotes.

19 (7 pts). A rectangular plot of land will be fenced into three equal portions by twodividing fences parallel to two sides. If the total fence to be used is 120 feet, express thetotal area of the plot as a function of one of the sides.

20 (12 pts). Find the exact value of each of the following:

a. sin&

13!4

'

= b. cos&

13!4

'

=c. tan

&

2!3

'

= d. csc&

2!3

'

=e. cot

&

!11!6

'

= f. sec&

!11!6

'

=

21. Suppose that sinx = !1/2 and ! < x <3!

2, and that tan y = 5 and 0 < y <

!

2. Find

the following.

a (3 pts). secx = b (3 pts). sin(2x) =c (3 pts). sin y = d (6 pts). cos(x+ y) =

22. Find the exact value of the following or state that it doesn’t exist.

a (2 pts). sin!1(

!"

22

)

= b (3 pts). cot&

cos!1&

!25

''

=

c (2 pts). sin(arcsin(5!)) = d (2 pts). tan!1(#3) =

e (3 pts). arccos&

cos&

7!5

''

= f (2 pts). cos2(.37) + sin2(.37) =

23 (9 pts). Sketch one cycle of the graph of y = 3 sin(2x+ !3). Label hashmarks along the

x-axis so as to clearly indicate the locations of all maximums, minimums, and every placewhere the sine is zero.

24 (5 pts). Find g!1(x), if g(x) = ln(10x+5).

25. Find all real solutions x to the given equation.

a (8 pts). cos(2x)! cosx = 0 b (6 pts).#3 sin(3x) = cos(3x)

26 (5 pts). Rewrite cos(arctanx) as an algebraic expression in x. (That is, expresscos(arctanx) entirely in terms of polynomials and radicals.)