Math 111, Final Exam
Name:
215 pts, 3 hours
April 27, 2013
No notes, books, electronic devices, or outside materials of any kind.
Page 1 of 3
Read each problem carefully. Unless otherwise indicated, supporting work will be required
on every problem; one-word answers, or answers which simply restate the question, will
receive no credit.
Give all answers in exact form (i.e., “ready-to-calculate form”). You are not required to
rationalize denominators but should simplify answers down to rational numbers or integers
whenever possible. You are expected to know the values of all trigonometric functions at
multiples of π/4 and of π/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.
2 (4 pts). Find the domain of f (x) =
!
|6 − 5x| ≥ 10
x−3
. Write your answer in interval notation.
x+6
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 to
x − 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 two
reference points on your graph.
a. f (x) = x2/3
b. g(x) = −x2/3 + 2
6a (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
p. 2 of 3
No calculators
y
7 (8 pts). Give a piecewise definition of
the function whose graph is given at right.
4
3
8 (7 pts). Find the difference quotient
√
f (x + h) − f (x)
if f (x) = 2x + 1.
h
Simplify your answer until it is defined
at h = 0.
y=f(x)
2
1
x
-5
-4
-3
-2
-1
1
2
3
4
9 (8 pts). Sneezium is a radioactive
-1
substance. If 12 mg of Sneezium de-2
cays to 10 mg in 15 hours, what is its
half-life?
-3
10. Solve for x.
-4
a (6 pts). log6 (x + 4) + log6 (x − 1) = 1
x
3−2x
b (6 pts). 4 = 5
11. Evaluate the expression. Express your answer as a rational number (possibly an
integer) in lowest terms.
"√ #
3 4
1
e
b (4 pts). ln √
a (2 pts). log2
32
e
12 (5 pts). Write as a single logarithm. − ln(2x − 1) + 2 ln(3x + 1) − 3 ln(4x + 1)
x
1
and g(x) =
. Find the following functions and their domains.
13. Let f (x) =
x−3
x−3
Simplify your answers. Express domains in interval notation.
$ %
f
(x)
a (6 pts).
g
b (7 pts). (f ◦ g)(x)
y
14 (8 pts). Determine the polynomial of degree 4 whose
graph is shown at right. Express your answer in factored
20
form.
16
15 (8 pts). Find all zeros of the function p(x) = 2x4 − 5x3 + 5x2 − 20x − 12.
(Hint: x = −2i is a zero.)
16 (5 pts). Find the equations of all
horizontal and slant asymptotes of the
x3 − 3x2
. (If they
graph of y = 2
x + 4x + 3
don’t exist, write none.)
12
8
(1,12)
4
x
-4
-3
-2
-1 -4
1
2
3
4
-8
-12
-16
17 (4 pts). Ship A is 3 km due east from Charleston, and ship B is 2 km southeast from
Charleston. How far apart are the ships?
(Here “southeast” refers to the direction exactly halfway between south and east.)
5
111 Final Exam 4/27/2013
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No calculators
18 (15 pts). Find the following and sketch an accurate graph of w(x) =
a. Domain of w(x) (in interval notation).
3(x + 2)(x + 1)
.
(x − 1)(x + 2)
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 two
dividing fences parallel to two sides. If the total fence to be used is 120 feet, express the
total area of the plot as a function of one of the sides.
20 (12 pts). Find the exact value of each of the following:
'
&
'
&
=
b. cos & 13π
=
a. sin &13π
4 '
4'
2π
2π
d. csc& 3 =
c. tan & 3 =
'
'
−11π
e. cot
=
f. sec −11π
=
6
6
π
3π
, and that tan y = 5 and 0 < y < . Find
21. Suppose that sin x = −1/2 and π < x <
2
2
the following.
a (3 pts). sec x =
c (3 pts). sin y =
b (3 pts). sin(2x) =
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(5π))
=
d (2 pts). tan−1 ( 3) =
&
& 7π ''
=
f (2 pts). cos2 (.37) + sin2 (.37) =
e (3 pts). arccos cos 5
23 (9 pts). Sketch one cycle of the graph of y = 3 sin(2x + π3 ). Label hashmarks along the
x-axis so as to clearly indicate the locations of all maximums, minimums, and every place
where 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) − cos x = 0
b (6 pts). 3 sin(3x) = cos(3x)
26 (5 pts). Rewrite cos(arctan x) as an algebraic expression in x. (That is, express
cos(arctan x) entirely in terms of polynomials and radicals.)