Final Exam Math 1113 sec. 010
Fall 2012
Name:
Your signature (required) confirms that you agree to practice academic honesty.
Signature:
Problem Points
1
2
3
4
5
6
7
8
9
10
MC
INSTRUCTIONS: There are 10 problems
worth 10 points each. You may eliminate
any one problem, or I will count your best
9 out of 10. The multiple choice portion is worth 10 points, and can not be
eliminated. No use of notes or books is
allowed. You may use a TI-83/84 calculator, or a scientific calculator without graphing capabilities. Cell phones,
tablets, or other ”smart” devices MAY
NOT substitute for a calculator. Illicit
use of notes, book, or any smart device
will result in a grade of zero on this exam.
To receive full credit, you must clearly
justify your answer.
Potentially Useful Formula: The n complex nth roots of a number are
p
θ + 2πk
θ + 2πk
n
+ i sin
, k = 0, 1, . . . , n − 1
wk = |z| cos
n
n
(1) Use the given information to evaluate the expressions exactly. (Decimal or other
calculator produced answers will not be considered for credit.) It is not necessary
to rationalize denominators or simplify radicals.
Given:
(a) cos(θ − φ)
(b) sin
θ
2
(c) tan(2φ)
2
sin θ = ,
5
π
< θ < π,
2
and
1
cos φ = ,
4
0<φ<
π
.
2
(2) The graph of the rational function has an oblique asymptote. Find the equation of the
oblique asymptote.
x2 − x − 2
f (x) =
x+2
(3) Find all values of the variable x such that the vectors u and v are orthogonal where
u =< 1, −3 >
and v =< x, 4 > .
(4) Use Cramer’s to solve the system of equations. Cramer’s rule must be used to
receive credit.
4x + 3y = −2
−x + 2y = 2
(5) State the domain and range of the given functions. You may use set or interval notation,
your choice. Read each function carefully.
(a) y = tan−1 x
Domain:
Range:
(b) y = cos−1 x
Domain:
Range:
(c) y = csc x
Domain:
Range:
(6) Solve the system of equations. Use any applicable method. (Solutions with no supporting work will not be considered for credit.)
x
+ 2z = 4
x − y + z = 3
3x + 2y
= 6
(7) Determine the (i) Amplitude, (ii) Period, (iii) Phase Shift, and (iv) Vertical Shift of each
function. If the function does not have one of these properties, write ”none”. Indicate the
direction of any phase or vertical shift. Note: You do not have to produce any graphs.
(a) y = −3 cos(2x)+2
Amplitude
Period
Phase Shift
left right
Vertical Shift
up down
(b) y =
π
5
sin x −
2
4
Amplitude
Period
Phase Shift
left right
Vertical Shift
up down
(8) Construct an augmented matrix for the following system of equations. Use the traditional order (x, y, z) for the variables. Note that you are not being asked to solve the
system but rather to construct a matrix associated with it.
3x − 6
= 2y − z
2x + 2y − 1 = −3z
4 + 2y − 4z = 1 − x
(9) The points P and Q are given in polar coordinates. Convert them to Cartesian (a.k.a.
rectangular) coordinates, and find the equation of the line through P and Q.
√ π π
and Q =
2,
P = 2,
2
4
(10) Find all solutions of the trigonometric equation on the interval [0, 2π).
1 − cos2 x + sin x = 0