Complex Fractions:
Simplify each of the following, please complete 4/7 (minimum) of the problems below.
25
a
1. a
5 a
4
x2
2.
10
5
x2
12
2x  3
3.
15
5
2x  3
x
1

4. x  1 x
x
1

x 1 x
2x
3x  4
5.
32
x
3x  4
1 6
1  2
x x
6.
4 3
1  2
x x
2
4
1
3 10
 2
x
x
7.
11 18
1  2
x x
1
1
Summation Notation:
Find the sum of each of the following series, please complete 2/4 (minimum) of the problems #8-11.
5
8.
 (2n  3)
n 1
7
9.
 (i  2)
i 1
(1)n1
11. 
n 3 n  2
i 1
10. 
i
i 3
6
5
Write each series in expanded form, please complete 2/3 (minimum) of the problems # 12-14.
5
12.
 (2 x
n
)
n 1
13.
xi

i 1 i  1
14.
xi

i 2 i
4
5
2
Operations on Rational Expressions
Simplify each of the following by factoring, please complete 4/6 (minimum) of the problems #15-20.
15.
x 2  x  6 x 2  x  20

12  x  x 2 x 2  4 x  4
x3  y 3
2 x 2  5xy  3 y 2
17.

2 x 2  xy  3 y 2 x 2  4 xy  y 2
19.
6 x 2  23x  4 4 x 2  20 x  25

6 x 2  17 x  3 2 x 2  11x  15
6x2  x  2 2 x2  9 x  4

6x2  7 x  2 4  7 x  2x2
16.
2 x 2  13x  20 6 x 2  13x  5
18.
 2
8  10 x  3x 2
9 x  3x  2
3x 2  10 x  8 2 x 2  9 x  10

6 x 2  13x  6 4 x 2  4 x  15
20.
Simplify each of the following operations of addition or subtraction, please complete 2/3 (min).
21.
x
2
6

 2
4x 1 2x  1 8x  2 x 1
23.
x 1
x  3 10 x 2  7 x  9


1  2 x 4 x  3 8 x 2  10 x  3
x
3x  2 7 x 2  24 x  28


3x  4 x  5 3x 2  11x  20
22.
3
Fractional and Integral Exponents
Simplify each of the following. Leave all answers with POSITIVE exponents. Complete ALL #24-30.
24.
3 x
2
2
y
 3
3 2
1
3
x y
2
2
 9ab 2   3a 2b 
26.  2   2 2 
 8a b   2 a b 
28.
 27m n   m
3
6 1 3

3
2
 y 2 3 y 5 6 
27. 

19
 y


1 3 5 6 6
n
 x 2 y 4 
25.  2 
 x y 
9
29. y 2 3  y1 3  y 2 3 
30. a1 6  a 5 6  a 7 6 
4
Functions
Let f ( x)  2 x  1 and g ( x)  2 x 2  1 . Find each. Please complete ALL.
31. f (2)  ____________
32. g ( 3)  _____________
33. f (t  1)  __________
34. f  g (2)  __________
35. g  f (m  2)  ___________
36.
f ( x  h)  f ( x )
 ____
h
Let f ( x)  x 2 , g ( x)  2 x  5, and h( x)  x 2  1 . Find each. Please complete ALL.
37. h  f (2)  _______
38. f  g ( x  1)  _______
5
39. g  h( x 3 )   _______
Find
f ( x  h)  f ( x )
for the given function f. Please complete ALL.
h
40. f ( x)  9 x  3
41. f ( x)  5  2 x
Derivatives and Critical Points
Find the derivative for each of the following functions, please complete 3/5 (min) for #42-46.
42. f ( x)  5x 4  3x 2  2
43. f ( x)  6 x3  5x 2  4 x  2
45. f ( x)  7
46. f ( x)  x5  2 x3  5 x  4
44. f ( x)  2 x  4
Find the slope of the line tangent to each of the following functions at the given point. Choose 1/2.
47. f ( x)  x 4  2 x3  5x 2  8 at the point (-2, 44)
48. f ( x)   x 2  x  2 at the point (0.5, 1.25)
6
Find the equation of the line, in slope intercept form (y=mx+b) or point-slope form ((y-y1)=m(x-x1)),
tangent to each of the following functions at the given point. Choose 1/2.
49. f ( x)  2 x 2  3x  10 at the point (1, 11)
50. f ( x)  3x3  2 x 2  4 x  2 at the point (-2, -42)
Find the critical point(s) (max/mins) for each of the following functions. Please complete ALL (both).
51. f ( x)  3x3  9 x  5
52. f ( x)  x 2  2 x  15
7
Proving Trigonometric Identities
Prove each of the following identities. Please complete 4/8 (min) for problems #53-60.
53. sin x  sin 3 x  cos 2 x sin x
54. sin x  cos x 
1  cos x  1  cos x 

55.

1  cos x  sin x 
56.
2
sec x  csc x
csc x sec x
1
1

 2 csc2 2 x
1  cos 2 x 1  cos 2 x
57. sin 2x  2sin x cos x
58. cos 4 x  cos 2 2 x  sin 2 2 x
59. sin( x  y ) cos y  cos( x  y ) sin y  sin x


60. sin  x    cos x
2

8
Trigonometric Equations:
Solve each of the following equations for 0 < x < 2. Please complete 5/10 (min) in #61-70.
1
2
61. 2cos 2 x  3
62. cos 2 x 
63. sin x  sin 2x
64. 2 cos 2 x  1  cos 2 x
65. cos 2 x  1  cos x  0
66. sin 2 x  cos 2 x  cos x  0
67. sin x  cos x  0
68. 4 cos 2 x  3  0
69. cos x tan x  sin 2 x  0
70. tan 2 x  1  0
9
Logarithmic Functions:
Evaluate each of the following logarithms. Please complete 3/9 (min) in problems #71-79.
71. log 4 16  ______
72. log 2 32  ______
73. log1000  ______
74. log6 216  ______
75. Ln (e) = _______
76. log3 7  ______
77. log6 28  ______
78. log5 12  ______
79. log12 9  ______
Solve each of the following for “x”. Please complete 3/8 (minimum) in problems #80-87.
80. log 4 x  3
81. ln x = 3ln 2
82. log 4 x  log 4 2  log 4 3
1
83. log x  log 27
3
84. log9 x  5log9 2  log9 8
85. log3 ( x  1)  2
86. ln (x+3) + ln (2x-4) = ln 3
87. log( x 2  3)  log( x  1)  log 5
10
88.
89.
90.
91.
92.
93.
94.
95.
11
Formula Sheet
Reciprocal Identities:
csc x 
1
sin x
sec x 
1
cos x
Quotient Identities:
tan x 
sin x
cos x
cot x 
cos x
sin x
Pythagorean Identities:
sin 2 x  cos 2 x  1
Sum Identities:
sin( x  y )  sin x cos y  cos x sin y
tan( x  y) 
tan 2 x  1  sec 2 x
tan( x  y ) 
1
tan x
1  cot 2 x  csc 2 x
cos( x  y )  cos x cos y  sin x sin y
tan x  tan y
1  tan x tan y
sin( x  y )  sin x cos y  cos x sin y
Difference Identities:
cot x 
cos( x  y )  cos x cos y  sin x sin y
tan x  tan y
1  tan x tan y
Logarithms:
y  log a x
Product property:
log b mn  log b m  log b n
Quotient property:
log b
Power property:
log b m p  p log b m
Change of base formula:
log a n 
Derivative of a Function:
Slope of a tangent line to a curve or the derivative: lim
x  ay
m
 log b m  log b n
n
Slope-intercept form: y  mx  b
Standard form:
is equivalent to
log b n
log b a
h 
Point-slope form: y  y1  m( x  x1 )
Ax + By + C = 0
12
f ( x  h)  f ( x )
h