Cary-Grove High School HPC 2-Day Algebra Review WS
Complex Fractions: Simplify each of the following.
25
12
a
4
2x  3
1. a
2.
15
5 a
5
2x  3
x

3. x  1
x

x 1
1
x
1
x
Operations on Rational Expressions: Simplify each of the following by factoring:
x 2  x  6 x 2  x  20
2x 2  13x  20 6x 2  13x  5


4.
5.
12  x  x 2 x 2  4x  4
8  10x  3x 2
9x 2  3x  2
Simplify each of the following operations of addition or subtraction.
x 1
x  3 10x 2  7x  9
x
3x  2 7x 2  24x  28

 2


6.
7.
1  2x 4x  3 8x  10x  3
3x  4 x  5 3x 2  11x  20
Fractional and Integral Exponents:
Simplify each of the following. Leave all answers with POSITIVE exponents
 x 2 y 4 
8.  2 
 x y 
2
2
 9ab2   3a 2 b 
9.  2   2 2 
 8a b   2a b 
1
3

10. 27m3n6
 m
13
1 3 5 6
n

6

11. y 2 3 y1 3  y 2 3

12. 1252/ 3
13. 815/ 4
Functions: Let f (x)  2x  1 and g(x)  2x 2  1 . Find each.
14. g(3)  _____________
15. f (t  1)  __________
16. g  f (m  2)   __________
f (x  h)  f (x)
 _____________
h
17.

h  f 2  _______

Let f x  x 2 , g(x)  2x  5, and h x  x 2  1 . Find each.
18.
Find

 
f xh  f x
h
21. f x  9  3x


19. f  g x  1   _______

20. g  h x 3   _______
for the given function f.

22. f x  3x 2  5x  9
Find the equation of the line, in slope intercept form
23. m = – 4 at the point (– 1, 11)
24. m = 3/4 at the point (2, 9)
2
Proving Trigonometric Identities: Prove each of the following identities.
25. sin x  sin x  cos xsin x
3
28.
2
sec x  csc x
26. sin x  cos x 
csc x sec x
1
1

 2csc2 2x 29. sin2x  2sin xcos x
1 cos 2x 1 cos 2x
1  cos x  1  cos x 

27.
1  cos x  sin x 
30. cos4x  cos2 2x  sin2 2x
Trigonometric Equations: Solve each of the following equations for 0 ≤ x < 2.
31. 2cos 2x  3
32. 4sin2 x  3  0
33. 2cos2 x  1 cos2x
34. cos 2 x  2  cos x  0
35. tan2 x  1  0
36. cos x tan x  sin2 x  0
For each of the following give the value without a calculator.


2
12 
37. tan  arccos 
38. sec  sin 1 
3
13 


3
2

12 
39. sin  2arctan 
5

Logarithmic Functions: Evaluate each of the following logarithms.
40. log 4 16  ______
41. log 2 32  ______
42. log1000  ______
43. log 6 28  ______
44. log5 12  ______
45. log12 9  ______
46. log 4 x  3
47. log 4 x  log 4 2  log 4 3
1
48. log x  log 27
3
49. log 9 x  5log 9 2  log 9 8
50. log 3 (x  1)  2
51. log(x  3)  log(2x  4)  log3
52. x 2  3x  4  14
53.
Solve each of the following for x.
x  5  9  0
2
Determine the points of intersection
55. y  x 2  3x  4 and y  5x  11
54. 2x 2  5x  8
56. y  cos x and y  sin x
Graph:(Use graph paper!)
57. x 2  y 2  16
58. y  5sin x (2 periods)
59. f x  cos 2x  3 (2 periods)
60. y  e x
61. y  x
62. y  3 x
63. y  ln x
64. y  x  3  2
65. y  x 2  3
66. y 
1
x
 x2
if x  0

67. y   x  2 if 0  x  3

if x  3
 4
4

Find each derivative. Leave answers in simplest form.
1
68. f ( x)  x7
69. y  5
x
70. f ( x)  4 x
71. y  x  11
72. f ( x)  2 x3  x 2  3x  7
73. f ( x) 
1
 3sin x
x
77. f ( x)  3x(6 x  5 x 2 )
75. y  x 2  3x  3x 2
80. f ( x)  x3 cos x
81. f ( x) 
74. y 
78. f ( x)  ( x 2  3)( x 2  4 x)
x2  4
5x  3

sin x  cos x
2
4 x3  3x 2
76. f ( x) 
x
2
79. y  x  x  8 
82. y 
x3  5 x  3
x2 1
sin x
x3
84. y   csc x  sin x
85. f ( x)  x 2 tan x
86. y   4 x  1
87. f ( x)  3  4  9 x 
88. y  x 2  2 x  1
83. f ( x) 
89. f ( x) 
3
5
 x  3
90. y  x 2  x  2 
3
4
4
91. f ( x) 
1
x 2
2
92. y  cos 4 x
93. f ( x)  5 tan  3x 
94. y  sec x 2
95. f ( x)  5cos 2 8 x
1
96. y  sin 2 2 x
4
cos  x  1
99. f ( x ) 
x
97. f ( x)  sin(tan 2 x)
98. y 
 x
2
 3  x
5

2
100. y  26  sec3 4 x
Find each derivative. Then evaluate the derivative at the given point. Leave answers in simplest form.
101. f ( x) 
8
, (2, 2)
x2
104. y  4sin x  x, (0, 0)
1 7
1

102. y    x3 ,  0,  
2 5
2

105. f ( x)  2 cos x  5, ( , 7)
 
107. f ( x)  sin x(sin x  cos x),  ,1
4 
5
103. f ( x)   4 x  1 , (1,9)
2
106. y   x  3  x 2  2  , (2, 2)
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
tan 2 x  1  sec 2 x
Sum Identities:
sin x  y  sin x cos y  cos x sin y


1
tan x
1  cot 2 x  csc 2 x
Difference Identities:
sin x  y  sin x cos y  cos x sin y
 
cos x  y  cos x cos y  sin x sin y
tan x  y 
cot x 
 
cos x  y  cos x cos y  sin x sin y
tan x  tan y
1 tan x tan y


tan x  y 
Double Angle Identities:
sin 2x  2 sin x cos x
Half-Angle Identities:
sin
Logarithms:
y  log a x
tan x  tan y
1 tan x tan y
 cos2 x  sin 2 x

cos 2x   2cos2 x  1
 1 2sin 2 x

x
1  cos x

2
2
cos
is equivalent to
x
1  cos x

2
2
tan 2 x 
tan
x
1  cos x

2
1  cos x
x  ay
Product property of logarithms:
log b mn  log b m  log b n
Quotient property of logarithms:
log b
Power property of logarithms:
log b m p  p log b m
Property of equality of logarithms:
If log b m  log b n, then m  n
Change of base formula of logarithms:
log a n 
m
 log b m  log b n
n
log b n
log b a
Derivative of a Function:
Slope of a tangent line to a curve or the derivative: lim
Slope-intercept form:
y  mx  b
Point-slope form:
y  y1  m( x  x1 )
Standard form:
ax  by  c
h 
6
2 tan x
1  tan 2 x
f ( x  h)  f ( x )
h
DERIVATIVE RULES – Let f, g and u be differentiable functions of x and let c and n be non-zero
constants.
Constant Multiple Rule:
Product Rule:
Sum or Difference Rule:
d
 fg   fg ' gf '
dx
Constant Rule:
Chain Rule:
d
c  f   c  f '
dx
Quotient Rule:
d
c  0
dx
d
 f (u )  f '(u )  u '
dx
d  f  gf ' fg '

dx  g 
g2
(Simple) Power Rule:
d n
d
 x   n  x n 1 ,
 x  1
dx
dx
General Power Rule:
d n
u   n  u n 1  u '
dx
Trigonometric Derivatives:
d
sin x   cos x
dx
d
 tan x   sec2 x
dx
d
sec x   sec x  tan x
dx
d
 cos x    sin x
dx
d
cot x    csc2 x
dx
d
csc x    csc x  cot x
dx
7
d
 f  g   f ' g '
dx