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Summer Review Packet for Students Entering Calculus
Complex Fractions:
Simplify each of the following.
25
a
a
1.
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

x x2
7.
11 18
1  2
x x
1
1
Summation Notation:
Find the sum of each of the following series.
5
8.
7
 (2n  3)
9.
i 1

i
i 3
11.
 (i  2)
n 1
i 1
(1)n1

n 3 n  2
6
10.
5
Write each series in expanded form.
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:
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
6 x2  x  2 2 x2  9 x  4

6 x2  7 x  2 4  7 x  2 x2
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.
21.
x
2
6

 2
4x 1 2 x  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.
24.
3 x y  3
2
3 2
2
1
3
x y
2
2
 9ab 2   3a 2b 
26.  2   2 2 
 8a b   2a b 
28.
 27m n   m
3 6 1 3
 x 2 y 4 
25.  2 
 x y 

3
 y 2 3 y 5 6 
27. 

19
 y


9
29. y 2 3  y1 3  y 2 3 
1 3 5 6 6
n
2
30. a1 6  a 5 6  a 7 6 
Functions
Let f ( x)  2 x  1 and g ( x)  2 x 2  1 . Find each.
31. f (2)  ____________
32. g ( 3)  _____________
33. f (t  1)  __________
34. f  g (2)  __________
35. g  f (m  2)  ___________
36.
4
f ( x  h)  f ( x )
 ____
h
Functions – Continued
Let f ( x)  x 2 , g ( x)  2 x  5, and h( x)  x 2  1. Find each.
37. h  f (2)  _______
Find
38. f  g ( x 1)  _______
39. g  h( x3 )   _______
f ( x  h)  f ( x )
for the given function f.
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.
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  5x  4
5
44. f ( x)  2 x  4
Find the slope of the line tangent to each of the following functions at the given point.
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)
Find the equation of the line, in slope intercept form, tangent to each of the following functions at the
given point.
49. f ( x)  2 x2  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) for each of the following functions.
51. f ( x)  3x3  9 x  5
52. f ( x)  x 2  2 x  15
6
Proving Trigonometric Identities
Prove each of the following identities.
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

7
Trigonometric Equations:
Solve each of the following equations for 0 < x < 2.
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
8
Inverse Trigonometric Functions:
For each of the following, express the value for “y” in radians.
71. y  arcsin
 3
2
72. y  arccos  1
73. y  arctan(1)
For each of the following give the value without a calculator.
2

74. tan  arccos 
3

12 

75. sec  sin 1 
13 

7

77. sin  2sin 1 
8

1


78. sin  sin 1  tan 1  3 
2


9
12 

76. sin  2 arctan 
5

Logarithmic Functions:
Evaluate each of the following logarithms.
79. log4 16  ______
80. log2 32  ______
81. log1000  ______
82. log 6 216  ______
83. log5 125  ______
84. log3 7  ______
85. log 6 28  ______
86. log5 12  ______
87. log12 9  ______
88. log 4 x  3
89. log8 x  3log8 2
90. log 4 x  log4 2  log 4 3
1
91. log x  log 27
3
92. log9 x  5log9 2  log9 8
93. log3 ( x  1)  2
Solve each of the following for “x”.
94. log( x  3)  log(2 x  4)  log 3
95. log( x2  3)  log( x  1)  log 5
10
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 ) 
Difference Identities:
cot x 
tan 2 x  1  sec 2 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
tan( x  y ) 
1
tan x
cos( x  y )  cos x cos y  sin x sin y
tan x  tan y
1  tan x tan y
cos 2 x  cos 2 x  sin 2 x
Double Angle Identities:
 1  2 sin 2 x
sin 2x  2 sin x cos x
 2 cos 2 x  1
tan 2 x 
2 tan x
1  tan 2 x
x
1  cos x

2
2
Half-Angle Identities:
sin
Logarithms:
y  log a x
Product property:
log b mn  log b m  log b n
m
log b  log b m  log b n
n
log b m p  p log b m
Quotient property:
Power property:
Property of equality:
Change of base formula:
Derivative of a Function:
cos
x
1  cos x

2
2
is equivalent to
tan
x
1  cos x

2
1  cos x
x  ay
If log b m  log b n , then m = n
log b n
log a n 
log b a
Slope of a tangent line to a curve or the derivative: lim
Slope-intercept form: y  mx  b
Standard form:
Ax + By + C = 0
h 
Point-slope form: y  y1  m( x  x1 )
11
f ( x  h)  f ( x )
h
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