PENNCREST HIGH SCHOOL
SUMMER REVIEW PACKET
For students in entering CALCULUS (AP, Regular, or Intro.)
Name: __________________________________________________________
Assigning Teacher:
 Mrs. Carter
 Mr. Graham
 Mrs. Scherer
 Mrs. Sudall
 Ms. Matlock
 Other: _____________________________________
1.
This packet is to be handed in to your Calculus teacher on the first day of the school year.
2.
All work must be shown in the packet OR on separate paper attached to the packet.
3.
Completion of this packet is worth one-half of a major test grade and will be counted in your first marking period grade.
1

Summer Review Packet for Students Entering Calculus (all levels)
Complex Fractions:
Simplify each of the following.
1.
25
 a a
5
 a
2.
2

5

4 x

10
2 x

2
4. x x x

1 x

1


1 x x
1
5.
1
 x

2 x
3 x

4
32
3 x

4
3 10
1
7. x x 2
1

11

18 x x
2
2
3.
4

5

12
2 x

3
15
2 x

3
6.
1
1
1 6 x x
2
4 3 x x
2

Summation Notation:
Find the sum of each of the following series.
8. n
5 

1
(2 n

3) 9. i
7 

1
( i

2)
10. i
6 

3 i
 i
1
11. n
5 

3
Write each series in expanded form.
12. n
5 

1 x n
(2 )
 n

1 n

2
13. i
4 

1 i x i

1
14. i
5

2 x i  i
3

Operations on Rational Expressions
Simplify each of the following by factoring:
15. x
12
2   x x
6
2
 x x
2
2
 

4 x

20
4
17.
2 x
2 x
3  y
 xy

3
3 y
2

2 x x
2
2


5
4 xy xy


3 y
2 y
2
16.
6
6 x x
2
2

 
7 x

2

2 x

2  x
9
 x
2
 x
4
2
18.
2 x
2


13 x x


3 x
20
2

6
9 x x
2
2


13
3 x x


2
5
19.
6 x
2 
23 x

4 4 x
2 
20 x 25
6 x
2 
17 x

3

2 x
2 
11 x


15
20.
3 x
2 
10 x

8

2 x
2 
9 x

10
6 x
2 
13 x

6 4 x
2 
4 x

15
Simplify each of the following operations of addition or subtraction.
21.
4 x x

1

2 x
2

1

8 x
2 
6
2 x

1
22. x
3 x

4

3 x x


5
2

7
3 x x
2
2


24
11 x x


28
20
23. x


1 x

4 x x


3
3

10
8 x
2 x 2


7
10 x x


9
3
4

Fractional and Integral Exponents
Simplify each of the following. Leave all answers with POSITIVE exponents.
24.

3
2 2 x y

3
 
3

1 3 x y

2

25.


2 x y

4

2 x y



2
26.


9 ab

2
8

2 a b
 
 
3

2 a b
2
2 a b

2
 3
28.

27
3 m n

6 m

1 3 5 6 n

6
27.

 y
2 3 y

5 6 y
1 9
 9
29. y
2 3
 y
1 3  y

2 3

30. a
1 6
 a
5 6  a

7 6

Functions
Let ( )

2 x

1 and g x

2 x
2 
1 . Find each.
31. f (2)

____________
34.

( 2)
 
__________
32. g ( 3)
35.

_____________ 33. ( 1)
(

2)
 
___________ 36.
__________
( h ) f x

____ h
5

Functions – Continued
Let f x
 x
2
, ( )

2 x

5,
37.

( 2)
 
_______
( )
 x
2 
1 . Find each.
38.

(

1)
 
_______
Find
( h ) f x
for the given function f. h
40. ( )

9 x

3 41. ( ) 5 2 x
Derivatives and Critical Points
Find the derivative for each of the following functions.
42. f x

5 x
4 
3 x
2 
2 43. f x

6 x
3 
5 x
2 
4 x

2
45. ( )

7 46. f x
 x
5 
2 x
3 
5 x

4
6
39.
3
( ) _______
44. ( )

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 x
3 
5 x
2 
8 at the point (-2, 44)
48. 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 x
2 
3 x

10 at the point (1, 11)
50. f x

3 x
3 
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

3 x
3 
9 x

5 52. ( )
 x
2 
2 x

15
7

Proving Trigonometric Identities
Prove each of the following identities.
53. sin x
 sin 3 x
 cos 2 x sin x
55. x x
 

 sin x x
 2
57. sin 2 x
 x x
59. sin( x
 y ) cos y
 cos( x
 y ) sin y
 sin x
54. sin x
 cos x
 sec x
 csc x csc sec x
56.
1
1 cos 2 x


1 x
 2
2 csc 2 x
58. x
 2 x
 2 cos 4 cos 2 sin 2 x
60. sin x


2

 cos x
8

Trigonometric Equations:
Solve each of the following equations for 0 < x < 2

.
61. 2cos 2 x

3 62. cos 2 x


1
2
63. sin x
 sin 2 x 64. 2 cos
2 x
  x
65. cos
2 x
  x

0 66. sin
2 x
 cos 2 x
 cos x

0
67. sin x
 cos x

0
69. cos tan x
 sin
2 x

0
68.
70.
4 cos
2 x
  tan
2 x
 
0
0
9

Inverse Trigonometric Functions:
For each of the following, express the value for “y” in radians.
71. y
 arcsin

2
3
72. y
 arccos
 
For each of the following give the value without a calculator.
74. tan arccos

2
3 
77.
 sin 2 sin

1
7
8
75. sec sin


1
12
13 
78.

1
1
2
 tan

1 
3
73. y
 
76.
 sin 2 arctan
12
5
10

Logarithmic Functions:
Evaluate each of the following logarithms.
79. log 16
4

______ 80. log 32
2

______
82. log 216
6

______
85. log 28
6

______
83. log 125
5

______
86. log 12
5

______
Solve each of the following for “x”.
88. log
4 x

3 89. log
8 x

3log 2
8
81.
84.
87. log1000

______ log 7
3

______ log 9
12

______
90. log
4 x
 log 2 log 3
4

4
91. log x

1 log 27
3
94. log( x 3) log(2 x

4)
 log 3
92. log
9 x

5log 2 log 8
9

9
95.
93. log (
3 x log( x
2   x
  log 5
 
2
11

Reciprocal Identities: csc x

1 sin x sec x

1 cos x cot x

1 tan x
Quotient Identities: tan x
 sin x cos x cot x
 cos x sin x
Pythagorean Identities:
Sum Identities: sin
2 x
 cos
2 x

1 tan
2 x

1
 sec
2 sin( x
 y )
 sin x cos y
 cos x sin y tan( x
 y )
 tan
1
 x tan
 tan x tan y y x 1
 cot
2 x
 csc
2 x cos( x
 y )
 cos x cos y
 sin x sin y
Difference Identities: sin( x
 y )
 sin x cos y
 cos x sin y tan( x
 y )
 tan
1
 x tan
 tan x tan y y cos( x
 y )
 cos x cos y
 sin x sin
Double Angle Identities: sin 2 x

2 sin x cos x cos 2 x
 cos
2 x
 sin
2 x

1

2 sin
2 x

2 cos
2 x

1 tan 2 x

2 tan
1
 tan
2 x x
Half-Angle Identities: sin x
2
 
1
 cos x
2 x cos
2
 
1
 cos x
2 tan x
2
 
1
 cos x
1
 cos x
Logarithms:
Product property:
Quotient property:
Power property:
Property of equality:
Change of base formula: y
 log b log a mn
 x is equivalent to log b m
 log b n x
 a y log b m
 log b m n log
If m p  b log m
 b p log log b b
 n m log b n
, then m = n log n
 a log log b b n a
Derivative of a Function: Slope of a tangent line to a curve or the derivative: h lim
  f ( x
 h )
 f ( x ) h y
Slope-intercept form: y
 mx
 b
Standard form: Ax + By + C = 0
Point-slope form: y
 y
1
 m ( x
 x
1
)
12