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.
2.
3.
This packet is to be handed in to your Calculus teacher on the first day of the school year.
All work must be shown in the packet OR on separate paper attached to the packet.
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.
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
− 2
x
x
6.
4 3
1− + 2
x x
2−
4−
1−
1−
3 10
−
x x2
7.
11 18
1+ + 2
x x
1−
2
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
5
6
10.
Write each series in expanded form.
5
12.
∑ (2 x
n
)
n =1
xi
∑
i =1 i + 1
4
13.
xi
14. ∑
i =2 i
5
3
Operations on Rational Expressions
Simplify each of the following by factoring:
16.
6 x2 + x − 2 2 x2 + 9 x + 4
•
6 x2 + 7 x + 2 4 − 7 x − 2 x2
x3 − y 3
2 x 2 + 5 xy + 3 y 2
• 2
17.
x + 4 xy + y 2
2 x 2 + xy − 3 y 2
18.
2 x 2 + 13 x + 20 6 x 2 − 13 x − 5
÷ 2
8 − 10 x − 3 x 2
9 x − 3x − 2
6 x 2 + 23x − 4 4 x 2 + 20 x + 25
÷
19.
6 x 2 + 17 x − 3 2 x 2 + 11x + 15
3 x 2 − 10 x − 8 2 x 2 − 9 x + 10
÷
20.
6 x 2 + 13 x + 6 4 x 2 − 4 x − 15
15.
x 2 + x − 6 x 2 + x − 20
•
12 + x − x 2 x 2 − 4 x + 4
Simplify each of the following operations of addition or subtraction.
21.
x
2
6
+
+ 2
4x −1 2x + 1 8x + 2x −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
3 x + 2 7 x 2 + 24 x + 28
+
−
3 x + 4 x − 5 3 x 2 − 11x − 20
22.
4
Fractional and Integral Exponents
Simplify each of the following. Leave all answers with POSITIVE exponents.
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
⎛ 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.
5
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( x 3 ) ⎤⎦ = _______
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) = 5 x 4 − 3 x 2 + 2
43. f ( x) = 6 x 3 − 5 x 2 + 4 x − 2
45. f ( x ) = 7
46. f ( x) = x 5 − 2 x 3 − 5 x + 4
6
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 x 3 + 5 x 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 x 2 + 3 x + 10 at the point (1, 11)
50. f ( x) = 3 x3 − 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) = 3x 3 − 9 x + 5
52. f ( x) = 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
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 csc 2 2 x
1 + cos 2 x 1 − cos 2 x
57. sin 2 x = 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
π⎞
⎛
60. sin ⎜ x + ⎟ = cos x
2⎠
⎝
8
Trigonometric Equations:
Solve each of the following equations for 0 < x < 2π.
−1
2
61. 2 cos 2 x = 3
62. cos 2 x =
63. sin x = sin 2 x
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
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
⎝
⎠
10
12 ⎞
⎛
76. sin ⎜ 2 arctan ⎟
5⎠
⎝
Logarithmic Functions:
Evaluate each of the following logarithms.
79. log 4 16 = ______
80. log 2 32 = ______
81. log1000 = ______
82. log 6 216 = ______
83. log 5 125 = ______
84. log 3 7 = ______
85. log 6 28 = ______
86. log 5 12 = ______
87. log12 9 = ______
88. log 4 x = 3
89. log8 x = 3log8 2
90. log 4 x = log 4 2 + log 4 3
1
91. log x = log 27
3
92. log 9 x = 5log 9 2 − log 9 8
93. log 3 ( x + 1) = 2
Solve each of the following for “x”.
94. log( x + 3) + log(2 x − 4) = log 3
95. log( x 2 + 3) − log( x + 1) = log 5
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 ) =
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:
sin 2 x = 2 sin x cos x
= 1 − 2 sin 2 x
= 2 cos 2 x − 1
tan 2 x =
2 tan x
1 − tan 2 x
x
1 − cos x
=±
2
2
tan
x
1 − cos x
=±
2
1 + cos x
sin
Logarithms:
Product property:
Power property:
Property of equality:
y = log a x is equivalent to
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
If log b m = log b n , then m = n
Change of base formula:
log a n =
Derivative of a Function:
Slope of a tangent line to a curve or the derivative: lim
Quotient property:
cos
x
1 + cos x
=±
2
2
Half-Angle Identities:
x = ay
log b n
log b a
Slope-intercept form: y = mx + b
Standard form:
Ax + By + C = 0
h→∞
Point-slope form: y − y1 = m( x − x1 )
12
f ( x + h) − f ( x )
h