AP CALCULUS AB
Summer Review Packet Welcome to AP Calculus AB! This packet contains prerequisite material that you must have mastered by this point in your math career in order to be successful in Calculus AB. The Calculus student who struggles, usually struggles because of poor Algebra/Algebra 2 skills. There is a lot of information here- some of it you may know very well and some of it you may need to spend time reviewing. Either way, your time and effort spent working through these problems will directly benefit you next year. All of the problems in this packet should be done without the use of a calculator. This may seem like quite a challenge, however ≈ 65% of the AP test is without a calculator and therefore many of your tests and quizzes in this class will be noncalculator as well. It’s time to start preparing now! On the other hand, the calculator portion of the AP test will expect that you are skilled at utilizing the graphing calculator to solve various types of problems. We will be practicing these skills in the beginning of the year and throughout the year. You will want to invest in a graphing calculator ASAP if you don’t already have one. (Recommended models: TI-84 Plus, TI-84 Plus C Silver Edition, or TI-84 Plus CE ←new!) By the end of the summer ALL PROBLEMS in this packet need to be completed and understood. Your best work should be shown. (attach separate sheets, if needed!) This packet will be collected and you will be tested on this information during the first week of school. If you need help, please feel free to email us. You can also visit the AP Summer Assignment tab on www.mrsronan.com for helpful example problems. We look forward to meeting you (or reconnecting with you!) in September! Get ready to eat, breathe, and sleep CALCULUS!
Mr. Cook
[email protected] Mrs. Ronan
[email protected] 1
OPERATIONS WITH FRACTIONS
1
1. Perform the indicated operation without the use of a calculator.
2
a.
3 4 • 11 5
b.
2 •4 11
c.
5 25 ÷ 12 36
d. 24 ÷
e.
18 ÷2 5
f.
5 7 + 12 24
g.
0 5 + 1 6
h.
3 8
18 4 − 25 35
AREA & VOLUME
Find the area. 5 cm
2.
3.
6 cm
9 cm
Find the volume. Leave π in your answer. 4.
5. 3 cm
6. The figure below is the shape of a washer with an outer radius of 18 ft and an inner radius of 15 ft. If the figure is 3 ft in height, find the volume of the washer.
2
3
FUNCTIONS
7. If f (x ) = x 2 − 5 , find: a. f ( x + h )
b. f ( x + h ) − f ( x )
c.
f ( x + h) − f ( x) h
8. Use the given functions to compute each composition below. f (x ) = − x 2 + 2 x
j ( x) = x − 3
− 1 + 3 x − 1, x ≥ 2 0≤ x<2 n(x ) = x 2 − 4, − x, x<0
g ( x ) = 3x − 1
k ( x ) = sin x
h ( x ) = 4x
m ( x ) = {( 3, 2 ) , ( 4,3) , (1, 6 )}
p (x )
q(x )
3π b. g k 2
c. (h f )(− 1)
d. p (q (8))
e. ( f m )( 3)
f. j −1 ( g (− 2))
g. n(n(− 9))
h. f ( g ( x ) )
i. f ( k ( x ) )
a. f (h(− 1))
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4
GENERAL GRAPHS
Graph each. Include any asymptote lines. State the domain and range. 9. f ( x ) = sin x
10. f ( x ) = cos x
11. f ( x ) = tan x
D: ____________
D: ____________
D: ____________
R: ____________
R: ____________
R: ____________
12. f ( x ) = x
13. f ( x ) = x2
14. f ( x ) = x3
15. f ( x ) = x
D: ____________
D: ____________
D: ____________
D: ____________
R: ____________
R: ____________
R: ____________
R: ____________
16. f ( x ) = x
17. f ( x ) = 3 x
18. f ( x ) =
1 x
19. f ( x ) =
1 x2
D: ____________
D: ____________
D: ____________
D: ____________
R: ____________
R: ____________
R: ____________
R: ____________
20. f ( x ) = ex
21. f ( x ) = ln x
D: ____________
D: ____________
R: ____________
R: ____________
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5
FUNCTION TRANSFORMATIONS
22. Given the function y = f ( x ) , describe the transformation(s) applied in each case. a. y = f ( x ) − 4
b. y = f ( x − 4 )
c. y = f ( 2 x )
d. y = f ( x )
e. y = 5 f ( x ) − 3
f. y = − f ( x + 1)
23. To the right is the graph of y = f ( x ) . Sketch each transformation.
a. y = 2 f ( x )
d. y = f ( x + 2 ) + 1
b. y = − f ( x )
c. y = f (− x )
e. y = f ( x )
f. y = f ( x )
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Graph each. Include any asymptote lines. 24. f ( x ) = 4 − x
25. f ( x ) = ln( x + 1)
26. f ( x ) =
2
( x − 2)
2
4 x 2 , x < 0 27. f ( x ) = 2 − x, 0 ≤ x ≤ 3 4, x>3
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ASYMPTOTES
For each function, find the equation(s) of the vertical and horizontal asymptotes if they exist. 28. g ( x ) =
31. y =
x x−3
2 x3 x3 − 1
29. y =
x+4 x2 −1
32. f ( x ) =
x −1 x + x−2 2
30. h ( x ) =
x+4 x2 + 1
33. f ( x ) =
5 x + 20 x 2 − 16
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7
EXPONENTIAL & LOGARITHMIC EXPRESSIONS
Evaluate each without using a calculator. 34. log 3 81
1 35. log 5 25
38. ln e 5
1 39. ln 2 e
42. Condense:
36. log16 4
37. ln 1
40. 6 log 6 4
1 log x − 2 log 5 − 10 log y 3
43. Expand: log 4
41. e 2 ln x
a5 64
8 RATIONAL EXPONENTS Re-express each in radical form. 44. 2 x −
3
1 2
1
45. x 4 y 4
46.
(xy )−
3 4
Evaluate without a calculator. 1
3
47. 27 3
48. 16 2
3
(
5
50. Simplify: x 2 x + x 2 − x 2
49. 8
− 43
) 7
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FACTORING
Factor each expression completely. 51. x3 − 25 x
52. 30 x − 9 x 2 − 25
54. 9a 4 − a 2b 2
56. 4 x 3 ( x − 2 )2 + 8 x 2 ( x − 2 )3
10
53. 3x3 − 5 x 2 + 2 x
55. x3 + 1
(gcf!)
57. 3x(2 x + 5) 2 + 3(2 x + 5) 2 −1
1
(gcf!)
LINES
Answers in point-slope and slope-intercept form. 58. Write the equation of the line that passes through ( 2, −4 ) and is parallel to the line 5 x − 2 y − 4 = 0 .
59. Write the equation of the line that passes through ( 7, −3) and is perpendicular to the line 2 x = 5 y + 8 .
60. The function f is a line. If f (3) = 5 and f (4) = 9 , find the equation of f .
61. The function g is a line. If the slope of g (x ) is
2 3
, and g (1) = 1 , then find g ( 32 ) .
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RATIONAL & RADICAL EXPRESSIONS
Re-express each in simplest form. 62.
5− x x 2 − 25
63.
x3 − 3x 2 − 3x + 1 x2 − 4x − 5
18 3 65. x 4 x8
3 3 − 66. x + h x h
Rationalize the numerator.
Rationalize the denominator:
68.
3− 9− x x
69.
x 1 + 2x − 1
64.
(4 + x )2 − 16 x
2 3x 67. 4 x− 9x 1−
70.
4+ 5 3+ 2 5
9
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TRIGONOMETRIC EXPRESSIONS
Evaluate the following without a calculator. 5π 71. cos 6
7π 72. sin 4
75. sin (2π )
1 76. cos −1 − 2
5π 79. sin 4 2
2 5π − cos 3
2π 73. tan − 3
7π 74. sec 6
77. tan −1 ( −1)
2π 78. sin −1 cos 3
π 80. 6sec π − 4 cot 2
2
π 2π 81. 4 cos − sin 6 3
−2
Re-express each using trigonometric identities. 82. sin θ (csc θ − cos θ tan θ )
84. Circle ALL equivalent to cos 2 x .
83.
(cos x )2
cos x 2
secθ − cosθ sin 2 θ sec 2 θ
cos 2 x
1 + sin 2 x
1 sec2 x 10
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SOLVING EQUATIONS
Solve each without a calculator. 85. 4t 3 − 12t 2 + 8t = 0
86. 2 x 2 − 3x = −3
87. 4e2 x = 5
88. − ln ( x + 2 ) = 3
89. 3 x − 2 − 8 = 8
90. 33 x + 5 = 92 x +1
91. 2 x + 1 = x + 3
92.
x +1 x − =0 x x +1
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For #93-96, Solve for x, using the domain 0, 2π ) . Use exact answers. 93. 4 sin 2 x = 1
94. cos 2 x = cos x
95. 2 cos x + 3 = 0
96. sin 2 x − cos 2 x = 0
For #97-100, Solve for z. 97. 4 x + 10 yz = 0
99. xz + y = 1 + z
98. y 2 + 3 yz − 8 z − 4 x = 0
100. ln z = 2 ln x + ln 10
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