Name:
Summer Packet AP Calculus
Mr. DeStefano
AP Calculus
Welcome to AP Calculus AB! The following packet should be completed during the summer.
Completing and understanding the topics in this packet will help us to complete the course in a timely
manner, have ample time to practice for the AP Exam, and will help us gauge whether you have the
appropriate skills necessary to be successful in the AP Calculus class and to achieve the highest score
possible on the AP Calculus Exam.
Please complete all the problems on separate pieces of paper and label the problem number and your
work clearly. This packet is due the first week we return to school and will be reviewed for 2 days. On
the third day of school the packets will be collected and graded. Expect your first test that will cover all
these topics that same day. If you find yourself struggling do not hesitate to contact me at
ddestefa@k12.wv.us . I plan on having at least 1 day (possibly two 8am-12am sessions between August
10th and 11th) of face to face help on this packet.
I will be placing online, interactive tutorial videos to help with each topic on the packet. Go to
http://www.berkeleycountyschools.org/martinsburghigh click on the Teachers link at the top. Scroll
down to DESTEFANO, DAVID and click on my name. There is a link on the side for AP Calculus Summer
Pack Help. Click on the topic you need help with and watch the videos. Feel free to pause, rewind and
replay the video as many times as you need. I would also suggest using resources such as Khan
Academy or iTunesU to refresh your memory or skills. I have listed the work in the packet by topic and
searching for those key words should be sufficient to finding appropriate resources. The direct link is:
http://www.berkeleycountyschools.org/Page/6538
This packet should be done without the use of a calculator and all work should be shone except for the
problems that indicate calculator use. You may use a calculator to check your answers. For android
users the app Wabbitemu is a good substitute for a graphing calculator. For iOS users Desmos is a good
substitute for a graphing calculator.
Remember most of the AP Calculus Exam is done without the use of a calculator. Everyone taking this
class is expected to take the AP Calculus Exam, Thursday May 5th, 2016 at 8 a.m.! If you do your work
and give yourself time to work through difficulties, I will make sure that you are as prepared as possible.
Mr. DeStefano
Name:
Summer Packet AP Calculus
Mr. DeStefano
AP Calculus
Topic 1: Fractional and Negative Exponents: Simplify using only positive exponents.
1.
√4𝑥−16
2.
4
√(𝑥−4)3
3
1
(2𝑥+5)−2
2
3
2
1
4
1
−
3. (𝑥 −2 + 𝑥 −1 𝑦 −1 + 𝑦 −2 )
1
2
Topic 2: Domain. Find the domain of the following functions. Use interval notation.
𝑥 2 −5𝑥−6
4. 𝑦 = 𝑥 2 −3𝑥−18
5. 𝑦 =
√3𝑥−7
2𝑥+9
6. 𝑦 = √𝑥 2 − 5𝑥 − 14
7. 𝑦 = log⁡(2𝑥 − 8)
Topic 3: Inequalities:
8. Write the following as a piecewise function: 𝑦 = |4𝑥 − 2|
Solve the following inequalities:
9. |𝑥 − 5| ≤ 3
10. |3𝑥 − 4| > −2
11. 𝑥 2 − 3𝑥 ≥ 10
12. 𝑥 3 + 4𝑥 2 − 𝑥 ≥ 4
13. 2𝑠𝑖𝑛2 𝑥 ≥ sin 𝑥,⁡⁡⁡0 ≤ 𝑥 < 2𝜋
14.
2𝑥−1
3𝑥−2
≤1
Topic 4: Even and Odd Functions: Show algebraically whether the relation is even, odd or neither
15. 𝑓(𝑥) = 2𝑥 2 − 7
16. 𝑓(𝑥) = −4𝑥 3 − 2𝑥
17. 𝑓(𝑥) = 4𝑥 2 − 4𝑥 + 4
Topic 5: Function Transformation: if 𝒇(𝒙) = 𝒙𝟐 − 𝟏, describe in words the transformations that would
take place to the graph of the following:
18. 𝑓(𝑥) − 4
19. 𝑓(𝑥 − 4)
20. −𝑓(𝑥 + 2)
21. 5𝑓(𝑥) + 3
22. 𝑓(2𝑥)
23. |𝑓(𝑥)|
Topic 6: Factor Theorem (p over q method/synthetic division): Use synthetic division to factor the
polynomial and solve 𝑷(𝒙) = 𝟎
24. 𝑃(𝑥) = 𝑥 3 − 6𝑥 2 + 3𝑥 + 10
Topic 7: Factoring. Factor completely.
25. 64𝑥 3 − 125𝑦 3
26. 𝑥 4 + 11𝑥 2 − 80
27. 2𝑥 2 + 50𝑦 2 − 20𝑥𝑦
28. 𝑥 2 + 12𝑥 + 36 − 9𝑦 2
29. (𝑥 − 3)2 (2𝑥 + 1)3 + (𝑥 − 3)3 (2𝑥 + 1)2
30. (3𝑥 + 4)−3 (2𝑥 − 5)3 + (3𝑥 + 4)−2 (2𝑥 − 5)2
31.
1
(2𝑥
10
5
1
3
+ 1)2 − 6 (2𝑥 + 1)2
Name:
Summer Packet AP Calculus
Mr. DeStefano
AP Calculus
Topic 8: Solving by factoring or Quadratic Formula. Solve each equation.
32. 𝑥 2 + 6𝑥 + 4 = 0
35.
1
𝑥2
33. 2𝑥 2 − (𝑥 + 2)(𝑥 − 3) = 12
1
34. 𝑥 − 10√𝑥 + 9 = 0
36. 𝑥 3 + 2𝑥 2 − 3𝑥 − 6 = 0
−𝑥 =6
Topic 9: Asymptotes. For each function, find the equations of both the vertical and horizontal
asymptote(s) if they exist. Put in the form of x=? and y=?
37. 𝑦 =
𝑥+4
𝑥 2 +1
38. 𝑦 =
𝑥 2 −9
39. 𝑦 =
𝑥 3 +3𝑥 2 −18𝑥
2𝑥 3
𝑥 3 −1
Topic 10: Complex Fractions. Simplify.
40.
3 4
−
𝑥 𝑦
4 3
−
𝑥 𝑦
41.
2
3𝑥
4
𝑥−
9𝑥
1−
42.
𝑥2 −𝑦2
𝑥𝑦
𝑥+𝑦
𝑦
Topic 11: Composition of Functions: if𝒇(𝒙) = 𝒙𝟐 , 𝒈(𝒙) = 𝟐𝒙 − 𝟏, 𝒂𝒏𝒅⁡𝒉(𝒙) = 𝟐𝒙 , find the following.
1
43. 𝑓(𝑔(2))
44. ℎ(𝑓(−1))
45. 𝑔 (𝑓 (ℎ (2)))
46. 𝑔(𝑓(𝑥))
47. 𝑔(𝑔(𝑥))
48. 𝑓(ℎ(𝑥))
Topic 12: Rationalizing Denominators. Simplify.
49.
3
50.
4
√(3𝑥)3
5
5√2+√5
Topic 13: Solving Rational Equations. Solve for x.
𝑥−5
3
51. 𝑥+1 = 5
53.
2
1
+ 𝑥−5
𝑥+5
16
= 𝑥 2 −25
52.
60
60
− 𝑥−5
𝑥
54.
2𝑥−3
𝑥+1
2
=𝑥
10
+ 𝑥 2 −1 =
2𝑥+3
𝑥−1
Topic 14: Solving Trigonometric Functions. Solve each equation on the interval[𝟎, 𝟐𝝅). Give exact
answers if possible. *you may use a calculator
55. 𝑐𝑜𝑠 2 𝑥 = cos 𝑥
56. 4 𝑠𝑖𝑛2 𝑥 = 1
57. 2𝑠𝑖𝑛2 𝑥 + sin 𝑥 = 1
58. 2 sin 𝑥 cos 𝑥 + sin 𝑥 = 0
59. 𝑠𝑖𝑛2 𝑥 − 𝑐𝑜𝑠 2 𝑥 = 0
*60. 8𝑐𝑜𝑠 2 𝑥 − 2 cos 𝑥 = 0
Topic 15: Logarithms. Simplify.
61. 32𝑙𝑜𝑔3 5
62. 2𝑙𝑜𝑔4 9 − 𝑙𝑜𝑔2 3
Name:
Summer Packet AP Calculus
Mr. DeStefano
AP Calculus
Topic 16: Solving Exponential Equations. Leave answers in terms of ln.
63. 5𝑥+2 = 2
64. 6𝑒 4𝑥 = 21
65. 𝑒 2𝑥 + 𝑒 𝑥 − 12 = 0
66. 𝑒 𝑥 𝑥 2 + 6𝑥𝑒 𝑥 = 0
Topic 17: Solving Logarithmic Equations.
67. 𝑙𝑜𝑔3 (𝑥 − 2) = 4
68. 5 + 4𝑙𝑜𝑔4 (3𝑥) = 13
69. 𝑙𝑜𝑔5 (𝑥 − 3) + 𝑙𝑜𝑔5 (𝑥 + 4) − 𝑙𝑜𝑔5 𝑥 = 1
Topic 18: Equations of Lines. Determine the equation of a line with the following characteristics.
70. Passes through the points (−1, 3)𝑎𝑛𝑑(2, −4)
71. Passes through the point (−1, 2) and is perpendicular to the line 2𝑥 − 3𝑦 + 5 = 0
Topic 19: Trigonometry. Evaluate the following without a calculator. Draw the reference triangle.
7𝜋
72. sin ( 6 )
73. csc(60°)
74. cos(120°)
𝜋
75. tan ( 2 )
Topic 20: Inverse Functions. Find the inverse of each function.
𝑥+2
76. 𝑓(𝑥) = 2𝑥 + 3
77. 𝑓(𝑥) = 5𝑥−1
Topic 21: Piecewise functions. Graph the piecewise function by hand. You may use graph paper
𝑥 2 − 5, 𝑥 < −1
78. 𝑓(𝑥) = { 0, 𝑥 = −1
6 − 4𝑥, 𝑥 > −1
Topic 22: Geometry.
79. Find the volume of the washer to the right with outer radius 1.8 in. and
inner radius 1.5 in, and height 0.3 in.
80. Three sides of a fence and an existing wall form a rectangular enclosure.
The total length of fencing used for the three sides is 240 ft. Find the two matching sides, x, if the area
enclosed is 5500 ft2
81. A cone-shaped water tank is 10m high and has a radius of 3m at the top. If the water is 6m deep in
the middle, what is the surface area of the top of the water?
Topic 23: Graphing Calculator Skills
For problems 82-85, 𝑓(𝑥) = 2𝑥 2 − 5𝑥 + 3⁡𝑎𝑛𝑑⁡𝑔(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑥 − 1
82. Find the zero(s) for f(x).
83. Find the intersection point of f(x) and g(x)
84. Find the maximum for f(x)
85. Find the minimum of g(x) on [0,3]