2015
Bishop Kelley High School
Summer Math Program
Course: AP Calculus AB & BC
NAME:
DIRECTIONS:





Show all work in the packet.
This material will be collected, graded, and points awarded at the discretion of each teacher on the first day of
the math class.
A test on this material will be administered during the first week of the class.
An additional resource for help with this packet is http://www.khanacademy.org. It provides videos of about 10
minutes in length on hundreds of different math topics.
Feel free to email me with questions and or concerns ttokarchik@bishopkelley.org
Math Teachers will be available in C-1 the following dates/times if you need help.
Date
Time
Monday July 27th
9-10:30am
Thursday July 30th
9-10:30am
Monday August 3rd
9-10:30am
AP CALCULUS
Summer Assignment
Name_______________________________
“Contrary to common belief, the calculus is not the height
of the so-called „higher mathematics.‟ It is, in fact, only the
beginning.” -Morris Kline
AP Calculus Summer Assignment
Due Date: The beginning of class on the first Monday of the school year.
The purpose of this assignment is to have you practice the mathematical skills necessary to
be successful in AP Calculus. All of the skills covered in this packet are skills from Algebra 2
and Pre-Calculus. If you need to, you may use reference materials to assist you and refresh
your memory (old notes, textbooks, online resources, etc.). While the graphing calculators will
be used in class, there are no calculators allowed on this packet. You should be able to do
everything without a calculator.
AP Calculus is a fast paced course that is taught at the college level. There is a lot of
material in the curriculum that must be covered before the AP exam in May. Therefore, we
cannot spend a lot of class time re-teaching prerequisite skills. This is why you have this
packet. Spend some time with it and make sure you are clear on everything covered in the
packet so that you will be successful in Calculus. You can also use the packet throughout the
year as a review tool for those concepts as they arise. Of course, you are always encouraged
to seek help from your teacher if necessary.
This assignment will be collected and graded the first Monday of the school year. Be sure
to show all appropriate work to support your answers. In addition, there may be a quiz on this
material during the first quarter.
Good Luck and we look forward to a great year next year!
Section 1: Algebra Review
Show all work – No credit will be given for answers missing appropriate work. No calculators.
Identify the following statements as true or false.
x y x y
1
1
1.
_____
2.
2
2 2
p q p
a
b
4. 3
3a
_____
b
a b
c
5. 3
1
_____
q
3.
3a b
_____
c
2k
2x h
6. a 2
k
x h
b2
_____
a b _____
For questions 7-15, identify the statement as true or false over the set of real numbers. Give a counter
example for any false statement.
7. x 3 1 x 3
8. x 3 x x 3
9. x 2 0
10. x 2
x
11. 2 x
13.
0
14.
x
16. Solve xy '
18. Factor: y 3
y 1
1
x
y ' for y ' .
x
12. x
x
15. x
17.Solve ln y
0
x
kt for y .
19. Factor: x 2 ( x 1) 4( x 1)
27
Simplify each expression.
3
1
x2 x
20.
x7
3
22.
5x h
h
5x 3
_________________
21. x
_________________
23.
3
x x6
2
3x h
h
_________________
3x 2
_________________
x2 1
24. x
x 1
x3
26.
1
1 2a
1 4
2
25. x x
1
3
x
__________________
2
1 2a
6a 2
__________________
4a 2 1
a
27. 2 x h
h
__________________
a
2x
__________________
Simplify, using factoring of binomial expressions. Leave answers in factored form.
Example:
( x 1) 3 (4 x 9) (16 x 9)( x 1) 2
( x 6)( x 1)
( x 1) 2 ( x 1)(4 x 9) (16 x 9)
( x 6)(x 1)
( x 1) 2 (4 x 2 5 x 9 16 x 9)
( x 6)(x 1)
( x 1) 2 (4 x 2 21x 18)
( x 6)(x 1)
( x 1) 2 (4 x 3)(x 6)
( x 6)(x 1)
( x 1)(4 x 3)
28. ( x 1) 3 (2 x 3) (2 x 12)( x 1) 2
_______________________________
( x 1) 2 (3 x 1) 2( x 1) 3
( x 1) 4
_________________________________
( x 1) 3 (2 x 3) (4 x 1)( x 1) 2
30.
( x 1) 2 (2 x 1)
_________________________________
29.
Simplify by rationalizing the numerator.
Example:
x 4 2
x
x 9
x
31 .
3
x 4 2
x
x 4
2
x 4
2
x 4 4
x x 4
_____________________
32 .
x
2
1
x x 4
x h
h
2
x
x 4
2
_____________________
Solve each equation or inequality for x over the set of real numbers.
33. 2 x 4
35.
3x 3
2x 2
3x 5
( x 1)(x 4 7)
37. 2 x 3
14
0 _____________________
0 _____________________
_____________________
Solve each of the systems.
39. x y 8
____________________
2x
y
7
34.
2x 7
x 1
36. x 2
38. x 2
2x
_____________________
x 4
9
x 1 _____________________
2x 8
40. y x 2 3x
y 2x 6
0 _____________________
_______________________
Section 2: Precalculus Review
Use your knowledge of the unit circle to evaluate each of the following. Leave your answers in radical form.
2
41. sin(30  ) _____
42. cos
_____
43. tan 45 _____
3
44. sin
_____
6
45. tan ______
47. cos 90  _____
50. cos 1
48. cos
1
_____
2
51. sin
Solve each trigonometric equation for 0
3
53 . sin x
_____________________
2
55 . cos
x
2
46. csc
3
_____
4
1
49. tan
2
_____
2
5
_____
6
6
_____
52. tan 1 (1) _____
x 2 .
54 . tan 2 x 1 _____________________
2
_____________________
2
56 . 2 sin 2 x sin x 1 0 _____________
For each trigonometric function identify the amplitude and any horizontal or vertical shifts from the basic
function.
1
x
57. y
cos
3 amplitude:_________ period:________ vertical shift:________
2
2
58. y
2 sin(2 x
59. y
tan 3x
) amplitude:_________
period:________ horizontal shift:_______
period:___________
Solve each exponential or logarithmic equation.
3
60 . 5 x
125 __________
61 . 8 x
1
16 x __________
62 . 81 4
x _________
63. 8
2
3
x __________
66. log 4 x
64. log 2 32
3 __________
x __________
67. log 3 ( x 7)
65. log x
1
9
2 __________
log 3 (2 x 1) ____________
Expand each of the following using the laws of logs.
68 . log 3 5 x 2 ________________________
69 . ln
5x
_______________________
y2
Complete each of the following using trigonometric identities and formulas.
70. sin
2
73. tan x
x
___________
_______________
71. sin 2 x cos2 x
74. 1 cot2 x
_________
____________
72. sin 2u
__________________
75. 1 cos2 x
_______________
76. A right triangle has a base of 5 and a hypotenuse of 7. Find the height.
Section 3: Graphing Review
Sketch the following functions. State the domain and range of each. Draw and label your own axes.
77. f ( x)
x
78. f ( x)
x2
79. f ( x)
x3
80. f ( x)
x
81. f ( x)
1
x
82. f ( x)
x
83. f ( x)
ex
84. f ( x)
ln x
86. f ( x)
sin x
85. f ( x)
9 x2
87. f ( x)
cos x
88. f ( x)
tan x
89. f ( x)
csc x
90. f ( x)
sec x
91. f ( x)
cot x
AP Calculus AB/BC – Summer Math Packet Answers
1.
2.
3.
4.
5.
6.
True
False
False
True
False
False
31.
32.


8. True  x | x   0,  
9. True  x | x    ,  
10. True x | x    ,0   1,  
11. True  x | x   0,  
12. True  x | x   0,  
13. True  x | x   0,  
14. True  x | x   1,0   1,  
15. True  x | x    ,  
7. True  x | x    ,  
( y  1)
x 1
kt
17. y  e
16. y ' 
18. ( y  3)( y 2  3 y  9)
19. ( x  1)( x  2)( x  2)
20. 1
21. x
22. 15x2  15xh  5h2
23. 6 x  3h
33.
34.
35.
36.
37.
38.
39.  5,3
40.  3,0   2, 2 
41.
42.
43.
44.
45.
46.
47.
25.
26.
3
4a  1
1
2
1
2
1
1
2
0
2
0
48.
 2
2
49.
3
3
2
24. x  x  1
 x  4
x  3x  1
1
x9 3
1
xh  x
1
x  0, ,  2
2
x  28
5
x
3
x5
17 11
x ,
2 2
x | x   2, 4
50. 60 or
2
27. 27.
a
2x  2x  h
28.  x  1 ( x  1)(2 x  9)
51. 45 or
52. 45 or
2
29.
3x 2  4 x  5
 x  1
30. x  4
3
53.

3

4

4
 2
3
,
3
 3 5 7
, ,
54. ,
4 4 4 4

55.
2
AP Calculus AB/BC – Summer Math Packet Answers
56.
 5 3
, ,
6 6 2
1
57. , 4 , down 3
2
58. 2,  , Right
59.

2

3
60. 3
61. 3
62. 27
63.
64.
65.
66.
67.
68.
1
4
5
3
64
x 8
log3  5  2log3  x 
82.
83.
84.
85.
86.
87.
69. ln  5  ln( x)  2ln  y 
70. cos( x)
71. 1
72. 2sin(u) cos(u)
73.
88.
sin( x)
cos( x)
74. csc2 ( x)
89.
75. sin 2 ( x)
76. 4 6
90.
77.
78.
91.
79.
80.
81.