AP Calculus AB and BC – Summer Packet
NAME____________________________________________
Last Year’s Course _____________________ Last Year’s Teacher :__________________________
This packet is due the first day back in school in the fall. It will be graded for correctness. Spend some
quality time on this packet this summer. Work needs to be shown when needed. Also, do not rely on the
calculator. Half of your AP exam next year is taken without the calculator, so use paper and pencil
techniques only.
If you do not have the skills addressed in this packet, you will find that you will get problems incorrect
next year, even though you understand the calculus concepts. It is frustrating for students when they
are tripped up by the algebra and not the calculus. This summer packet is intended for you to brush
up and possibly relearn these topics.
If you need help on some of these topics, the following websites are good sources for instruction:
The Math Page: http://www.themathpage.com/
Paul’s Online Math Notes: http://tutorial.math.lamar.edu/sitemap.aspx
Virtual Math Lab: http://www.wtamu.edu/academic/anns/mps/math/mathlab//
Khan Academy Teaching Videos: https://www.khanacademy.org/
Also refer to the AP Calculus teachers’ websites for this document and other documents and links.
Section 1: Trigonometry – Know all unit circle values.
Determine the exact value of each without using a calculator:
1. sin 0
4. cos
7. tan

4
3
4
3. sin
7
4
5. cos3
6. cos
11
6
7
6
9. tan
5
3
2. sin
8. tan

3
 2
10. sin 1 

 2 
 3
11. cos 1 

 2 
12. arctan  1

 2 
13. cos  sin 1 
 

2



   
14. cos 1  tan   
  4 

 3 
15. sin  arctan    
 4 

16. List the Pythagorean Trigonometric Identities: _________________________
_________________________
_________________________
17. List the Double Angle Trigonometric Identities: sin 2x  ____________________________
cos 2x  ____________________________
18. Find all the exact solutions to 2sin 2 ( x)  3sin( x)  2  0 on the interval 0, 2  .
19. Solve the equation: 2sin 2 ( x) cos( x)  cos( x) on the interval 0, 2  .
20. Use Trigonometric Identities to simplify:  csc( x)  tan( x)  sin( x)cos( x)
21. Graph the following from 0, 2 
a. y  sin 
b. y  cos 
Section 2: Exponential Functions and Logarithms
Simplify:
22. e3 ln x
23. eln 3
24. e 3ln x
25. ln e3
26. ln e2 x
27. ln1
28. log 1 8
29.
2
x13
x6
30.
x3
x
1
31. 27

2
3
5
x2
34.
x
2 3


32. 125x 3 


 x 
35. 

4 3
 x 
33.
6
36.
4
x5 x
e4 x
e3
Graph the following:
37. y  2 x
38. y  log 4 x
Section 3: Algebra Review
Simplify the following:
2
39.
4
3
5
x3
42. 2
x 9
1 1

x y
40.
xy
1
x
x
41.
1
x
x
x 2  4 x  12
43. 2
x  6 x  16
x3  7 x 2  8 x
44. 3
x  8 x 2  2 x  16
For #’s 45-52, graph each function and state the following:
A. zeros
B. y-intercept
45. f  x   9  x2
C. domain and range (interval notation or exclude values)
46. f  x  
x4
x 2  16
47. f  x   x3  5x2  14 x
 x 2 , x  2

48. f  x    x3 ,
2 x  2
2 x  1, x  2

49. f  x   x  4
50. f  x  
51. f  x   2  x
52. f  x   x  1  3
1
x
For #’s 53-56, write the equation of a line in point-slope form: y  y1  m  x  x1 
53. A line containing  2,5 and  3, 2 
54. A line containing  4, 1 and the origin.
55. A horizontal line with a y-intercept at -3.
56. A vertical line with a root at 5.
57. Expand the binomial  2 x  3
3
5


58. Simplify: x 2  x  x 2  x 2 


59. Use sign analysis to solve:
60. Given f  x   3x2  1 , find
3
x  12 4
 0
x3 x
f  x  h  f  x
h
(the difference quotient).
61. Use your graphing calculator to find the x-values of the point(s) of intersection for:
f  x   x2  4 x  32 and g  x   3x  5 (Round to 3 decimal places.) Draw a quick sketch and state the
intervals where g  x   f  x  and where f  x   g  x  .
62. Find the inverse to the following functions. Sketch the function and its inverse on the same graph and
state whether the inverse is a function. Support your answer.
a. 2 x  6 y  1
b. y  9  x 2
63. Find the inverse to y  1  x3 and show that f  f 1  x    x