Download AP Calculus Packet - East Penn School District

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.
This is a NON CALCULATOR packet, except for Question 61. 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/
More free video tutorials, from Algebra to Calculus
http://www.mrmathblog.com/ (This one specifically has AP Calculus)
http://patrickjmt.com/ (JMT = Just Math Tutorials)
http://www.mathispower4u.com/ (James Sousa, Arithmetic to advanced college Calculus courses)
Also refer to the AP Calculus teachers’ websites for this document and other documents and links.
AP Calculus Summer Packet 2016
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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 

AP Calculus Summer Packet 2016
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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 
AP Calculus Summer Packet 2016
b. y  cos 
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Section 2: Exponential Functions and Logarithms
Simplify:
22. e3ln x
23. eln 3
24. e3ln x
25. ln e3
26. ln e2 x
27. ln1
28. log 1 8
29.
2
x13
x6
30.
x3
x
1
31. 27
34.
x

2
3
5
2
x
2 3


32. 125x 3 


 x 
35. 

4 3
 x 
33.
6
4
x5 x
e4 x
36. 3
e
Graph the following (Know these graphs WITHOUT a calculator)
37. y  2 x
AP Calculus Summer Packet 2016
38. y  log 4 x
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Section 3: Algebra Review
Simplify the following: Note: #40 use multiply by reciprocal method; #41 use LCD method
2
39.
42.
4
1 1

x y
40. a.
xy
3
5
x3
x2  9
43.
xy
b.
1
x
41. x
1
x
x
1 1

x y
x 2  4 x  12
x 2  6 x  16
44.
x3  7 x 2  8 x
x3  8 x 2  2 x  16
For #’s 45-52, graph each function (WITHOUT a calculator) and state the following:
A. zeros
B. y-intercept
45. f  x   9  x 2
AP Calculus Summer Packet 2016
C. domain and range (interval notation or exclude values)
46. f  x  
x4
x 2  16
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47. f  x   x3  5x 2  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
AP Calculus Summer Packet 2016
1
x
Page 6 of 8
For #’s 53-54, 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
5


58. Simplify: x  x  x 2  x 2 


3
59. Use sign analysis to solve the rational inequality:
60. Given f  x   3x 2  1 , find
AP Calculus Summer Packet 2016
3
2
x  12 4
 0
x3 x
f  x  h  f  x
(the difference quotient).
h
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61. Use your graphing calculator to find the x-values of the point(s) of intersection for: f  x   x 2  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  (use interval notation).
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
AP Calculus Summer Packet 2016
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