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. 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 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:
1. e3 ln x
2. eln 3
3. e 3ln x
4. ln e3
5. ln e2 x
6. ln1
7. log 1 8
8.
2
x13
x6
9.
x3
x
1
10. 27

2
3
5
x2
13.
x
2 3


11. 125x 3 


 x 
14. 

4 3
 x 
12.
6
15.
4
x5 x
e4 x
e3
Graph the following:
16. y  2 x
17. y  log 4 x
Section 3: Algebra Review
Simplify the following:
2
1.
4
3
5
x3
4. 2
x 9
1 1

x y
2.
xy
1
x
x
3.
1
x
x
x 2  4 x  12
5. 2
x  6 x  16
x3  7 x 2  8 x
6. 3
x  8 x 2  2 x  16
For #’s 7-14, find the following for each function:
A. zero’s
B. y-intercept
7. f  x   9  x2
C. domain (interval notation) D. range
8. f  x  
x4
x 2  16
E. graph
9. f  x   x3  5x 2  14 x
 x 2 , x  2

10. f  x    x3 ,
2 x  2
2 x  1, x  2

11. f  x   x  4
12. f  x  

13. f x  2  x

1
x
14. f x  x  1  3
For #’s 15-18, write the equation of a line in point-slope form: y  y1  m  x  x1 



15. A line containing 2,5 and 3,2



16. A line containing 4,1 and the origin.
17. A horizontal line with a y-intercept at -3.
18. A vertical line with a root at 5.
19. Expand the binomial  2 x  3
3
5


20. Simplify: x 2  x  x 2  x 2 


21. Use sign analysis to solve:
22. Given f  x   3x2  1 , find
3
x4 4
 0
x3 x
f  x  h  f  x
(the difference quotient).
h
23. Find the point(s) of intersection for: f  x   x2  4 x  32 and g  x   3x  5 , also state the domain
where g  x   f  x  and where f  x   g  x 
24. Find the inverse to the following functions and show graphically whether the inverse is a function.
a. 2 x  6 y  1
b. y  9  x 2
25. Find the inverse to y  1  x3 and show that f  f 1  x    x