AP Calculus AB
Summer Assignment 2014
Name: __________________________________________
Directions: Complete this packet of review material in its entirety. Any student
enrolled in AP Calculus AB should be competent with the material presented in
this packet. You may use notes or other resources to help you complete this packet
of review material. You should not work together with other students nor
receive extensive help from a tutor.
Bring the completed packet with you on the first day of school,
This packet will be checked for completeness and conceptual comprehension. All
problems should be done or well attempted. Show work on every problem in the
space provided. Write neatly.
You will be assessed separately on the topics presented in this packet within the
first two weeks of school. You should use this packet as a study guide for the
assessment. Prior to the assessment you will be given an opportunity in class to
ask questions pertaining to the problems and concepts represented in this packet.
No formal instruction will be given on the topics presented in this packet. Again,
this is review material.
If you have significant trouble completing this packet you should contact your
guidance counselor to reconsider your course placement.
I hope you are looking forward to a fun and challenging year in AP Calculus AB!
See you in September!
Please contact me with any further questions.
Mr. Cassidy
jcassidy@wmrhsd.org
1. Given: f ( x)  x 2  3x  4 , find the following: Show all work.
a) f (1)
b) f ( x  2)
c)
f ( x  2)  f ( 2)
x
2. Sketch the graph of the equation.
3
y  2   x  1
2
d)
f ( x  h)  f ( x )
,h  0
h
3. Write an equation of the line that passes
through the point (2, 1) and is
Show all work.
a) parallel to 4x – 2y = 3
b) perpendicular to 4x – 2y = 3
4. A line is represented by the equation ax + by = 4.
a) When is the line parallel to the x-axis? b) When is the line parallel to the y-axis?
c) Give values for a and b such that the line has a slope of
5
8
.
d) Give values for a and b such that the line is perpendicular to y  2 5 x  3 .
e) Give values for a and b such that the line coincides with the graph of 5x + 6y = 8.
1
, g ( x)  x 2  1 , and h( x)  x  5 .
1 x
a) State the domain and range of each function.
5. Let f ( x) 
b) Find the following and state the domain of each.
i.
iv.
h 1 ( x)
 f  g (x)
ii.
g
 x 
h
 f  g (x)
iii. 
v. g (h( x))
6. Assume g(x) and h(x) are unknown functions. However it is known that g (0)  1 , g (1)  3 ,
g ( 2)  5 , g (7)  2 , h(1)  7 , h( 2)  1 , h(5)  0 . Evaluate:
a) ( g  h)( 2)
b) g ( g (h(1)))
c) h( g (h(5)))
d) g 1 (5)
e) ( g 1  h 1 )(1)
f) (h 1  g 1 )( 3)
7. If F ( x)  f  g  h , identify a set of possible functions for f, g, and h.
a) F ( x)  2 x  2
b) F ( x)  4 sin( 1  x)
8. The domain of function f is [-6, 6]. Complete the graph of f given that f is
a) even
b) odd
9. Algebraically determine whether the functions are even, odd, or neither. Show all work.
x2
x3  x
a) g ( x)  x 4  2 x  2
b) h( x)  2
c) g ( x)  2
2x  1
x 4
10. Consider the function f ( x)  x 3  2 x 2  x .
a) Describe the end behavior of the function. (Consider the leading coefficient test.)
b) Find the zeros of the function and their multiplicity.
c) Sketch a graph of the function without using your graphing calculator.
11. Sketch a possible graph of the situation.
a) The speed of an airplane as a function
of time during a 5-hour flight.
b) The value of a new car as a function of
time over a period of 8 years.
12. For the following rational functions, state the equations of the vertical, horizontal, or slant
asymptotes.
3x
2x 2  1
2 x 2  11x  15
a) f ( x)  2
b) f ( x)  4
c) f ( x) 
x  16
2x  3
x 4
13. Let log 10 P  x , log 10 Q  y , and log 10
 P
R  z . Express log 10 
3
 QR
14. Solve the equation: log 6 x  1  log 6 ( x  1)
2

 in terms of x, y, and z.

15. The mass m kg of a radio-active substance at time t hours is given by m  4e 0.2t . If the
mass is reduced to 1.5 kg., how long does it take?
16. The function f is given by f ( x)  e x11  8 . Find f 1 ( x) and its domain.
17. Use the given graph of f to sketch the graph of the transformations of f.
a) y   f (x)
b) y  f ( x)
c) y  2 f ( x)
c) y  
1
f ( x  3)
2
d) y  f ( x  2)  4
18. The graph of y = f(x) and 6 transformations (a, b, c, d, e, g) are given. Match each of the
transformation to one of the functions (i – vi) listed below.
i) y  f ( x  5) _____
ii) y  f ( x)  5 _____
iii) y   f ( x)  2 _____
iv) y   f ( x  4) _____
v) y  f ( x  6)  2 _____
vi) y  f ( x  1)  3 _____
19. Sketch  in standard position and find EXACT values for the 6 trig functions of  .
5
a)   495
b)    
6
20. Find the values of the other 5 trig functions under the given conditions.
6
sec   and tan   0
5
21. Find two degree angles 0    360
sec  2
22. Verify the given identities. Show ALL steps.
1
1

 2 sec 2 
a)
1  sin  1  sin 
b) 2 tan 2 x  2 tan 2 x csc 2 x  sin 2 x  cos 2 x  3
23. Solve the given equations on the interval 0,2  . Give the answers in radians. Show all
work.
a) sin 2 x   3  sin 2 x 
b) 3 sec 2 x  4
24. Evaluate each expression. Give the answer in radians. Reminder: The range for inverse trig
functions is restricted to the following intervals:
  
  

,
y = arcsinx
y
=
arccosx
y
=
arctanx


0
,

 , 
 2 2 
 2 2
  
y = arccscx
y = arcsecx
0,  , y≠  y = arccotx 0,  
 2 , 2  , y≠0
2
 2 3

a) arctan 3
b) arccos1
c) arc csc 
d) arc cot(1)
3 

 
e) arc sec(1)
f) arcsin( 1)
g) arc sec( 2)
 1
h) arcsin   
 2
i) arctan( 1)

3

j) arc cot  

3



2

k) arccos 

2


l) arc csc 2
 
25. Sketch a graph and use it to evaluate the limit. If the limit does not exist, state the reason
why.
x 3 , x  2
x
x
a) lim 
b) lim
c) lim
x4 x  4
x  x  4
x 2 5,
x2

d) lim
x 0
2
x2
e) lim cos
x 0
1
x
f) lim x  2
x 2
26. Use the cancellation method to evaluate the following limits. Show all work.
1
1

t3  8
a) lim
b) lim 5  x 5
x 0
t  2 t  2
x
27. Use the rationalization method to evaluate the following limits. Show all work.
x5 3
4  18  x
a) lim
b) lim
x4
x2
x4
x2
28. Find the following limits at infinity. If the limit does not exist, state the reason why.
4
2x2  6
4x4
a) lim
b) lim
c) lim 2
x 2 x  3
x    x  12
x  x  1
29. Find the value(s) of x for which f ( x) 
2x  6
x2  9
is discontinuous and label these
discontinuities as removable or non-removable.
30. Graph and solve the following quadratic inequalities:
a.) x 2  6 x  8  0
b.) x 2  4 x  60