AP Calculus AB
Summer Assignment 2012
Name: __________________________________________
Directions: 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 full day of school.
This packet will be checked for completeness. All problems should be done or
well attempted. Show work on every problem in the space provided. Write neatly.
You will be assessed on the topics presented in this packet at the beginning of the
school year. We will review this packet and you will be given an opportunity to
ask questions in class in the days prior to the assessment. If you have significant
trouble completing this packet you should contact your guidance counselor to
reconsider your course placement.
If you have not yet learned limits, or need additional review of this topic, please
refer to the following instructional videos from KhanAcademy:
http://www.khanacademy.org/math/calculus/v/introduction-to-limits
http://www.khanacademy.org/math/calculus/v/limit-examples--part-1
http://www.khanacademy.org/math/calculus/v/limit-examples--part-2
http://www.khanacademy.org/math/calculus/v/limit-examples--part3
http://www.khanacademy.org/math/calculus/v/limit-examples-w--brain-malfunction-on-first-prob--part-4
http://www.khanacademy.org/math/calculus/v/more-limits
Please contact me with any further questions.
(salvesen@wmchs.org)
1. Given: f ( x)  x 2  3x  4 , find the following:
a) f (1)
c)
f ( x  2)  f ( 2)
x
2. Sketch the graph of the equation.
3
y  2   x  1
2
b) f ( x  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
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. Let 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.
x2
x3  x
4
a) g ( x)  x  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. Write the function without using absolute value signs. (Hint: Make a piece-wise function.)
f ( x)  x  x  2
12. 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.
13. For the following rational functions, state the equations of the vertical, horizontal, or slant
asymptotes.
3x
2x 2  1
2 x 2  11x  15
f
(
x
)

a) f ( x)  2
b)
c) f ( x) 
x 4  16
2x  3
x 4
14. Let log 10 P  x , log 10 Q  y , and log 10
 P
R  z . Express log 10 
3
 QR
15. Solve the equation: log 6 x  1  log 6 ( x  1)
2

 in terms of x, y, and z.

16. 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?
17. The function f is given by f ( x)  e x11  8 . Find f 1 ( x) and its domain.
18. 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
19. Use the graph of y = f(x) to match the function with its graph.
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
20. Sketch  in standard position and find EXACT values for the 6 trig functions of  .
5
a)   495
b)    
6
21. Find the values of the other 5 trig functions under the given conditions.
6
sec   and tan   0
5
22. Find two degree angles 0    360
sec  2
23. Find two radian angles 0    2
cos  0.4667
24. 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
25. Solve the given equations on the interval 0,2  . Give the answers in radians.
a) sin 2 x   3  sin 2 x 
b) 3 sec 2 x  4
26. Solve the given equations. Find all solutions. Give the answers in radians.
a) cot 2 x  3 csc x  3  0
b) 2 cot 2 x  6  0
27. Solve the equation 2 cos 2 ( x)  sin( 2 x) for 0  x   giving your answers in terms of π.
28. 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
 
29. Complete the table of values for each function. Determine if the limit exits based on your
table of values. If the limit does exist, give its value. If the limit does not exist, state the reason
why.
x3
 x  1, x  2
a) lim
b) lim 
x 2 2 x  3,
x 3 x  3
x2

x
f(x)
2.9
2.99
2.999 3.0001 3.001 3.01
x
f(x)
1.9
1.99
1.999 2.001
2.01
30. Use the direct substitution method to evaluate the following limits. If the limit does not
exist, state the reason why.
x
x
tan x
a) lim
b) lim
c) lim arccos
1
x e ln x
x
1
2
2
x
4
31. 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
32. If lim f ( x)  2 and lim g ( x) 
x c
x c
1
x
 f ( x)  2 g ( x) 
1
, find lim 
.
x c
2
3 f ( x)


f) lim x  2
x 2
2.1
33. Use the cancellation method to evaluate the following limits.
1
1

3
t 8
a) lim
b) lim 5  x 5
x 0
t  2 t  2
x
34. Use the rationalization method to evaluate the following limits.
x5 3
4  18  x
a) lim
b) lim
x4
x

2
x4
x2
35. Graph the function and evaluate each limit (if it exists). f ( x) 
a) lim
x 1
x 1
x 1
 x 2  1,


2 x  3,
36. Let f ( x)  
b) lim
x 1
x 1
c) lim
x 1
x 1
x 1
x 1
x 1
x 1
x0
. Find each limit (if it exists).
x0
a) lim f ( x)
x0
b) lim f ( x)
x0
37. 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
x 2 x  3
x    x  12
x  x 2  1
38. Find the value(s) of x for which f ( x) 
2x  6
x2  9
discontinuities as removable or non-removable.
is discontinuous and label these