AP Calculus Mandatory Summer Practice
“It is not enough to have a good mind, the main thing is to use it well”-Descartes
Math is like exercise. The longer you go without practice, the more difficult it is to get back to
your original strength. There are certain skills that have been taught to you over the previous
years that are essential towards your success in AP Calculus. If you do not have these skills,
you will find that you will consistently get problems incorrect next year, even though you may
feel like you understand the Calculus concepts.
On a separate piece of paper, you must show all of your work, number your problems, and
justify your solutions when indicated. You may not use a calculator. If you are unsure of how
to approach a problem or would like to know if an answer is correct, please look online for help,
ask a peer, or send me an email. A great resource to use is www.khanacademy.org. Please take
these problems seriously. As stated before, students who are weak in these skills have an
extremely difficult time succeeding in Calculus without them.
I believe you will benefit the most from this packet by starting it as soon as the school year
ends. You should try to complete a few problems each day, as if it was a daily journal. Do not
do all of it now, and do not wait and do it a week before we start school in August. You are
more likely to retain the information if you spread it out.
All problems are due August 4, 2014 (first day of school). If for some reason you will be
absent on August 4th, you need to notify the front office and myself to arrange another date
before August 4th to turn in your assignment and to pick up materials that you will miss.
Absences and tardies are unacceptable in my class.
If you have any questions, please email me at LMarcelino@laalliance.org.
I look forward to working with you next year.
Mrs. Marcelino
Simplify as an expression with only positive exponents.
 2   2 
3. 2 

2
 2  x    2  x  
1/ 2
3
2. 5    4  9 x   9 
2
1. 3 x
x1/ 2
sin x
2
4. 16x 2 y 
5. 
1
3/ 2
 2 x  5
7. 2
3
2
 1
4
1 
8.  2  1 1  2 
x y
y 
x
3/ 4
4 x  16
6.
4
 x  4
3
1/ 2
Find the domain of the following functions. Use interval notation (ex: [0, 3)).
x2  4
2x  4
9. y  log  2 x 12
10. y 
13. y  x  3  x  3
2x  9
14. y 
2x  9
17. y  x  5 x  14
3x  2
18. y 
4x 1
2
11. y 
x2  5x  6
x 2  3x  18
x 2  8 x  12
15. y  4
x5
19. y 
3
22 x
x
12. y 
16. y  tan x
x6
20. y 
x  x  30
2
x
cos x
Factor completely.
21. x 5  11x 3  80 x
22.
 x  3  2 x  1   x  3  2 x  1
2
3
3
2
23. 2 x 2  50 y 2  20 xy
Solve the following inequalities.
24. x 2  16  0
25. x 2  6 x  16  0
26. x 2  3 x  10
27. 2 x 2  5 x  3
28. x 3  4 x 2  x  4
29. 2sin 2 x  sin x
Describe, in words, the transformations on f  x  in each of the following.
30. f  x   4
31. f  x  4
32.  f  x  2 
33. 5 f  x   3
34. f  2 x 
35. f  x 
Determine if each function is even, odd, or neither. Justify your solution.
36. f  x   2 x2  7
37. f  x   4x3  2x
38. f  x   4 x2  4 x  4
39. f  x   x 
1
x
Solve each equation.
40. 7 x 2  3x  0
41. 4 x  x  2  5x  x 1  2
42. x 2  6 x  4  0
43. 2 x 2  3 x  3  0
44. 2 x2   x  2 x  3  12
45. x 
46. x 4  9 x 2  8  0
47. x  10 x  9  0
48.
1 13

x 6
1 1
 6
x2 x
Find the equations of all vertical and horizontal asymptotes (if they exist).
49. y 
x
x 3
50. y 
x4
x2 1
51. y 
x4
x2  1
52. y 
x2  9
x3  3x 2  18 x
53. y 
2 x3
x3  1
54. y 
x
2 x  10
2
Simplify the following expressions.
55.
1
4
x
56.
1
2
x
x
x
1
2
1
x
57.
1
x
x
x
x2  y2
x
1 x

3
x x
xy
x
58.
59. 2
60. 1  x
1 x
x
x y
x 1

y
x
1 x
2
x
If f  x   x , g  x   2 x 1, and h  x   2 , find the values of the following.
61. f  g  2  
62. g  f  2  
63. f  h  1 

64. g 

  1 
f  h 
  2 
Solve each equation.
65.
2 5 1
 
3 6 x
66. x 
68.
2
1
16

 2
x  5 x  5 x  25
69.
6
5
x
60 60
2


x x 5 x
67.
x 1 x 1

1
3
2
70.
x5 3

x 1 5


Solve each equation on the interval 0, 2  . Give exact values  ex:  if possible.
3

71. sin x 
1
2
72. cos 2 x  cos x
73. 2cos x  3  0
74. 4sin 2 x  1
75. 2sin 2 x  sin x  1
76. cos 2 x  2 cos x  3
77. 2sin x cos x  sin x  0
78. 8cos 2 x  2 cos x  1
79. sin 2 x  cos 2 x  0
Answer the following questions over a variety of topics.
80. Let f be a linear function where f  2   5 and f  3  1 . Find f  x  .
81. Find an equation for the line, in point-slope form, that contains  5,1 and is perpendicular to 6 x  3 y  2 .
82. Use the table to calculate the average rate of change from t = 1 to t = 4.
t
x t 
0
8
1
7
2
5
3
1
4
2
83. Order the points A, B, and C, from least to greatest, by their rates of change.
84. Find the distance between the points 8, 1 and  4, 6 .
85. If g  x  
x
, find g 1  x  (the inverse of g).
x3
86. Find the points of intersection in the graphs of y  x  1 and y 2  2 x  6 .
87. Rewrite
1
ln  x  3  ln  x  2   6 ln x as a single logarithmic expression.
2
88. Evaluate the following.
 7 
a) sin 

 6 
b) csc  60
c) cos 120
 2 
d) sec  

 3 
 
e) tan  
2
f) cot  135
 x 2  5, x  1

x  1 .
89. Sketch a graph of the piecewise function f  x   0,
6  4 x, x  1

90. Describe the left and right end-behavior of the function f  x   3x .
91. Find the domain and range of each function (without a calculator).
a) f  x    x  3  2
2
b) f  x   2 x  4  3
c) f  x   3 1  x


e) f  x   tan  x  
f) f  x   e x
4


92. The circle below has a radius of 6 ft. Find the area and circumference of the circle, then find the length of
arc s.
d) f  x   5sin x
93. Find the area of the trapezoid.
94. Find the missing sides and angles of the triangle. Then find its area.
95. Find the volume of a washer with outer radius of 18 ft., inner radius of 15 ft., and height of 3 ft.
96. Rewrite log5  x  3 into an equivalent expression using only natural logarithms.
97. Three sides of a fence and an existing wall form a rectangular enclosure. The total length of fence used
for the three sides is 240 ft. Find x if the area enclosed is 5500 ft2.
98. The number of elk after t years in a state park is modeled by the function P  t  
1216
.
1  75e 0.03t
a) What was the initial population?
b) When will the number of elk be 750?
c) What is the maximum number of elk possible in the park?
99. Simplify the expression csc x  tan x sin x cos x .
100.Use long division, or synthetic division, to rewrite the expression
101.Rewrite y  3x 2  24 x  11 in vertex form
x3  7 x 2  14 x  8
.
x4
 x  k   h by completing the square.
2
 x 2 ,  2  x  1

x 1
102.Sketch a graph of the piecewise function f  x   2,
.
3 x  5, 1  x  3

103.Use a graphing calculator or www.wolframalpha.org to solve e2 x  3x 2 .
104.Do the lines  x  5 y  22 and 7 x  2 y  19 intersect? Justify your solution.
105.The function f  x  is graphed below. Find the following.
a) f  2  =?
106.Find the value of the following:
  3  x  6  x  
a) lim  3

x 
 x  2x  1 
b) f  0  =?
x2 1
b) lim
x  5 x  6
c) f  x   0 , x=?
5e x  1
c) lim x
x   e  5 x
107.Simplify the following expression: eln(5x)
108.Simplify the following expression: 5ln(ex)
109. Find the equation of the line with slope -1 that satisfies the initial condition f(1)=5.
110.Solve the equation for z in terms of x and y. y2+2xyz+z=0
111.Simplify the difference quotient of the function f(x)=x3-4x2-2. (Difference quotient =
112.Rewrite the following functions as piecewise functions.
a. f x   5x  6
b. g x   x 2  x  2
f x  h   f x 
)
h