Download Calculus Summer Review Packet

AP Calculus Summer Review Packet
TO: All 2011/2012 AP Calculus Students
FROM: AP Calculus Teacher
Ms. Kane—East Room 239, Ms. Radek-Carreon—West
We are pleased that you have chosen to complete your math sequence by enrolling in AP Calculus
for next year. To help ensure your success in AP Calculus next year, we have created a summer
review work packet. This packet contains material that you must have knowledge of on the 1 st day of
the course. The packet has questions from Algebra, Advanced Algebra, and Pre-Calculus.
This packet is to be completed in the following manner:

Work must be clearly enumerated and in consecutive order on separate paper.

Each problem with work counts toward the total grade.

Each problem must have WORK! NO WORK! NO CREDIT!

This packet is worth the equivalent of one test.

The packet with work is due on the first FULL day of school.

The packet is due whether or not you are physically present in school.
Resources you may want to use:
 Your Algebra, Advanced Algebra, or Pre-Calculus notes
 Calculus: Graphical, Numerical, Algebraic, Third Edition, Finney, Demana, Waits & Kennedy
 www.interactmath.com
 college.hmco.com/mathematics/larson/calculus_analytic/7e/students/index.html
 college.hmco.com/mathematics/resources/students/ace/college_algebra/ace.html
 mathforum.org
Below is a list of supplies you will need for AP Calculus. Shop for the items when they are on sale
and be prepared the first day of school. AP Calculus is a college-level course and you are expected
to be prepared with your materials each day.
AP Calculus Supplies Needed:
Pencils/Erasers
Binder
Paper
Graphing Calculator (TI-83, TI-83 Plus Silver Edition, TI-84, or TI-84 Plus)
Calculus Summer Review Packet
This packet is a review of information you learned in Algebra, Advanced Algebra & PreCalculus. You
need to know this information to be successful in AP Calculus. Therefore, this packet is due on your
FIRST DAY IN CALCULUS. It is to be completed CORRECTLY, NEATLY, and on a SEPARATE
sheet of paper.
Your Calculus teacher will collect your work on your FIRST DAY IN CALCULUS. Failure to turn in
your completed work on your FIRST DAY IN CALCULUS may jeopardize your ability to remain in the
course.
I. Simplify.
x6
1. 2
x  5x  6
5.
2x  6
2. 2
x  6x  9
cos3 x  cos2 x
1  sin 2 x
6.
tan 3 x
sec2 x  1
II. Trigonometric Identities
9. Pythagorean Identities (there are 3)
_________________________
_________________________
_________________________
11. sin 2x =
7.
sin 2 x
2 sin x cos x
8.
sec x
tan x
10. cos 2x = ? (there are 3)
_________________________
_________________________
_________________________
_________________________
III. Simplify each expression
3
2
2
1
1

12. x
13. x  4
14.
12
10
x 5 x
7
2
x
x  16
IV. Solve for a.
17. 15ab  3
4
3
22.
n
n 1
1
1

15. x  3 3
x
16.
19. c2  2ac  7a  5b  0
18. 7c  3ab  0
V. Expand and simplify
n2
21. 
n0 3
x 2  x  30
4.
x 2  25
4 x
3. 2
x  16
1
3
n
2
4x
 2
x  6x  9 x  2 x  3
2
20. 2 cos a  1, 0  a  2
VI. Simplify
x
23.
x
24.
eln8
F1I
28. log GJ
H5K
29. log 3 9
33. 82 / 3
34. (8a 2 /5 )(2a 5/ 2 )
5
25.
e( 2ln x )
F1 I
30. lnGJ
Ha K
26. ln 1
31. e
4ln x
5
27. ln e
4 xy 5
32.

1
4
10 x y 2
35. (4a 3/5 )5/ 2
VII. Using the point-slope form: y  y1  m( x  x1 ) , write an equation for the line:
36. with slope –4, containing the point (3,2)
37. containing the points (1,-4) and (-5,7)
38. parallel to 2x – 5y =7 and passing through (6,1) 39. perpendicular to the line 2x – 5y = 7 and passing
through (6,1)
VIII. Without using a calculator, determine the exact value of each expression.
40. sin 0
45. tan

2
41. cos

2
46. tan 
42. cos
47. tan
3
4

4
43. sin 
44. sin

6
1
48. sin(cos1 )
2
IX. Solve for x, where x is a real number. Show the work that leads to your solution.
49. x  6x  2  18
x4  4
50.
0
x3
51. x  3  16
53. ( x  4)( x  4)  0
54. x 2  2 x  35  0
55. sin 2 x  sin x, 0  x  
2
56. ( x  4) 2 ( x  3)  ( x  4)( x  3) 2  0
59. log x  log( x  9)  1
b g
2
57. 82 x  4 x 3
60. ln x  5
X. Miscellaneous: Follow the directions for each problem.
61. Find the x-intercept(s): 5x 2  7 y 2  3xy  20  0
62. Find all points of intersection of the graphs of x 2  x  y  6 and x  y  2 .
1
63. If the point ( 1, ) lies on the graph of the equation 2 x  ky  11, find k.
4
52. 2 x 2  5x  12
58. e5 x  3
64. Find the equation of the vertical line that passes through the point (5,4).
65. A business had annual sales of $120,000 in 2008 and $264,000 in 2011. Assuming that the annual increase
in sales followed by a linear pattern, what were the retail sales in 2010?
66. Given f ( x)  x  3  5 , find f (2)  f (7) .
67. Given f ( x)  x 2  3x  4 , find f ( x  1)  f (1) .
68. Find f ( x  h) for f ( x)  x 2  5x  6 .
69. Let f ( x) 
1  2 x,
R
S
T x ,
2
x 1
x 1
. Find f(4) and find f(-5).
70. Given f ( x )  4 x  3 and g( x)  x 2  9 , find the product of f ( x ) g ( x ) .
71. Given f ( x )  3x  6 and g( x)  x 2  4 , find
f ( x)
.
g( x)
72. Given f ( x)  2 x 2  5 and g( x)  x  3 , find ( f  g )( x ) .
73. Given f ( x )  3x  2 , find f 1 ( x ) .
XI. Graph each function, identify the x- & y-intercepts and identify the domain and range.
74. y  sin x
75. y  cos x
76. y  tan x
77. y  x 3  x 2  6x
78. y  x 2  6 x  8
79. y  e x
80. y  ln x
81. y  x
82. y  3 x
83. y  x  3  2
84. y 
R
x
|
x2
85. y  S
|T5
2
1
x
if x  0
if 0  x  4
if x  4
X. Use the drawing to the right to find the limit of each.
86. lim f ( x )
87. lim f ( x )
88. lim f ( x )
x 3
89. lim f ( x )
x 1
x 3
90. lim f ( x)
x 1
x 3
91. lim f ( x )
x 1