Wheeler High School
To:
AP Calculus AB Students
From: Mrs. Barry, Math Department Chair
Re:
Summer work
AP Calculus AB is a college level course covering material traditionally taught in the first semester of
college calculus. These courses are taught in one semester consisting of daily 90-minute classes.
Students need a strong foundation to be ready for the rigorous work required throughout the term.
Completing the review packet, which consists of topics studied in pre-requisite courses, will help
ensure a proper background. Students should expect to work approximately 10 hours on this
assignment. The packet will be given a grade based on completeness and a quiz with representative
problems from the packet will be given during the first week of school. Students are expected to
show all work with logical steps. You must show your work for problems on separate paper. Do
not list only an answer.
The packets are due to your instructor on your first day of class, August 3, 2015.
Students enrolled in AP Calculus AB and BC will be using a graphing calculator throughout the course.
A TI 89 calculator will be available to AP Calculus students for use during the semester if they do not
already have one. If you have the opportunity to purchase one, it will be a helpful tool throughout
both courses and the AP Exam.
AP Calculus AB students will have the opportunity to continue studying Calculus for a second term
through the BC Calculus course. The BC course will cover material traditionally taught in the second
semester of college Calculus. The AB course is a prerequisite for BC Calculus. Students taking both
the AB and BC courses and passing the AP Calculus BC test could earn up to 6 hours of college credit
depending upon the Advanced Placement policy at the college. Some colleges also award credit for a
passing score on the AP Calculus AB exam. Since the AP test is only offered in May, students taking
only the AB course during the fall semester should be prepared to study for the test independently.
The success of each student in the AP Calculus program depends upon diligent effort, consistent
completion of homework assignments, and practice of newly learned skills. Although a suggested
assignment is given for each lesson, completion of some assignments may be optional.
Calculus is a challenging, stimulating, and dynamic field of mathematics and we look forward to
sharing our enthusiasm for the topics with you. If you have any questions or need to contact
someone over the summer, please feel free to e-mail me at Lynn.Barry@cobbk12.org .
Sincerely,
Mrs. Lynn Barry
Math Department Chair
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Calculus Summer Packet
Work the following problems on separate paper. Show all necessary work and write your final answer on the
answer sheet found at the end of the packet. Turn in your work and your answer sheet to your instructor on
the first day of class.
PART 1: ALGEBRA
Exponents.
 8 x yz   2x 
Simplify completely:
1
3
3
1)
4x
1
3
 yz 
2
3
1
3
Factor Completely.
2) 9x2 + 3x - 3xy - y
3) 64x 6 – 1
4) 42x 4 + 35x 2 - 28
5) 15x
Rationalize the denominator / numerator.
x
6)
1 x  2
7)
5
2
 2x
3
2
 24x
1
2
x 11
x
Simplify the rational expression.
8)
( x  1)3 ( x  2)  3( x  1)2
( x  1)4
Solve algebraic equations and inequalities. Use synthetic division to help factor the following, state all factors
and roots.
10) p( x )  x 4  x 3  2x  4
9) p(x) = 6x3 - 17x 2 - 16x + 7
3
cannot be a root of
2
4x5 + cx3 - dx + 5 = 0, where c and d are
integers.
11) Explain why
12) ) Explain why x4 + 7x2 + x - 5 = 0 must
have a root in the interval [0, 1], ( 0 ≤ x ≤ 1)
Solve.
2
13) (x + 3) > 4
16)
x2  9
 0
x 1
Solve the system.
x  y  1 0
19)
y  x 2  5
14)
x 5
 0
x 3
15) x 3  2x 2  3x  0
18) 5 x  1  9
17) 0  x  2  8
20)
x 2  4x  3  y
x 2  6x  9  y
3
PART 2: GRAPHING AND FUNCTIONS
Linear Graphs - Write the equation of the line described below.
21) Passes through the point
(2, -1) and has slope -1/3.
22) Passes through the point
(4, -3) and is perpendicular to
3x  2y  4 .
23) Passes through (-1, -2) and
3
Is parallel to y  x  1.
5
Conic Sections - Write the equation in standard form and identify the conic.
24) x = 4y 2 + 8y - 3
25) x 2  y 2  4x  2y  4  0
26) 4x 2  16x  3y 2  24y  52  0
27) x 2  4y 2  2x  24y  19  0
Functions - Find the domain and range of the following.
domain restrictions - denominator ≠ 0, argument of a log or ln > 0, radicand of even index must be ≥ 0
range restrictions- reasoning, if all else fails, use graphing calculator
3
x 2
28) y =
29) y = log(x - 3)
2x  3
30) y = x 4 + x2 + 2
30) y =
32) y = | x - 5 |
33) Given f(x) below, graph over the domain
[ -3, 3]. What is the range?
if x  0
 x

f (x)   1
if -1  x < 0
 x  2 if x < -1

Compositions and Inverses - Find the compositions and inverses as indicated below.
Let:
34) g -1(x)
f(x) = x2 + 3x - 2
g(x) = 4x - 3
35) h -1(x)
36) f(g(x))
h(x) = lnx
w(x) =
37) h(g(f(1)))
x4
38) Does y = 3x2 9 have an inverse
function? Explain
your answer.
Basic Shapes of Curves:
Sketch the graphs on the answer sheet. You may use your graphing calculator to verify the graph,
but you should be able to graph the following by knowledge of the shape of the curve, by plotting a
few points, and by your knowledge of transformations.
39) y =
x
40) y = lnx
41) y = | x - 2 |
4
42) y =
1
x
1
x 2
46) y  3 x
43) y =
45) y  e x
44) y =
x
x 4
2
47)
 25  x 2

 x 2  25
f (x)  
x 5

0


if x  0
if x  0, x  5
if x  5
Asymptotes – Identify any horizontal, vertical, or slant asymptotes.
48) y 
1
x 1
49) y 
3x 2
2x  3 x  3
2
50) y 
5x 2  5x  1
x 1
51) y 
2x 2
3x  4x  1
3
Even and Odd Functions - Identify as odd, even, or neither. Show substitutions!
Even: f (x) = f (-x)
Odd: f (-x) = - f (x)
52) f(x) = x3 + 3x
54) f(x) = sin 2x
53) f(x) = x 4 - 6x2 + 3
Test for symmetry. Show substitutions.
Symmetric to y axis: replace x with - x and relation remains the same.
Symmetric to x axis: replace y with - y and relation remains the same.
Origin symmetry: replace x with - x, y with - y and the relation is equivalent.
55) y  x 4  x 2
56) y = sin(x)
57) y = cos(x)
58) y 
x
x5  x3
59) y 
x
x5  x3
PART 3: LOGARITHMIC AND EXPONENTIAL FUNCTIONS
Simplify Expressions.
1
60) log4  16

64) logw w 45
61) 3log3 3 65) ln e
3
4
log3 81 
1
1
log3  27

3
62) log9 27
66) ln 1
63) log125  51 
67) ln e2
Solve equations.
68) log6(x + 3) + log6(x + 4) = 1
69) log x2 - log 100 = log 1
70) 3 x+1 = 15
5
PART 4: TRIGONOMETRY
Unit Circle – find the following without using a calculator.
 
71) sec   
 6
 9 
72) tan 

 4 
 11 
73) cos 

 3 
 11 
74) sin 

 4 
75) cot 8
 5 
76) tan 

 2 
 5 
77) csc 

 6 
 7 
78) sin 

 3 
State the domain, range, and fundamental period for each function.
79) y = sin x
80) y = cos x
81) y = tan x
83) 1 - cos2 x
84) sec2x - tan2 x
Identities – simplify.
82)
(tan 2 x )(csc 2 x )  1
(cscx )(tan 2 x )(sinx )
Solve equations.
85) cos2x = cos x + 2
87) 4cos2 x  1
0 ≤ x ≤ 2π
x 
86) 2 sin(2x) =
0 ≤ x ≤ 2π
3
88) cos2x + sinx + 1 = 0


2
x
3
2
Graphing – State the amplitude and period of the following functions and graph.
89) y  2sin(2x )
90) y  1/ 4cos(4x  12)
PART 5: GEOMETRY
Triangle Trig - Find the value of x.
91)
50
x
10
92)
X
70o
70o
10
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CALCULUS SUMMER PACKET ANSWER SHEET
NAME ________________________________
1.
2.
3.
4.
5.
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