Massey Hill Classical High School
AB Calculus AP Prerequisite Packet
To:
From:
AP Calculus AB Students and Parents
Carolyn Davis, AP Calculus Instructor
The AP Course: AP Calculus AB is a college level course covering material traditionally
taught in the first semester of college calculus. The course is taught in one semester
consisting of 90 -minute classes.
The Prerequisite Packet: Students need a strong foundation and willingness to be ready for
the rigorous work required throughout the term. Completing the prerequisite packet should
prepare you for the material to be taught in the course. This packet consists of material
studied during Algebra II and Pre- Calculus Honors. Students should anticipate working
approximately 10 hours to complete it properly.
The packet will be collected on the third day of class and your grade will be determined on
neatness, completeness of solutions, and accuracy. In preparation for the AP test students
need to begin showing all work with logical steps. Do not list only an answer. Work neatly
and in an organized fashion.
Calculators: Students enrolled in AP Calculus AB will be using a graphing calculator
throughout the course. A graphing calculator is required on the AP test. We will be using a
TI 84 Plus and TI 89 in class;, I recommend a TI 84 Plus Silver Edition and/or the TI-89.
If you have a TI-83 calculator some of our needed programs can be downloaded onto your
calculator.
Since the AP test is offered only in May, students taking only the AB course should study for
the test independently and attend any after school review sessions or practice test sessions
that are offered. It is the student’s responsibility to find out when these sessions will be
held.
I anticipate a motivating and challenging year. Calculus is a stimulating and exciting field of
mathematics and I look forward to sharing the excitement with you. I will be there to help
and support you.
SKILLS NEEDED FOR CALCULUS
I. Algebra:
*A. Exponents (operations with integer, fractional, and negative exponents)
*B. Factoring (GCF, trinomials, difference of squares and cubes, sum of
cubes, grouping)
C. Rationalizing (numerator and denominator)
*D. Simplifying rational expressions
*E. Solving algebraic equations and inequalities (linear, quadratic, higher order
using synthetic division, rational, radical, and absolute value equations)
F. Simultaneous equations
II. Graphing and Functions
*A. Lines (intercepts, slopes, write equations using point-slope and slope
intercept, parallel, perpendicular, distance and midpoint formulas)
B. Conic Sections (circle, parabola, ellipse, and hyperbola)
*C. Functions (definition, notation, domain, range, inverse, composition)
*D. Basic shapes and transformations of the following functions (absolute value,
rational, root, higher order curves, log, ln, exponential, trigonometric.
piece-wise, inverse functions)
E. Tests for symmetry: odd, even
III. Geometry
A. Pythagorean Theorem
B. Area Formulas (Circle, polygons, surface area of solids)
C. Volume formulas
D. Similar Triangles
* IV. Logarithmic and Exponential Functions
*A. Simplify Expressions (Use laws of logarithms and exponents)
*B. Solve exponential and logarithmic equations (include ln as well as log)
*C. Sketch graphs
*D. Inverses
* V. Trigonometry
**A. Unit Circle (definition of functions, angles in radians and degrees)
B. Use of Pythagorean Identities and formulas to simplify expressions
and prove identities
*C. Solve equations
*D. Inverse Trigonometric functions
E. Right triangle trigonometry
*F. Graphs
VI. Limits
A. Concept of a limit
B. Find limits as x approaches a number and as x approaches ∞
* A solid working foundation in these areas is very important.
Calculus Prerequisite Problems
Work the following problems on your own paper. Show all necessary work.
I. Algebra
1
3
( 8 x yz ) ( 2x )
3
A. Exponents:
1)
4x
1
3
( )
yz
2
3
−1
3
B. Factor Completely:
2) 9x2 + 3x - 3xy - y
4) 42x 4 + 35x 2 - 28
6) x -1 -3x -2 + 2x -3
3) 64x 6 - 1
5) 15 x
5
2
− 2x
3
2
− 24 x
1
2
Hint: Factor GCF x 1/2 first.
Hint: Factor out GCF x-3 first.
C. Rationalize denominator / numerator:
7)
3− x
1− x − 2
8)
x+1 + 1
x
D. Simplify the rational expression:
(x + 1)3 (x - 2) + 3(x + 1 )2
9)
(x + 1)4
E. Solve algebraic equations and inequalities
10. – 11. Use synthetic division to help factor the following, state all factors and roots.
10) p(x) = x3 + 4x 2 + x - 6
11) p(x) = 6x3 - 17x 2 - 16x + 7
3
12) Explain why 2 cannot be a root of f(x) = 4x5 + cx3 - dx + 5 , where c and d are integers.
(hint: You can look at the possible rational roots.)
Solve: You may use your graphing calculator to check solutions.
13) (x + 3)2 > 4
16)
1
x < x
14)
x+5
≤ 0
x-3
15) 3 x3 − 14 x 2 − 5 x ≤ 0 (Factor first)
17)
x2 - 9
x+1
18)
≥ 0
1
4
x-1 + x-6
> 0
1
21) | 2x + 1 | < 4
F. Solve the system. Solve the system algebraically and then check the solution by
graphing each function and using your calculator to find the points of intersection.
20) x 2 < 4
23) x 2 - 4x + 3 = y
- x 2 + 6x - 9 = y
22) x - y + 1 = 0
y–x2= -5
II. Graphing and Functions:
A. Linear graphs: Write the equation of the line described below.
1
24) Passes through the point (2, -1) and has slope - 3 .
25) Passes through the point (4, - 3) and is perpendicular to 3x + 2y = 4.
3
26) Passes through ( -1, - 2) and is parallel to y = 5 x - 1.
B. Conic Sections: Write the equation in standard form and identify the conic.
27) x = 4y 2 + 8y - 3
28) 4x 2 - 16x + 3y 2 + 24y + 52 = 0
C. Functions: Find the domain and range of the following.
Note: 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
29) y = x - 2
32) y =
2x - 3
30) y = log(x - 3)
31) y = x 4 + x2 + 2
y= |x-5|
34) domain only:
33)
y=
x +1
x −1
35) 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

Find the composition /inverses as indicated below.
Let
f(x) = x2 + 3x - 2
g(x) = 4x - 3
h(x) = lnx
w(x) =
x-4
2
36) g -1(x)
37) h -1(x)
38) w -1(x), for x ≥ 4
39) f(g(x))
40) h(g(f(1)))
41) Does y = 3x2 - 9 have an inverse function? Explain your answer.
Let f(x) = 2x, g(x) = -x, and h(x) = 4, find
42) (f o g)(x)
44) Let
43)
s (x) =
4-x
(f o g o h)(x)
t(x) = x2, find the domain and range of (s o t )(x).
and
D. Basic Shapes of Curves:
Sketch the graphs. You may use your graphing calculator to verify your 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.
45) y =
x
1
47) y = x
46) y = lnx
1
49) y = x - 2
50) y =
 25 − x 2

 x 2 − 25
53) f ( x ) = 
x −5


0

x
-4
x2
48) y = | x - 2 |
J
52) y = 3 sin 2 (x - 6 )
51) y = 2-x
if x < 0
if x ≥ 0, x ≠ 5
if x = 5
E. Even, Odd, Tests for Symmetry:
Identify as odd, even , or neither and justify you’re answer. To justify your answer you
must show substitution using -x ! It is not enough to simply check a number.
Even:
54) f(x) = x3 + 3x
57) f(x) = sin 2x
60) f(x) =
1 + |x|
x2
f (x) = f (-x)
Odd: f (-x) = - f (x)
55) f(x) = x 4 - 6x2 + 3
58) f(x) = x2 + x
56) f ( x ) =
x3 − x
x2
59) f(x) = x(x2 - 1)
61) What type of function (even or odd) results from the product of two
even functions?
odd functions?
IV LOGARITHMIC AND EXPONENTIAL FUNCTIONS
A. Simplify Expressions:
1
67) log 4 ( 16
)
68) 3 log 3 3 -
70) log125 ( 51 )
3
4
log 3 81 +
71) logw w 45
1
1
log 3 ( 27
)
3
72) ln e
69) log 9 27
74) ln e 2
73) ln 1
B. Solve equations:
75) log6(x + 3) + log6(x + 4) = 1
76) log x2 - log 100 = log 1
77) 3 x+1 = 15
V TRIGONOMETRY
A. Unit Circle:
quiz.
Know the unit circle – radian and degree measure. Be prepared for a
78) State the domain, range and fundamental period for each function?
a) y = sin x
b) y = cos x
c) y = tan x
B. Identities:
Simplify:
79)
(tan2x)(csc2x) - 1
(cscx)(tan2x)(sinx)
80) 1 - cos2 x
81) sec2x - tan2 x
82) Verify : (1 - sin2x)(1 + tan2x) = 1
C.
Solve the Equations
83) cos2x = cos x + 2,
0 ≤ x ≤ 2J
84) 2 sin(2x) =
3,
0 ≤ x ≤ 2J
85) cos2x + sinx + 1 = 0,
D. Inverse Trig Functions:
Note: Sin -1 x = Arcsin x

2
87) Arcsin  −

 2 
86) Arcsin1
90.
0 ≤ x ≤ 2π

 3 
89) sin  Arccos 
 

2



 3
88) Arccos 

 2 
State domain and range for: Arcsin(x) , Arccos (x), Arctan (x)
E.. Graphs:
functions.
Identify the amplitude, period, horizontal, and vertical shifts of these
(
92) y = −π cos π2 x + π
91) y = - 2sin(2x)
)
F. Be able to do the following on your graphing calculator:
Be familiar with the CALC
commands; value, root, minimum, maximum, intersect. You
may need to zoom in on areas of your graph to find the information.
Answers should be accurate to 3 decimal places. Sketch graph.
93. – 95. Given the following function f(x) =
93. Find all roots.
2x 4 - 11x 3 - x 2 + 30x.
Note: Window
x min: -10 x max: 10 scale 1
y min: - 100 y max: 60 scale 0
94. Find all local maxima.
95. Find all local minima.
96. Graph the following two functions and find their points of intersection using the
intersect command on your calculator.
y = x3 + 5x2 - 7x + 2 and y = .2x2 + 10
VI. Functions and Models
Window: x min : -10
y min: - 10
x max: 10 scale 1
y- max: 50 scale 0
97. The graphs of f and g are given.
(a) State the values of f(-4) and g(3).
(b) For what values of x if f(x)=g(x)?
(c) Estimate the solution of the equation
block =-1.
(d) On what interval is f decreasing?
(e) State the domain and range of f.
(f) State the domain and range of g.
f(x) each
98. Find the domain of each function.
x
a)
f ( x) =
3x − 1
b) g (u ) = u + 4 − u
99. Biologists have noticed that the chirping rate of crickets of a certain species is
related to temperature, and the relationship appears to be very nearly linear. A
cricket produces 113 chirps per minute at 70° F and 173 chirps per minute at 80° F.
(a) Find a linear equation that models the temperature T as a function of the
number of chirps per minute N.
(b) What is the slope of the graph? What does it represent?
(c) If the crickets are chirping at 150 chirps per minute, estimate the temperature.
100. Classify each function as a power function, root function, polynomial (state its
degree), rational function, algebraic function, or logarithmic function.
(a) f ( x) = 5 x
(b) g ( x) = 1 − x 2
(c ) h ( x ) = x 9 + x 4
101. Find f+g, f – g, fg, and f/g and state their domains.
f ( x) = x3 + 2 x 2 ,
g ( x) = 3 x 2 − 1
(d ) r ( x) =
x2 + 1
x3 + x
102. Find the functions f o g ,
f ( x) = sin x,
go f,
f o f , and
g o g and their domains.
g ( x) = 1 − x
103 and 104, Determine an appropriate viewing rectangle for the given function
and use it to draw each graph.
103. f ( x) = 5 + 20 x − x 2
104. f ( x) = 4 81 − x 4
105 . Find all solutions of the equation correct to three decimal places.
x3 − 9 x 2 − 4 = 0
106. Under ideal conditions a certain bacteria population is known to double every
three hours. Suppose that there are initially 100 bacteria.
(a) What is the size of the population after 15 hours?
(b) What is the size of the population after t hours.
(c) Estimate the size of the population after 20 hours.
(d) Graph the population function and estimate the time for the population to reach
50,000.
For #107- 109 , find a formula for the inverse of the function.
107. f ( x) = 10 − 3 x
108. f ( x) = e x
3
109. y = ln(x+3)
For #135-136, find the exact value of each expression (non-calculator).
110. (a) log 2 64
(b) log 6
111. (a) log10 1.25 + log10 80
1
36
(b) log 5 10 + log 5 20 − 3log 5 2
112. Express the given quantity as a single logarithm.
2 ln 4 – ln 2
113. The graph shown gives a salesman’s distance from his home as a function of
time on a certain day. Describe in words what the graph indicates about his travels
on this day.