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Course: AP Calculus AB
Teacher Name/s: Veronica Moldoveanu, Ethan Batterman
Assignment Title: AP Calculus AB Summer Packet
Assignment Summary/Purpose:
The material in this packet should be a review of past material that will be necessary in the first few weeks of
the Calculus program. Completion of this packet will set each student up for continued success in the AP
Calculus program.
Due date (must be completed before school starts)
Assigned during: Summer break
Due Date: First day of class
Estimated time needed to complete the assignment: Approximately _6-8__ hrs. (not to exceed 8 hrs.)
Description of how the assignment will be assessed:
The material from this packet will be assessed through quizzes given during the first month of school. No
quizzes will be given during the first week of classes. Students will have an opportunity to get help on the
packet before the quiz.
Grade impact to overall course grade:
Content will be integrated into the coursework.
Tools/resources needed to complete the assignment:
TI-83, 84, or 89 graphing calculator
Contacts:
Name E-mail
Veronica Moldoveanu
Ethan Batterman
Summer AP Assignment
Coversheet
Falls Church High School
Falls Church High School
Summer Review Packet For students entering AP Calculus AB
Name: ________________________________
1. This packet is to be handed in to your Calculus teacher on the first day of the school year.
2. All work must be shown in the packet OR on separate paper attached to the packet.
3. Completion of this packet will help you to be prepared for the topics that will begin the
year in AP Calculus.
4. If you have questions over the summer, you may email Mrs. Moldoveanu at
[email protected] or Mr. Batterman at [email protected]
SHORT ANSWER SECTION
Objectives: For objectives #1 - #5 you should be able to do all without a calculator
1. Identify functions as even or odd Algebraically Graphically
2. Know key points and basic shapes of essential graphs
2 3, ,f x x f x x f x x
xf x e
lnf x x
sin , cosf x x f x x
3. Review Trigonometry Evaluate trig values Evaluate inverse trig values Solve trig equations
4. Understand characteristics of rational expressions (be able to sketch without a calculator) Vertical asymptotes Horizontal asymptotes Zeros Holes
5. Be able to use properties of natural logarithms to solve equations
6. Know your calculator
I. Symmetry – Even and Odd Functions
Quick Review
Even Function
Symmetric about the y axis
( ) ( )f x f x for all x
Example: 2y x
Odd Function
Symmetric about the origin (equivalent to a rotation of 180
degrees)
( ) ( )f x f x for all x
Example: 3y x
To determine algebraically if a function is even, odd, or neither, find f x and determine if it is equal to
f x , f x , or neither.
Example: Determine if 2
4
1
xf x
x
is even or odd.
2 2 2
4 4 4
1 11
x x xf x f x
x xx
Therefore, f x is an odd function.
Determine if the following functions are even, odd, or neither.
1. 2
4 3
xf x
x
2.
1
xf x
x
3. 2 41 3 3f x x x 4. 3 51 3 3f x x x
II. Essential Graphs For each graph, show two key points (label coordinates) and basic shape of the graph.
1. f x x
2. 3f x x
3. sinf x x
4. xf x e
5. lnf x x
6. cosf x x
III. Trigonometry Review On each right triangle with a hypotenuse of 1, label the lengths of the other sides: 45° = _____ radians 90° = _____ radians 30° = _____ radians 60° = _____ radians
In calculus we always use radians! Never degrees.
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
-2π -π π 2π
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
-2π -π π 2π
1. Evaluate the sine, cosine, and tangent of each angle without using a calculator. It is not necessary (ever
again!) to rationalize the denominator. (ie, an answer of 1
2 does not need to be written
2
2)
a) 4
b)
5
4
c)
6
d) 2
e)
5
3
f)
11
6
2. Find two solutions to the inverse trig equations. 0 2 .
1. Draw a triangle, if necessary 2. Label θ and sides 3. Determine reference angle
4. Notice restrictions 5. Place in correct quadrant
a) 2
cos2
b) 2
cos2
c) sec 2
d) tan 1 e) cot 3 f) 3
sin2
3. Solve for y . Notice the only difference between #2 and #3 is the domain restrictions #3.
a) 1 1sin
2y
b) 1 1
cos2
y
c) 1tan 3y
d) 1csc 2y e) 1cos 0y f) 1cot 1y
IV. Rational Functions
Rational functions are ratios of polynomials:
f xh x
g x
h x has a zero when 0h x (which occurs when 0f x and the factor does not cancel)
Ex. 2
2
2
1
x xh x
x
2 1
1 1
x xh x
x x
2 1
1 1
x xh x
x x
Therefore, 0h x when 2x .
h x has a vertical asymptote when 0g x and the factor that causes 0g x does not cancel
Ex. 2
2
2
1
x xh x
x
2 1
1 1
x xh x
x x
2 1
1 1
x xh x
x x
Therefore, h x has a vertical asymptote when 1x .
h x has a hole (is undefined but the limit exists) when 0g x and the factor that causes 0g x cancels
from both f x and g x .
Ex. 2
2
2
1
x xh x
x
2 1
1 1
x xh x
x x
2 1
1 1
x xh x
x x
Therefore, h x has a hole when 1x .
Note that
2
1
xh x
x
because these two functions do not have the same domain.
h x has a horizontal asymptote at y a when limx
h x a
or limx
h x a
. To determine limx
h x
consider first the largest exponent of f x and g x . If f x has the larger exponent, then limx
h x
(DNE). If g x has the larger exponent, then lim 0.x
h x
If the exponents are the same, consider the
leading coefficient.
Ex. 2
2
2
1
x xh x
x
Leading coefficients
1
1
Therefore, lim 1x
h x
and h x has a horizontal asymptote at 1.y
Once the basic characteristics of rational expressions are determined, the functions can be sketched without a calculator:
Ex. 2
2
2
1
x xh x
x
Zero at 2x . Vertical Asymptote: 1x Hole when 1x Horizontal Asymptote: 1y
Graph points as needed until you see the shape.
For each rational function below: a) Find all zeros
b) Write the equations of all vertical asymptotes c) Write the equations of all horizontal asymptotes d) Find the x value of any holes e) Sketch the graph (no calculator) showing all characteristics listed (Not all functions will have all characteristics listed above)
1. 2
1
xf x
x
2. 2 3
1
xf x
x
3. 2
1f x
x
4. 2
1xf x
x
5. 2
2
3 4
6
x xf x
x x
6. 2 4
xf x
x
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
-4 -2 2 4
4
2
-2
-4
V. Properties of Natural Logarithms
Recall that lny x and xy e (exponential function) are inverse to each other
Properties of the Natural Log:
ln ln lnAB A B Ex: ln 2 ln 5 ln 10
ln ln lnA
A BB
Ex:
6ln 6 ln 2 ln ln 3
2
ln lnpA p A Ex: 4ln 4lnx x and 33ln 2 ln 2 ln 8
ln xe x , ln 1e , ln 1 0 , 0 1e
Use the properties of natural logs to solve for x. Ex:
2 5 11 7
5 11
7 2
5 11ln ln
7 2
ln 5 ln 7 ln 11 ln 2
ln 5 ln 7 ln 11 ln 2
ln 5 ln 7 ln 11 ln 2
ln 11 ln 2
ln 5 ln 7
x x
x
x
x
x
x x
x x
x
x
1. 10 4x
2. 3 52 4x xe e 3. 3 710 5x xe
VI Calculator Skills No matter what type of calculator you own, you should be able to do the following problems. Use the index of the owner’s manual of your calculator to look up instructions or ask a classmate how to do each. Evaluations:
1) Fractions and more. Be careful to use enough parentheses.
a)
6 .379
3 2 b) 2
2) Logarithms and Exponents
a) ln 2 b) log 2 c) ln (3e) d) 45 e) 3e
f) 3 10 f) 218 g) 2
18 h) 2 .01343
3) Trigonometry Evaluations. Make sure you watch the mode – degree versus radians.
a) osin23 b)
sin5
c)
sec7
d) 1tan 1.5 in radians & degrees
Graphing Skill #1: You should be able to graph a function in a viewing window that shows the important features. You should be familiar with the built-in zoom options for setting the window such as zoom-decimal and zoom-standard. You should also be able to set the window conditions to values you choose.
1. Graph 2 3y x using the built in zoom-decimal and zoom-standard options. Draw each.
2. Find the appropriate viewing window to see the intercepts and the vertex defined by 22 14 4y x x .
Use the window editor to enter the x and y values. Window: Xmin = _______
Xmax = _______
Xscl = _______
Ymin = _______
Ymax = _______
Yscl = _______ 3. Find the appropriate viewing windows for the following functions:
1
104
y x 5y x 100 1.06x
y
1
10y
x
Xmin = _______ Xmin = _______ Xmin = _______ Xmin = _______
Xmax = _______ Xmax = _______ Xmax = _______ Xmax = _______
Xscl = _______ Xscl = _______ Xscl = _______ Xscl = _______
Ymin = _______ Ymin = _______ Ymin = _______ Ymin = _______
Ymax = _______ Ymax = _______ Ymax = _______ Ymax = _______
Yscl = _______ Yscl = _______ Yscl = _______ Yscl = _______
Graphing Skill #2: You should be able to graph a function in a viewing window that shows the x-intercepts (also called roots and zeros). You should be able to accurately estimate the x-intercepts to 3 decimal places. Use the built-in root or zero command.
1. Find the x-intercepts of 2 1y x x . Window [-4.7, 4.7] x [-3.1, 3.1]
(Write intercepts as points)
x-intercepts: ______________
2. Find the x-intercepts of 3 2 1y x x .
x-intercepts: ______________
Graphing Skill #3: You should be able to graph two functions in a viewing window that shows the intersection points. Sometimes it is impossible to see all points of intersection in the same viewing window. You should be able to accurately estimate the coordinates of the intersection points to 3 decimal places. Use the built-in intersection command. 1. Find the coordinates of the intersection points for the functions:
2( ) 3 ( ) 7f x x g x x x .
Intersection points: ______________
2. Find the coordinates of the intersection points of: 2( ) 4 ( ) 2xf x x g x .
Intersection points: ______________
Graphing Skill #4: You should be able to graph a function and estimate the local maximum or minimum values to 3 decimal places. Use the built-in max/min command.
1. Find the maximum and minimum values of the function 3 4 1y x x .
(Value means the y-value)
Minimum value: ______________
Maximum value: ______________
2. Find the maximum and minimum values of the function 3 24 4y x x x .
REVIEW: Do the following problems on your calculator.
1. 3
4 8.556
2 9
2. 12ln 4 1.32e
3. sin 48
4) 1cos .75 in radians and degrees
5) 5
tan6
6) Find the x-intercepts, relative maximum, and relative minimum of 3 22 1y x x
7) Find the coordinates of the intersection points for the functions 2( ) 2 9f x x x and
3( ) 3
4g x x
Multiple Choice Section
Directions: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice.
Show all work on this packet. If no work is required, explain how you arrived at your answer. Follow calculator instructions as given in each section. * A choice of “none” is short for “none of these”. A choice of DNE means “does not exist”. These problems are due the first class. You will be placed in a group to discuss the answers and questions will be answered during class and/or after school. Do not expect there to be time in class for ALL questions to be resolved. There will be a short quiz on the material during the 3rd class.
NO CALCULATOR Functions
1) Given 2( ) 2f x x x and ( ) 2 3g x x , find ( ( ))f g x .
a) 24 8 3x x b) 22 4 3x x c) 3 22 6x x x
d) 23x x e) none of these
2) Given 2( )f x x and ( ) 6g x x , find ( ( 1))f g .
3) If 1
( )2
f x x , find ( ) ( )f x h f x
h
.
a) 2 b) 1
2 c)
1
2x h
h
d) 1 e) none
4) Is the function 3 2( ) 2 3f x x x even, odd, or neither? Show why.
5) If f is a one-to-one function on its domain, the graph 1( )f x is a reflection of the graph of ( )f x with
respect to:
a) the x-axis b) the y-axis c) y = x d) y = -x e) none
6) In which graph does y not represent a one-to-one function of x?
a) b) c)
d) All of these are one-to-one functions of x. e) None of these are one-to-one function of x.
7) Given 3( ) 3 1f x x , find 1( )f x .
a) 3
1
3 1x b) 13 1x c) 3( 1)x d) 3
1
3
x e) none
NO CALCULATOR
Solving equations
8) Solve for x. 3 1
62 4
x x
a) 5 b) 23
5 c)
35
8 d)
1
2 e) none
9) Solve for x. 2
1 2 2
3 3 9
x
x x x
a) 1
2 b) 3 c) -3 d) -3 and 3 e) none
10) Solve for x. 7 2
92 2
x x
x x
a) 18
5 b)
2
3 c)
2
5 d)
5
18 e) none
11) Solve for p: 2
2
4 pg
r
.
12) Solve for x. 2
2 16x x
a) 8 2 15 b) 10 4 6 c) 10 2 26 d) 8 4 15 e) none
13) Solve for x. 2
3 1 25x
a) 4
3 , 2 b) -2, 2 c) 2 d) -2,
4
3 e) none
14) Solve for x. 3 23 24 21 0x x x
a) 7, 1 b) -7, -1 c) 0, 1, 7 d) 0, -1, -7 e) none
15) Solve for x. 2
2 34 25x
a) -5.8, 5.8 b) -4.6, 4.6 c) 21 d) -11, 11 e) none
16) Solve for x. 2 4 12x
a) 5 7
,2 2
b) 5 7
,2 2
c) 5 5
,2 2
d) 5
2 e) none
17) Solve by factoring. 22 4 9 18x x x
a) 9
2,2
b) 9
2,2
c) 9
2 d)
9
2 e) none
18) Solve by completing the square. 2 6 1 0x x
a) 3 26 b) 3 10 c) 3 17 d) 3 2 2 e) none
19) Solve for x. 2 1 4
11
x
x x
a) 1 b) -1 c) 1
, 13
d) 1
1,3
e) none
20) Solve for x. 23 6 2 0x x
a) 3 3
3
b) 1 3 c)
3 15
3
d)
1, 2
3 e) none
21) Solve for x. 24 12 135x x
a) 9 15
,2 2
b) 5 3
,2 2
c) 15 9
,2 2
d) 3 6
2
e) none
22) Solve the inequality algebraically. 3 2 9x
a) , 3 b) , 3 c) 3, d) 3, e) none
23) Find all the real zeros of the polynomial function 6 2( )f x x x .
a) 0 b) 0, 1 c) 1 d) 0, 1, -1 e) none
NO CALCULATOR Factoring and division
24) Which polynomial function has zeros of 0, -1 and 2?
a) ( ) 1 2f x x x x b) ( ) 1 2f x x x x
c) ( ) 1 2f x x x d) 2
( ) 1 2f x x x e) none
25) Use long division to find the quotient. 3 26 7 15 6 2 1x x x x
a) 2 17 53 2
2 2(2 1)x x
x
b) 2 1
3 5 5(2 1)
x xx
c) 2 113 5 5
(2 1)x x
x
d) 2 29 / 2
3 4 17(2 1)
x xx
NO CALCULATOR Graphs 26) Find the domain of the relation shown at the right.
a) , b) , 3 c) , 3
d) 3, e) none of these
27) Find the range of the function shown at the right.
a) , b) 8, 1 c) 3,
d) 1, 5 e) none of these
28) Find the domain of the function ( ) 5f x x .
a) , 5 b) , 5 c) 5, d) 5, e)none
29) Describe the transformation of the graph of ( )f x x which yields the graph of ( ) 20g x x .
a) vertical shift 20 units up b) vertical shift 20 units down
c) horizontal shift 20 units right d) horizontal shift 20 units left
30) Graph 2( ) ( 1)g x x using a transformation of the graph of 2( )f x x .
a) b) c) d)
31) Which sequence of transformations will yield the graph of 2( ) ( 1) 10g x x from the graph of 2( )f x x ?
a) horizontal shift 10 units right b) horizontal shift 1 unit left vertical shift 1 unit up vertical shift 10 units up c) horizontal shift 1 unit right d) horizontal shift 10 units left vertical shirt 10 units up vertical shift 1 unit up
32) Find the x-intercept(s) of 2 23 2 4 12 0x y xy
a) 6, 0 b) 2, 0 c) (4, 0) d) (6, 0) e) none
33) Find the intercepts of the graph of 3 7 21x y .
a) x-int: (0, 7) b) x-int: (0, 3) c) x-int: (3, 0)
y-int: (3, 0) y-int: (7, 0) y-int: (0, 7)
d) x-int: (7, 0) e) none y-int: (0, 3)
34) Find the x and y-intercepts: 2 5 4y x x
a) (0, -4), (0, 1), (4, 0) b) (0, 4), (4, 0), (1, 0) c) (0, -4), (-4, 0), (-1, 0)
d) (0, 4), (-4, 0), (-1, 0) e) none of these
35) Determine the left and right behaviors of the graph of 5 2( ) 2 1f x x x .
a) up to the left, down to the right b) down to the left, up to the right
c) up to the left, up to the right d) down to the left, down to the right
e) none of these
36) Determine the left and right behaviors of the graph of 4 3 2( ) 3 5f x x x x .
a) up to the left, down to the right b) down to the left, up to the right
c) up to the left, up to the right d) down to the left, down to the right
e) none of these
37) Which function is graphed?
a) 3 2( ) 6f x x x b) 3 2( ) 6f x x x x
c) 3 2( ) 6f x x x x d) 4 2( ) 6f x x x x
e) none of these
38) Graph the following: 2 2, 0
( )2, 0
x xf x
x x
39) Find the domain of the function 2
1( )
3 2f x
x x
.
a) , 2 , 2, 1 , 1, b) , 1 , 1, 2 , 2, c) ,
d) 1 12 2
, , , e) none of these
40) Find the domain of 2
2( )
3 2
xf x
x x
.
a) all real numbers except -2, 1, and 2 b) all real numbers except -2
c) all real numbers except 1 and 2 d) all real numbers e) none
41) Find the domain of 2
3 1( )
9
xf x
x
.
a) all real numbers b) all real numbers except 3
c) all real numbers except 1
3 d) all real numbers except
1, 3
3
42) Find the vertical asymptote(s) of the graph of 3
( )( 2)( 5)
xf x
x x
.
a) y = 2, y = -5, y = -3 b) x = 2, x = -5, x = -3, x = 1
c) x = 1 d) x = 2, x = -5 e) none
43) Find the horizontal asymptote(s) of the graph of 3 1
( )2
xf x
x
.
a) y = 0 b) x = -2 c) 1
3x d) y = 3 e) none
44) Find the horizontal asymptote(s) of the graph of 2
2
3 2 16( )
7
x xf x
x
a) 7x b) y = 3 c) 7y d) y = 0 e) none
45) Find all intercepts of the graph of 14
( )2 7
xf x
x
.
a) (0, -2), (14, 0) b) (-14, 0), 12
( , 0) c) (14, 0), 12
(0, )
d) (14, 0), 72
(0, ) e) none
46) Match the rational function with the correct graph. 2
( )2
xf x
x
a) b) c) d) 47) Match the graph with the correct function.
a) 3
( )1
xf x
x
b) ( ) 3f x x
c) 2
1( )
2 3
xf x
x x
d)
2 2 3( )
1
x xf x
x
e) None of these
48) What is the domain of ( ) 3 xf x e ?
a) 3, b) 0, c) , d) , 3 e) none
49) Without using a graphing utility, sketch the graph of ( ) 3 2xf x .
NO CALCULATOR Trigonometry
50) Give the exact value of 3
4cos
.
a) 2
2 b)
1
2 c)
3
2 d)
2
2 e) none
51) Find all solutions to 2cos 3 0x in the interval 0, 2 .
a) 11
,6 6
b)
5 7,
6 6
c)
5,
3 3
d)
2 4,
3 3
52) Give the exact value of 3
csc2
.
a) 2 b) undefined c) -1 d) 1 e) none of these
53) Find all solutions to 2sec sec 2x x in the interval 0, 2 .
a) 2 4 3
, , ,2 3 3 2
b)
5, ,
3 3
c)
2 4,
3 3
d)
11, ,
6 6
54) Find the exact value of 5
tan6
.
a) 3
2 b) 3 c) -1 d)
3
3
55) Evaluate sec3
.
a) 2
2 b)
3
2 c)
3
3 d) 2
56) Find all solutions of 2sin cos cos 0x x x in the interval 0, 2 .
a) 5 3
, , ,6 2 6 2
b)
7 3 11, , ,
2 6 2 6
c)
5 11,
6 6
d) 0, e) none of these
NO CALCULATOR Logarithms and natural logarithms
57) Solve for x. 27 81x
a) 3
4 b)
1
3 c)
4
3 d)
2
3 e) none
58) Evaluate. 1ln xe
a) 1 xe b) e c) 1 x d) ln (1 )x e) none
59) Simplify. 5 3ln e x
a) 3 1
ln5 5
ex b)
3ln
5 5
e x c)
3ln
5 5
x d)
3 1ln
5 5x e) none
60) Simplify. 3ln e
a) 3
ln2
b) 2
ln3
c) 3
2 d)
2
3 e) none
61) Solve for x. 2 1ln 9xe
a) 1 ln 9
2
b)
9 1
2ln 2e c) 23 d) 4 e) none
62) Simplify. 57 ln xe
a) 5 ln7x b) 7 5x c) ln 7
5x d) 35x e) none
63) Solve for x. 12 3x x
a) ln 2
ln 6 b)
1ln
3 c)
2ln
3 d) ln3 ln 2 e) none
64) Solve for x. ln (7 ) ln (3 5) ln (24 )x x x
a) 6
11 b)
7
3 c)
7, 5
3 d)
6,5
11 e) none
65) Find the domain of the function ( ) ln( 1) f x x .
a) , b) 0, c) 1, d) ,1 e) none
NO CALCULATOR Limits
66) Use the graph to estimate 2
lim ( )x
f x
.
a) DNE b) 0
c) -3 d) 2
e) none
67) Use the graph to find 1
lim ( )x
f x
, if it exists.
a) 1 b) -2
c) DNE d) -1
e) -3
68) Find 2
3lim ( 2 1)x
x
a) 37 b) 19 c) -17 d) 2 e) none
69) Find 3 2
21
3 4 5 2lim
2x
x x x
x x
.
70) Find 1
lim ( )x
f x
if 2 4, 1
( )2, 1
x xf x
x
71) If 1
lim ( )2x c
f x
and 2
lim ( )3x c
g x
, find lim ( ) ( )x c
f x g x
.
72) Find 2
1
5 6lim
1x
x x
x
.
a) 0 b) -7 c) d) e) none
73) Find 32
2lim
8x
x
x
.
a) 1
20 b) 0 c)
1
4 d)
1
12 e) DNE
74) Find the limit. 2 2
0
( ) 2( ) ( 2 )limx
x x x x x x
x
a) 4x b) -2 c) 2 2x d) DNE e) none
