AP Calculus Summer Packet
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Laramie County Community College *
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Course
2010 500
Subject
Mathematics
Date
Nov 24, 2024
Type
Pages
26
Uploaded by computerwizard697
Maranatha
High
School
Summer Packet –
202
3
Due the first day of school
Be sure to show all your work. Credit will not be given for answers not supported by adequate work. Topics A -
V
Correctness Score ____ / 5
Topics A -
V
Completeness Score ____ / 1
5
AP Calculus
AB
Isaac Newton
Gottfried Leibniz
Topic A: Functions
1.) If (
)
2
4
f
x
x
x
=
−
, find: a.) ( )
(
)
4
4
f
f
−
−
b.) 3
2
f
c.) (
)
(
)
2
f
x
h
f
x
h
+
−
2.) If ( )
3
4
3
V r
r
=
, find: a.) 3
4
V
b.) (
)
(
)
1
1
V r
V r
+
−
−
c.) (
)
( )
2
V
r
V
r
3.) If (
)
f
x
and (
)
g x
are given in the graph, find:
a.) (
)( )
3
f
g
−
b.)
( )
(
)
3
f
g
4.) If (
)
2
,
0
1,
0
2
2
2,
2
x
x
f
x
x
x
x
x
−
=
−
+
−
, find: a.) ( )
( )
0
2
f
f
−
b.) (
)
5
4
f
−
−
c.) ( )
(
)
3
f
f
Topic B: Domain and Range
Find the domain of the following functions using interval notation: 1.) (
)
3
f
x
=
2.) 3
2
y
x
x
x
=
−
+
3.) 3
2
x
x
x
y
x
−
+
=
4.) 2
4
16
x
y
x
−
=
−
5.) (
)
2
1
4
4
3
f
x
x
x
=
−
−
6.) 2
9
y
x
=
−
7.) (
)
log
10
y
x
=
−
8.) 2
2
14
49
x
y
x
+
=
−
Find the range of the following functions: 9.) 4
2
1
y
x
x
=
+
−
10.) 100
x
y
=
11.) 2
1
1
y
x
=
+
+
Find the domain and range of the following functions using interval notation. 12.) 13.) 14.)
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Topic C: Graphs of Common Functions Sketch each of the following as accurately as possible. You will need to be VERY familiar with each of these graphs throughout the year
. You may use a graphing calculator for some of them if you have access to one over the summer. If you plan on renting a TI-Nspire and thus do not have one for the summer, I strongly recommend you use try www.desmos.com
. There is an app for Desmos as well that is free that you can install on your phones. Again, these are VERY important graphs to know. Be very accurate with regards to “open circles” and “closed circles”
as those features may not be revealed on a graphing utility. For students who have not taken Trigonometry yet, do your best with #’s 9
-14. 1.
y
x
=
2. 2
y
x
=
3. 3
y
x
=
4.
y
x
=
5.
y
x
=
6. x
y
x
=
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
7. 1
3
y
x
=
8. 2
3
y
x
=
9.
sin
y
x
=
10.
cos
y
x
=
−
−
−
−
−
11.
tan
y
x
=
12.
cot
y
x
=
−
−
−
−
−
13.
sec
y
x
=
14.
csc
y
x
=
−
−
−
−
−
−
−
−
−
−
15. x
y
e
=
16.
ln
y
x
=
17. 1
y
x
=
18.
y
x
=
19. 2
1
y
x
=
20.
2
x
y
=
21.
2
4
y
x
=
−
22. sin
x
y
x
=
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Topic D: Even/Odd Functions and Symmetry
Show work to determine if the relation is even, odd, or neither. You may want to research how to determine evenness and oddness. 1.) (
)
7
f
x
=
2.) (
)
2
2
4
f
x
x
x
=
−
3.) (
)
3
3
2
f
x
x
x
= −
−
4.) ( )
1
f
x
x
=
+
5.) (
)
2
1
f
x
x
=
+
6.) (
)
8
f
x
x
=
Show work to determine if the graphs of these equations are symmetric to the x
-axis, y
-axis, or the origin. 7.) 4
1
x
=
8.) 2
4
2
6
y
x
=
+
9.) 2
3
3
4
x
y
=
10.) x
y
=
11.) x
y
=
12.) 2
2
1
x
y
y
=
+
+
Topic E: Function Transformations
If (
)
2
1
f
x
x
=
−
, describe in words what the following would do to the graph of (
)
f
x
: 1.) (
)
4
f
x
−
2.) (
)
4
f
x
−
3.) (
)
2
f
x
−
+
4.) (
)
5
3
f
x
+
5.) (
)
2
f
x
6.) (
)
f
x
Here is a graph of (
)
y
f
x
=
: Sketch the following graphs: 7.) (
)
2
y
f
x
=
8.) (
)
y
f
x
= −
9.) (
)
1
y
f
x
=
−
10.) (
)
2
y
f
x
=
+
11.) ( )
y
f
x
=
12.) (
)
y
f
x
=
Topic F: Special Factorization
Factor completely. 1.) 3
8
x
+
2.) 3
8
x
−
3.) 3
3
27
125
x
y
−
4.) 4
2
11
80
x
x
+
−
5.) ac
cd
ab
bd
+
−
−
6.) 2
2
2
50
20
x
y
xy
+
−
7.) 2
2
12
36
9
x
x
y
+
+
−
8.) 3
2
2
3
x
xy
x y
y
−
+
−
9.) (
) (
)
(
) (
)
2
3
3
2
3
2
1
3
2
1
x
x
x
x
−
+
+
−
+
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Topic G: Linear Functions
1.) Find the equation of the line in point-slope form, with the given slope, passing through the given point. a.) (
)
7,
3,
7
m
= −
−
−
b.) (
)
1
,
2,
8
2
m
= −
−
c.) 2
1
,
6,
3
3
m
=
−
2.) Find the equation of the line in point-slope form, passing through the given points. a.) (
)
(
)
3, 6 ,
1, 2
−
−
b.) (
)
(
)
7, 1 ,
3,
4
−
−
c.)
2
1
2,
,
, 1
3
2
−
3.) Find the equations of the lines through the given point that are a.) parallel and b.) normal to the given line. a.) (
)
5,
3 ,
4
x
y
−
+
=
b.) (
)
6, 2 , 5
2
7
x
y
−
+
=
c.) (
)
3,
4 ,
2
y
−
−
= −
4.) Find the equation of the line in general form, containing the point (
)
4,
2
−
and parallel to the line containing the points (
)
1, 4
−
and (
)
2, 3
.
5.) Find k
if the lines 3
5
9
x
y
−
=
and 2
11
x
ky
+
=
are a.) parallel and b.) perpendicular.
Topic H: Solving Quadratic and Polynomial Equations
Solve each equation for x over the real number system. 1.) 2
7
18
0
x
x
+
−
=
2.) 2
1
0
4
x
x
+
+
=
3.) 2
2
72
0
x
−
=
4.) 2
12
5
2
x
x
−
=
5.) 2
20
56
15
0
x
x
−
+
=
6.) 2
81
72
16
0
x
x
+
+
=
7.) 1
17
4
x
x
+
=
8.) 3
2
5
5
25
0
x
x
x
−
+
−
=
9.) 4
3
2
2
15
18
0
x
x
x
−
+
=
10.) If 2
y
x
kx
k
=
+
−
, for what values of k
will the quadratic have two real solutions?
Topic I: Asymptotes
For each function, find the equations of both the vertical asymptote(s) and horizontal asymptote (if it exists) and the location of any holes. 1.) 1
5
x
y
x
−
=
+
2.) 2
8
y
x
=
3.) 2
16
8
x
y
x
+
=
+
4.) 2
2
2
6
5
6
x
x
y
x
x
+
=
+
+
5.) 2
25
x
y
x
=
−
6.) 2
2
5
2
12
x
y
x
−
=
−
7.) 3
2
4
x
y
x
=
+
8.) 3
3
2
4
2
4
8
x
x
y
x
x
x
+
=
−
+
−
9.) 3
2
10
20
2
4
8
x
y
x
x
x
+
=
−
−
+
10.) 1
2
x
y
x
x
=
−
+
(Hint: Express with a common denominator)
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Topic J: Negative and Fractional Exponents
Simplify and write with positive exponents. 1.) 2
5
12
x
−
−
2.) (
)
2
5
12
x
−
−
3.) (
)
1
1
4
x
−
−
4.) 3
4
4
x
−
−
5.) 3
3
2
5
x
y
−
6.) (
)
2
3
1
x
−
−
7.) (
)
1
8
2
121
x
8.) (
)
4
2
3
8
x
−
9.) (
)
3
5
5
32
x
−
−
−
10.) (
)
(
)
3
2
4
1
16
32
4
x
x
−
11.) (
)
(
)
1
2
2
1
2
2
1
1
x
x
−
−
+
12.) (
)
1
2
2
2
x
−
−
−
+
Topic K: Complex Fractions
Eliminate the complex fractions: 1.) 5
8
2
3
−
2.) 2
4
9
4
3
3
−
+
3.) 7
3
2
2
5
3
5
4
+
+
−
4.) 1
1
x
x
x
x
−
+
5.) 1
2
1
1
x
x
−
−
+
−
6.) 1
1
x
y
x
y
−
−
+
+
7.) 2
1
2
1
x
x
x
x
−
−
−
+
+
−
8.) (
)
3
4
1
3
4
3
3
4
x
−
−
−
9.) (
)
(
)
(
)
1
1
2
2
2
2
2
1
2
2
1
2
1
x
x
x
x
x
−
−
−
−
−
Topic L: Inverses
Find the inverse of each of the following functions and use a graphing utility (TI-Nspire or Desmos) to show graphically that its inverse is a function. 1.) 2
6
1
x
y
−
=
2.) y
ax
b
=
+
3.) 2
9
,
0
y
x
x
=
−
4.) 3
1
y
x
=
−
5.) 9
y
x
=
6.) 2
1
3
2
x
y
x
+
=
−
Find the inverse of each of the following functions and show that ( )
(
)
1
f
f
x
x
−
=
7.) (
)
1
4
2
5
f
x
x
=
−
8.) (
)
2
4
f
x
x
=
−
9.) (
)
2
2
1
x
f
x
x
=
+
10.) Without finding the inverse, find the domain and range of the inverse to (
)
2
1
x
f
x
x
+
=
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Topic M: Adding Fractions and Solving Rational Equations
1.) Combine the following fractions: a.) 2
1
3
x
−
b.) 1
1
3
3
x
x
+
−
+
c.) 5
5
2
3
15
x
x
−
+
d.) 2
1
3
1
2
1
x
x
x
x
−
−
−
+
2.) Solve the equation for x
. a.) 2
1
5
3
6
x
−
=
b.) 2
1
1
10
3
3
9
x
x
x
+
=
−
+
−
c.) (
)
5
5
5
2
3
5
x
x
x
−
=
+
d.) 2
2
2
1
3
11
1
2
1
2
1
x
x
x
x
x
x
x
−
+
−
=
−
+
−
−
Topic N: Absolute Value Equations
Solve the following equations: 1.) 4
8
20
x
+
=
2.) 1
7
13
x
−
=
3.) 8
2
2
40
x
x
+
+
=
4.) 4
5
5
2
0
x
x
−
+
+
=
5.) 2
2
1
7
x
x
−
−
=
6.) 2
12
12
x
x
x
−
=
−
Topic O: Solving Inequalities
Solve the following inequalities: 1.) (
)
(
)
5
3
8
5
x
x
−
+
2.) 5
1
4
2
3
2
x
x
−
−
+
3.) 3
1
1
4
2
x
+
4.) 7
5
3
x
x
+
−
5.) (
)
2
2
25
x
+
6.) 3
2
4
x
x
7.) 5
1
6
2
x
x
−
+
8.) Find the domain of: 2
6
4
x
x
x
−
−
−
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Topic P: Exponential Functions and Logarithms
Simplify the following: 1.) 2
1
log
4
2.) 8
log 4
3.) 2
3
1
ln
e
4.) 5
log 40
5
5.) ln12
e
6.) 12
12
12
log
2
log
9
log
8
+
+
7.) 2
2
2
3
log
log
3
32
+
8.) 1
1
3
3
4
log
log 12
3
−
9.) (
)
5
3
log
3
Solve the following: 10.) (
)
5
log
3
8
2
x
−
=
11.) (
)
2
9
1
log
3
2
x
x
−
+
=
12.) (
)
log
3
log5
2
x
−
+
=
13.) (
)
(
)
2
2
log
1
log
3
5
x
x
−
+
+
=
14.) (
)
5
5
log
3
log
2
x
x
+
−
=
15.) 3
2
1
ln
ln
2
x
x
−
=
16.) 2
3
18
x
−
=
17.) 3
1
10
x
e
+
=
18.) 2
1
8
5
x
x
−
=
Topic Q: Geometry
1.) You will use each of the following forumlas in AP Calculus AB. Complete each of the following. Find the area between the x
-axis and f (
x
) from x
= 0 to x
= 5. Sketch the region to verify. 2.) ( )
4
f x
=
3.) ( )
f x
x
=
4.) ( )
3
f x
x
=
+
5.) 2
( )
9
f x
x
=
−
6.) 1,
2
( )
5
,
2
x
x
f x
x
x
+
=
−
7.) Fill in the four blanks.
Topic R: Basic Right Angle Trigonometry Solve the following
. If point P
is on the terminal side of θ
, find all 6 trigonometric functions of θ
. (Answers need not be rationalized.) 1.) (
)
2,4
P
−
2.) (
)
5, 2
P
−
3.) If 5
cos
13
= −
,
in quadrant II, 4.) If 2 10
cot
3
=
,
in quadrant III, find sin
and tan
. find sin
and cos
. 5.) State the quadrant in which each of the following is true. a.) sin
0 and cos
0
b.) csc
0 and cot
0
c.) tan
0 and sec
0
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Topic S: Special Angles
Evaluate each of the following. 1.) 2
2
sin 120
cos 120
+
2.) 2
2
2
2tan 300
3sin 150
cos 180
+
−
3.) 2
2
2
cot 135
sin 210
5cos 225
−
+
4.) (
)
(
)
cot
30
3tan600
csc
450
−
+
−
−
5.) 2
2
3
cos
tan
3
4
−
6.) 11
5
11
5
sin
tan
sin
tan
6
6
6
6
−
+
Determine whether each of the following statements is true or false. 7.) sin
sin
sin
6
3
6
3
+
=
+
8.) 2
5
5
cos
1
cos
3
3
5
5
tan
sec
1
3
3
+
=
−
9.) 3
3
3
2
sin
1
cos
0
2
2
2
+
+
10.) 3
2
4
4
cos
sin
3
3
0
4
cos
3
+
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Topic T: Trigonometric Identities
Verify the following identities: 1.) (
)(
)
2
1
sin
1
sin
cos
x
x
x
+
−
=
2.) 2
2
sec
3
tan
4
x
x
+
=
+
3.) 1
sec
sec
1
cos
x
x
x
−
= −
−
4.) 1
1
1
1
tan
1
cot
x
x
+
=
+
+
5.) (
)
csc
csc 2
2cos
x
x
x
=
6.) (
)
2
cos 3
1
4sin
cos
x
x
x
=
−
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Topic U: Solving Trigonometric Equations Solve each equation on the interval
)
0,2
. Do not use a calculator.
1.) 2
sin
sin
x
x
=
2.) 3
3tan
tan
x
x
=
3.) 2
2
sin
3cos
x
x
=
4.) cos
sin
tan
2
x
x
x
+
=
5.) sin
cos
x
x
=
6.) 2
2cos
sin
1
0
x
x
+
−
=
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Topic V. Graphical Solutions to Equations and Inequalities (Note: The following functions are for a TI-Inspire. As long as you get a calculator approved by CollegeBoard
, you may use it for the calculator portions of the AP Exam. You will have to research the following functions if you are not going to use a TI-Inspire.
Acceptable graphic calculators can be found here: https://
apstudents.collegeboard.org/exam-policies-guidelines/calculator-policies
You have a shiny new
graphing calculator. So when are we going to use it? So far, no mention has been made of it. Yet, a graphing calculator is a tool that is required on the AP Calculus exam. For about 25% of the exam, a calculator is permitted. So it is vital you are comfortable using it. There are several settings of the calculator you should make. First, so you d
on’t get into rounding difficulties, it is suggested you set your calculator to show at least three decimal places (and preferably more). This is sandard on the AP Calculs exam, so it’s best we start getting used to it. To do this, press the button, followed by 5
Settings, then 2: Document Settings… Be sure the Display Digits option is set to Float 6. This will ensure that you always see 6 digits across the screen. (There may be times that this can be a problem –
i.e. when you have a decimal answer w
ith four or more digits to the left of the decimal. We’ll deal with this later.) Also, be sure that your calculator’s Angle Setting is in Radian mode throughout the year. To make these changes “stick” select Make Default at the bottom.
You must know how to graph functions on your TI-Nspire. The best way to graph a function is to press the Calculator key (located in the upper left corner of your handheld just below ) twice on the left side. Notice how each time you press this button, your screen toggles between a calculator page and a graphing page. While on a graphing page, select e
to bring up the function entry line. Next input the expression you wish to graph and press Enter. How to find zeros (solutions or x
-intercepts) of a function.
Step 1
: Enter the left side of the equation that’s alreay set equal to zero (for example, 2
2
9
3
0
x
x
−
+
=
) into the f1
(
x
) entry line. Step 2: Select followed by 6: Analyze Graph and 1: Zero Step 3: Move the cursor to a position you feel is left of the zero you wish to find first and press Enter. Then move the cursor to the right of the desired zero you wish to find and press Enter. You will notice that the ordered pair for the zero will show up automatically once it falls within the range of your lower and upper bound. You can find relative Maximum and Minimum values in a similar way. How to find the intersection of two functions.
Step 1: Enter each side of the equation (for example, 3
2
3
x
x
=
−
into the f1
(
x
) and f2
(
x
) entry lines. Step 2: Select followed by 6: Analyze Graph and 4: Intersection Step 3: Repeat Step 3 above. This problem could also be solved by setting the above equation equal to zero, and using the following procedure “
6: Analyze Graph and 1: Zero” in
stead. Note: You can always move things around on your screen and place them in positions that may make them easier to read. To do this, rub your finger over the Touchpad until the cursor appears. Move the cursor to the item (i.e. intersection ordered pair from the example above) you want to move. An open hand should appear.
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Press Enter to close the hand. By rubbing the Touchpad, the ordered pair should move. Press Enter when you have found a suitable place to put it. We will learn much more about the functionality of this calculator in the first several days of class. While many of these problems on this page can be done with other graphing utilities (desmos, etc.), I highly recommned that you wait to do these problems when you have a TI-Nspire. Topic V: Using the TI-Nspire (Continued) Use your TI-Nspire to find the zeros of each of the following functions. Make sure each equation is set equal to zero first. 1.) 3
3
5
0
x
x
−
−
=
2.) 2
2
1
2
x
x
−
=
3.)
)
2ln(
1)
5cos
on
0,2
x
x
+
=
Use your TI-Nspire to find the solution (intersection) of the given system of equations. 4.) 2
4
2
2
( )
6.5
6
2
( )
1
x
x
f x
x
x
x
g x
x
e
−
=
−
+
+
=
+
+
Use your TI-Nspire to find both a relative maximum and a relative minimum point of the given function. 5.) 5
4
( )
2
3
4
h x
x
x
x
=
−
+
−
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