The Brief but Handy AP Calculus AB
Book
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By Jennifer Arisumi
&
Anug Saha
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Table of Contents
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What is a limit?……………………………………………..…………………….4
What does it mean for a limit to exist?...................................5
How limits fail to exist………………………………………………………...6
Finding the limit graphically…………………………………………………7
Finding the limit numerically and algebraically .…………………8
Definition of Continuity at a Point……………………………………….9
How limits impact the continuity of a function ..………....10-11
Derivatives ………………………………………………………………….12-13
Examples and Practice…………………………………………….…………14
Antiderivatives ..………………………………………………………….……15
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Table of Contents
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Indefinite and Definite Integrals ……………………………………….16
Fundamental Theorems of Calculus..…………………………………17
Common Integrals and Basic Properties/Formulas/Rules…..18
Examples and Practice ……………………………………………………..19
Application Problem …………………………………………………………20
About the Authors ……………………………………………………….21-22
Bibliography ………………………………………………………………….....23
The End……………………………………………………………………………..24
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What is a limit?
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• The limit of f(x) as x approaches a is L
lim 𝑓 𝑥 = 𝐿
𝑥→𝑎
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Let y=f(x) be a function. Suppose that a and L
are #’s such that:
• Whenever x is close to a but not equal to a,
f(x) is close to L
• As x gets closer and closer to a, but not
equal to a, f(x) gets close to L
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What does it mean for a limit to exist?
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For a limit to exist, three criteria must be met
if
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1. lim 𝑓 𝑥 exists → lim from the right exists
𝑥→𝑎+
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2. lim 𝑓(𝑥) exists → lim from the left exists
𝑥→𝑎−
3. lim 𝑓(𝑥) = lim 𝑓(𝑥) = L
𝑥→𝑎+
𝑥→𝑎−
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How limits fail to exist
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• A limit fails to exist when x approaching to
a is not the same from the left and the right
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Finding the limit graphically
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𝑥 2 +1
lim
𝑥→−1 𝑥+1
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= -1
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Finding the limit numerically and algebraically
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Numerically:
𝑥 2 −3𝑥+2
lim
𝑥→2 𝑥−2
x
1.75
1.9
1.99
1.999
2
2.001
2.01
f(x)
.75
.9
.99
.999
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1.001
11.01
=
(𝑥−2)(𝑥−1)
(𝑥−2)
Algebraically:
1.Factor
2.Simplify
3.Substitute
𝑥 2 −3𝑥+2
lim
𝑥→2 𝑥−2
lim 𝑥 − 1
𝑥→2
=2-1
=1
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= x-1
2.1
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2.25
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1.25
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Definition of Continuity at a Point
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A function is continuous at a point c if:
(a) f(c) is defined
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(b) lim 𝑓(𝑥) exists
𝑥→𝑐
(c) lim 𝑓(𝑥) = f(c)
𝑥→𝑐
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How limits impact the continuity of a function
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Continuity at a point and on an open interval:
• f is continuous means: no interruption in
the graph of the function of “f” at “c”
• the f is continuous also means: there is no
gap, holes, or jumps in the graph
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Continuity Continue…
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𝑥→4
continuous because it has a gap at point 4
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f(x) = 𝑥 3 − 6𝑥 2 − 𝑥 + 30
lim 𝑥 3 − 6𝑥 2 − 𝑥 + 30 has a limit but is not
Derivatives
0011
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The derivative
is the
instantaneous
to one of its variables or the slope of a point on a given function. In algebra,
you can determine the slope of a line by taking two points on that line and
plug them into the slope formula:
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• But in Calculus we no longer want to find the slope of a line but the slope of a
point of a given curve. To do that we would set up the definition of the
Derivative which is a slightly modified version of the slope formula but the
only difference is that we’re finding the slope between two points that are
infinitely close to each other.
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Derivatives Continue…
• Finding the derivative using the definition of the derivative is a process that is
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very time consuming and sometimes involves a lot of complex algebra.
Fortunately, there’s a short cut to finding a derivatives!
• Here are some common derivatives:
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Examples and Practice
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Solution:
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2.
Solution:
3.
Solution:
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Antiderivatives
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• Antiderivatives are exactly how they sound- they are the
opposite of derivatives. Or they can be know as the step
before you take the derivative. It basically means that
you take the function that you are given and say that it
is the derivative, and figure out what function it is the
derivative of.
• Integrals are basically anti-derivatives, set into a formula
designed to tell you to take the antiderivative. There are
two types of integrals, indefinite and definite integrals.
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Indefinite and Definite integrals
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• Indefinite integrals: Simply an antiderivative.
Example:
• Definite integrals: The definite integral of f(x) is a
number and represents the area under the curve
of f(x) from a to b on the x axis.
Example:
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Fundamental Theorems of Calculus
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• First FTC: f(x) is continuous on [a,b] , F(x) is an
antiderivative of f(x) then
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• Second FTC: If f(x) is continuous on [a,b] then
F´(x)= f(x) when
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Common Integrals and Basic
Properties/Formulas/Rules
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Examples and Practice
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1.
Solution:
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2.
Solution:
3.
Solution:
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Application Problem
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The graph of the velocity function is shown in Figure 9.3-2.
1. When is the acceleration 0?
2. When is the particle moving to the right?
3. When is the speed the greatest?
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Solution:
1. a(t) = v'(t) and v'(t) is the slope of tangent to the graph of v. At t =1 and t =3,
the slope of the tangent is 0.
2. For 2 < t < 4, v(t) > 0. Thus the particle is moving to the right during 2 < t < 4.
3. Speed = |v(t)| at t =1, v(t) = – 4.
Thus, speed at t = 1 is |–4| = 4 which is the greatest speed for 0 ≤ t ≤ 4.
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About the Author
Anug Saha
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Anug Saha is a Indian-Bangali male who
was born and raised in Astoria, NY. He
still has no clue in what he wants to do
with his life but he knows he wants
math to be involved. He hopes that the
University of Wisconsin - Madison can
shape him into the person his parents
would be proud of. He loves food and
the presence of his friends. He knows he
will be a big deal one day.
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About the Author
Jennifer Arisumi
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Jennifer Arisumi is a junior at the High
School for Environmental Studies. She
is a very outgoing bright student. Her
favorite subject is math and hopes
that in her future math would be
involved. She hasn’t decided yet what
she wants to be in the future but she
knows it wants to be something
related to animals. She loves food and
making new friends.
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Bibliography
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• http://archives.math.utk.edu/visual.calculus/1/definition.
6/
• http://physics.info/kinematics-calculus/problems.shtml
• https://benchprep.com/blog/ap-calculus-topics-limits/
• Textbook
• class notes
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The End
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