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The Brief but Handy AP Calculus AB Book

By Jennifer Arisumi&

Anug Saha

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Table of ContentsWhat is a limit?……………………………………………..…………………….4What does it mean for a limit to exist?...................................5How limits fail to exist………………………………………………………...6Finding the limit graphically…………………………………………………7Finding the limit numerically and algebraically .…………………8Definition of Continuity at a Point……………………………………….9How limits impact the continuity of a function ..………....10-11Derivatives ………………………………………………………………….12-13Examples and Practice…………………………………………….…………14Antiderivatives ..………………………………………………………….……15

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Table of Contents

Indefinite and Definite Integrals ……………………………………….16Fundamental Theorems of Calculus..…………………………………17Common Integrals and Basic Properties/Formulas/Rules…..18Examples and Practice ……………………………………………………..19Application Problem …………………………………………………………20About the Authors ……………………………………………………….21-22Bibliography ………………………………………………………………….....23The End……………………………………………………………………………..24

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What is a limit?• The limit of f(x) as x approaches a is L

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?

For a limit to exist, three criteria must be met if

1. exists lim from the right exists2. exists lim from the left exists3. L

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How limits fail to exist

• 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

-1

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Finding the limit numerically and algebraically

x 1.75 1.9 1.99 1.999 2 2.001 2.01 2.1 2.25

f(x) .75 .9 .99 .999 1 1.001 11.01 1.1 1.25

Numerically:

Algebraically: = x-1

1.Factor 2.Simplify = 2 - 1

3.Substitute = 1

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Definition of Continuity at a Point

A function is continuous at a point c if:

(a) f(c) is defined(b) exists(c) = f(c)

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How limits impact the continuity of a function

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…

f(x) = has a limit but is not continuous because it has a gap at point 4

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Derivatives• The derivative is the instantaneous rate of change of a function with respect

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:

• 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

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 Practice1.

Solution:

2.

Solution:

3.

Solution:

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Antiderivatives• 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

• 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

• First FTC: f(x) is continuous on [a,b] , F(x) is an antiderivative of f(x) then

• 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

1.

Solution:

2.

Solution:

3.

Solution:

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Application Problem

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?

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 AuthorAnug Saha

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 AuthorJennifer Arisumi

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

• 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