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AP Calculus Cheat Cheat Book. By: Millie, Shi and Mei Ying. About the authors.

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AP Calculus Cheat Cheat Book

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AP CalculusCheat Cheat Book

By: Millie, Shi and Mei Ying


About the authors

I’m Shi Huang, a AP Calculus AB student. I am a senior from HSES. Math is my favorite subject because is challenging and fun; therefore, I took AP calculus during my senior year. I will major in math and science in college and hope to become a professional doctor in the future.


I am Millie Tang. I am a senior in Ms. Zhao’s AP Calculus AB class. I am a bit shy at school but I am really loud at home. One of my academic goals is to graduate and get a degree from Baruch College. Another goal is of course to have above a 3.0 GPA in College.


I am Mei Ying Chen, or call me Joey. I am a senior in Ms. Zhao's AP Calculus AB class and I am the class monitor. I considered myself as a friendly and optimistic person Next year, I am going to City College of New York. I didn’t decide my major yet but I want to study abroad. Lastly, I hope I receive a “5” on the AP Calculus Exam…


Chapter 1 – Limits and Continuity

What is limits?

Limit is the approximated value of f(x) in a given function as x approaches to a certain point.


Limits exist when…

the limit approach from both the

left and right sides are equal.

If :

then limit exist


Limit does not exist when…

Limit approach from the left side and

right side are not equal.

Ex: = DNE

= 2

= 4

Then = DNE


What is continuous?

A graph is continuous when there is

no interruption, no gaps, no holes or no

jumps in the graph of the function.

A graph is continuous when …

~ f(a) is define

~

~ = f(a)


What is discontinuous?

A graph is discontinuous when there is

a gap, hole or jump. It is also discontinuous

when limit does not equal to value of f(x).

Ex: = 1

f(2)= 1.5

Therefore it is discontinuous


You can find the limit by …

~ Graphing

To find the limit graphically, need to trace a certain

point from the left side and right side of the function.

Ex:

Then


~ Algebraically

To find the limit algebraically, need

to simplify, cancel out common factors,

then substitute.

Ex: = -

Simplify ----------------- >

Cancelation -------------- >

Substitution --------------- > = -


~ Numerically

To find the limit numerically, need to find

Closest values to x from both positive and

negative side. Substitute the x values

and find the closest to value of f(x).

Ex: = 0.25


Chapter 2 - Derivatives

What are derivatives?

Derivatives tell s the change of the rate of any x-value in the function. Derivatives is just another word for finding the slope at that specific point in the function.

Finding the derivative by Limit Process:


  • After we went over the general rules of finding derivative, let’s solve a derivative related question:


Chapter 2 - Continued

There are several rules to find the derivative which are the following:

  • Product Rule:

  • Quotient Rule:


  • Sum and Difference Rule:

  • Power Rule:

  • Chain Rule:


These are some common derivatives


Chapter 3 -Antiderivative

In calculus, an antiderivative, also known as the indefinite integralof a function f -is a differentiable function F whose derivative is equal to f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative.


More about Antiderivative

  • The notation used to refer to antiderivative is the indefinite integral. means the antiderivative of f with respect to x.

  • If F is an antiderivative of f , we can write

    = F + C

    C is the constant of integration

  • A Definite Integral can be evaluate by using the Fundamental Theorem of Calculus.


The Two Fundamental Theorem of Calculus

  • The First Fundamental Theorem of Calculus.

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

    then

  • The Second Fundamental Theorem of Calculus

  • If f(x) is continuous on [a,b] then g(x)=

    Is also continuous on [a,b] and g’(x)=


Let’s do some practices

  • Find the antiderivative of f(x)=

    Step 1: Indefinite integral

    Step 2: Apply rule ( *recall: )

    =

    = + C


2. Evaluate

Step 1: Separate definite Integral

Step 2: find antiderivative

= - [ + [9x]

Step 3: Substitute values

= [- - ] + [(9)(3)-(9)(-3)]

Step 4: Simplify

= 0 + 36 = 36


Evaluating Definite Integrals with Substitution


Alright, last example 

4. Evaluate

  • Substitute 2 with (t +1)

    = (2+1)(4x) -- multiply it by the derivative of 2

    = 8


Let’s learn some common antiderivatives…

The antiderivative of a function is a function plus a constant of integration(represent by letter C).


Citation

  • http://www.nr.edu/chalmeta/271DE/Section%203.2.pdf

  • http://www.analyzemath.com/calculus/continuity/continuous_functions.html

  • Ms.Zhao notes.

  • http://mszhao.com/files/4.5%20-%20How%20do%20we%20evaluating%20definite%20integrals%20with%20substituiton.pdf

  • http://www.wyzant.com/help/math/calculus/integration/antiderivatives


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