AP Calculus Cheat Cheat Book. By: Millie, Shi and Mei Ying. About the authors.
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AP CalculusCheat Cheat Book
By: Millie, Shi and Mei Ying
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…
What is limits?
Limit is the approximated value of f(x) in a given function as x approaches to a certain point.
the limit approach from both the
left and right sides are equal.
If :
then limit exist
Limit approach from the left side and
right side are not equal.
Ex: = DNE
= 2
= 4
Then = DNE
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)
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
~ 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:
Chapter 2 - Continued
There are several rules to find the derivative which are the following:
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.
= F + C
C is the constant of integration
then
Is also continuous on [a,b] and g’(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
4. Evaluate
= (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).