By: Millie, Shi and Mei Ying. About the authors I’m Shi Huang, a AP Calculus AB student. I am a...
-
Upload
madison-allen -
Category
Documents
-
view
215 -
download
1
Transcript of By: Millie, Shi and Mei Ying. About the authors I’m Shi Huang, a AP Calculus AB student. I am a...
![Page 1: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/1.jpg)
AP CalculusCheat Cheat BookBy: Millie, Shi and Mei Ying
![Page 2: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/2.jpg)
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.
![Page 3: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/3.jpg)
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.
![Page 4: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/4.jpg)
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…
![Page 5: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/5.jpg)
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.
![Page 6: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/6.jpg)
Limits exist when…the limit approach from both theleft and right sides are equal. If :
then limit exist
![Page 7: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/7.jpg)
Limit does not exist when…Limit approach from the left side and right side are not equal.Ex: = DNE= 2 = 4 Then = DNE
![Page 8: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/8.jpg)
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)
![Page 9: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/9.jpg)
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)= 1.5Therefore it is discontinuous
![Page 10: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/10.jpg)
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
![Page 11: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/11.jpg)
~ Algebraically To find the limit algebraically, need to simplify, cancel out common factors, then substitute.Ex: = - Simplify ----------------- > Cancelation -------------- > Substitution --------------- > = -
![Page 12: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/12.jpg)
~ Numerically To find the limit numerically, need to find Closest values to x from both positive andnegative side. Substitute the x values and find the closest to value of f(x).Ex: = 0.25
![Page 13: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/13.jpg)
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:
![Page 14: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/14.jpg)
After we went over the general rules of finding derivative, let’s solve a derivative related question:
![Page 15: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/15.jpg)
There are several rules to find the derivative which are the following:Product Rule:
Quotient Rule:
Chapter 2 - Continued
![Page 16: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/16.jpg)
Sum and Difference Rule:
Power Rule:
Chain Rule:
![Page 17: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/17.jpg)
These are some common derivatives
![Page 18: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/18.jpg)
In calculus, an antiderivative, also known as the indefinite integral of 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.
Chapter 3 -Antiderivative
![Page 19: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/19.jpg)
More about AntiderivativeThe 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 integrationA Definite Integral can be evaluate by
using the Fundamental Theorem of Calculus.
![Page 20: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/20.jpg)
The Two Fundamental Theorem of CalculusI. The First Fundamental Theorem of
Calculus. f(x) is continuous on [a,b], F(x) is an
antiderivative of f(x), then
II. 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)=
![Page 21: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/21.jpg)
Let’s do some practices1. Find the antiderivative of f(x)= Step 1: Indefinite integral
Step 2: Apply rule ( *recall: ) = = + C
![Page 22: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/22.jpg)
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
![Page 23: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/23.jpg)
Evaluating Definite Integrals with Substitution
![Page 24: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/24.jpg)
Alright, last example
4. Evaluate Substitute 2 with (t +1) = (2+1)(4x) -- multiply it by
the derivative of 2
= 8
![Page 25: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/25.jpg)
Let’s learn some common antiderivatives…
The antiderivative of a function is a function plus a constant of integration(represent by letter C).
![Page 26: 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.](https://reader036.fdocuments.in/reader036/viewer/2022070401/56649f1e5503460f94c35cd5/html5/thumbnails/26.jpg)
Citation http://
www.nr.edu/chalmeta/271DE/Section%203.2.pdfhttp://
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