<h1>
Calculus Guide (AP)
</h1>

#include <iostream>

using namespace std;

#define int long long

signed main(){
    cout << "Hello World!\n";
}

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A
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B
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C
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D
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E
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| Column 1 | Column 2 | Column 3 |
| -------- | -------- | -------- |
| Text | Text | Text |

## Table of content
[toc]
<h2>
<span style="font-family: 'Comic Sans MS', cursive;">Intro</span>
</h2>

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Calculus is widely recognized for its complexity and challenging concepts. In this guide, you will not only learn the knowledge required to succeed on the AP Calculus exam, but also discover helpful tips and lesser-known insights that can boost your confidence during the AP exams. Also if you look at this I will expect you to already be familiar with Pre-Calc knowledge.
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<h2>
<span style="font-family: 'Comic Sans MS', cursive;">Limits and Continuity</span>
</h2>

If you took your school's class, you might have been taught about many different ways and things you have to be careful of. Here are some main techniques that would need in your AP exam.

| Types | Examples |
|-------------|------------------|
| Direct Substitution|$\lim_{x \to 2}(3x^2-1)$ when it is a simple function|
| L'Hopital|$\lim_{x \to 0} \frac{sinx}x$ used when direct substitution=$\frac{0}{0}$|

If you want a more organized way you can follow this diagram given by Khan Academy
<img src="https://cdn.kastatic.org/ka-perseus-images/81d6e0612aabe88e862df34aff87b1718e9224ad.png" height=300 >

There are also other methods for certain problem types, but in the AP exams limits are usually very easy.
### Important Terms
|Vocabs|Meaning|
|--------------------------|----------------------|
|removable discontinuity|You will see a gap at a certain point, on a graph it will be a non filled circle on the point|
|jump discontinuity|The functino is sliced of from a certain point, making two non connecting segments|
|infinite discontinuity|In tests it will be written as asymptotes, where the curve will lead to infinity, just like how tan(x) graphs will|

#### Asymptotes
<img src="https://www.andrews.edu/~rwright/Precalculus-RLW/Images/02-07F02b.svg" height=300 >
<h2>
<span style="font-family: 'Comic Sans MS', cursive;">Differentiation, Basic rules</span>
</h2>

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The most fundenmental basics you have to learn in differentiation are the rules for addition, deduction, multiplication, and the division rules. Later in integration, you also have to reverse it.
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### Basics
The derivative of a function is also knowned as the average rate of change of that point. It is often used in physics to find the instantaneous velocity or acceleraton in a certain point. When you differentiate a function, it becomes the function of tangent lines at any point.

To write differential at a point with limit would be calculating the slope of the tangent line to the function at that point. Here are a few examples

**Normal**

$$
f(x)=x^2+1
$$

Since we know at its vertex (0,1) the slope of the tangent line will be zero, we can test it out.
$$
f'(x)=2x
$$
If we plug in x=0, f'(x) is also 0, which is correct.
Now if we want to find the slope of tangent line at other points, such as (4,17 ). Again we just plugin x=8, and we'll get 8.

<img src="https://hackmd.io/_uploads/HJTXMYMcge.png" height=300>

Graph of $$ f(x)=x^2+1 $$ and its tangent line at (4, 17)

### How to find the derivative? (Power rule)
To differentiate a function, break it into individual terms and differentiate each one separately. For each term, multiply the coefficient by the exponent, then reduce the exponent by 1.
$$ f(x)=3x^4+2x^3-5x^2+6x-3 $$
$$ f'(x)=(3 \cdot 4)x^{4-1}+(2 \cdot 3)x^{3-1}-(5\cdot2)x^{2-1}+(6\cdot1)x^{1-1}-(3\cdot0)x^{0-1}$$
$$ =12x^3+6x^2-10x+6$$

### Quotient rule
Quotient rule is a rule you'll likely get confused with at the end of the course. The key to it is if
$$ h(x)=\frac{f(x)}{g(x)} $$
$$ h'(x)=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{g(x)^2}$$

### Product rule
Useful for certain implicit differentiation
$$\frac{d}{dx}(f(x)\cdot g(x))=f'(x)\cdot g(x)+f(x)\cdot g'(x)$$
### Chain rule
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This is especially useful for composite functions
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$$ h(x)=f(g(x))$$
$$ h'(x)=f'(g(x))\cdot g'(x) $$

One of a question set that might be on the AP exam would be
|x|g(x)|f'(x)|g'(x)|
|------|-----|-----|------|
|0|8|10|3|
|1|13|5|7|
|3|5|1|4|
|5|2|8|1|
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Find the value of F' if F=f(g(3))
to find F' you do f'(g(3)) x g(3) which is f'(5) x 5=40 *#*
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### Derivatives of Inverse functions F(x)
We define $$g(x)=F^{-1}(x)$$
$$F(g(x))=x$$
and take the derivative of both sides using the chain rule (Note that **RHS is x**)
$$F'(g(x))\cdot g'(x)=1$$
$$g(x)=F^{-1}(x)=\frac{1}{F'(g(x))}$$

### Implicit Differentiation **IMPORTANT**
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This part is especially important as it will be crossed over to applications of integration and differentiation later on
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In this kind of differentiation we differentiate both x and y, normally when we see f(x)=3x, we differentiate like this
$$y=3x$$
$$dy\cdot 1=dx\cdot 3$$
$$\frac{dy}{dx}=3$$
This is easier since the derivative of LHS is always 1, but for some functions, such as the circle you have to differentiate both x and y at the same time.
$$x^2 + y^2= 9$$
An easy way to do it is to clearly divide out the dx and the dy
$$dx\cdot 2x+dy\cdot 2y=0$$
$$2x\cdot dx=-2y\cdot dy$$
$$\frac{dy}{dx}=\frac{-x}{y}$$
notice for circles RHS is always constant, therefore it always becomes 0.
#### Harder Examples
$$4x^5 y^7=9$$
For this example, you have to use product rule
$$dx\cdot 20x^4 y^7+dy\cdot 28x^5 y^6=0$$
yea... thats it

### Special Differentiation (MEMORIZE)
For these special cases, it is usually determined through graphing, if you want proof go on youtube or try it out.
#### Logarithms and Exponents
$$\frac{d}{dx}lnx=\frac{1}{x}$$
$$\frac{d}{dx}log_a x=\frac{1}{ln(a)\cdot a}$$
$$\frac{d}{dx}e^x=e^x$$
$$\frac{d}{dx}a^x=ln(a)\cdot a^x$$
#### Trigonometry
If you look closely you can probably see pairs, each pair are opposite of each other.
$$\frac{d}{dx}sinx=cosx$$
$$\frac{d}{dx}cosx=-sinx$$
$$\frac{d}{dx}tanx=sec^2x$$
$$\frac{d}{dx}cscx=-cscxcotx$$
$$\frac{d}{dx}secx=secxtanx$$
$$\frac{d}{dx}cotx=-csc^2x$$
### Shortcuts (Unit 5)
#### Critical Points
Inflection point
absolute minima
absolute maxima
relative minima
relative maxima

[Series + Euler's identity](https://drive.google.com/file/d/1EBh6_a9UFjTOphoG9eTgyhqPnvFQ2ojC/preview)

### Applications of differentiation
There are many occasions to use differentiation, one of the most popular type is rate of change that requires many steps. For example, what is the rate of change water in a hemisphere bowl if water is poured into it 0.3L/s and the radius of the sphere is 50cm at h=25.
### Integration
After understanding differentiation, integration is just simply the opposite of differentiation which is why its often called antiderivative. Many of the rules you'll see in this chapter can mostly be proved by reversing the derivative formulas.
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Here are the 4 basic integration techniques
U-sub (anti chain rule)
Integration by parts (anti product rule)
Long division
Trigonometry
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