<h1> Calculus Guide (AP) </h1> ```cpp= #include <iostream> using namespace std; #define int long long signed main(){ cout << "Hello World!\n"; } ``` :::success A ::: :::info B ::: :::warning C ::: :::danger D ::: :::spoiler E ::: | 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> :::success 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. ::: <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. ### Other vocabs |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> :::warning 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. ::: ### 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 $$ h(x)=\frac{f(x)}{g(x)} $$ $$ h'(x)=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{g(x)^2}$$ ### Chain rule $$ h(x)=f(g(x))$$ $$ h'(x)=f'(g(x))\cdot g'(x) $$ 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| 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 *#*