View source for Calculus

=Calculus=
==[[Introduction to calculus: what is derivative and Integral ]]==
This is a topic ties every things about (functions and graphs) together. 

We're mainly concerned with two parts:
#Derivative (Differentiation)
#Integrals (Integration)

Derivative is equal to rate of change. Most of the time we will us <math> dy \over dx </math> to present how one variable changes with another. 
The derivative is the gradient of a tangent line 

But from the beginning we are going to talk about where do you want the derivative, the other way to say is I will give you a point and then tell me what's the derivative or gradient for that point.

From the beginning here is the concepts you need for calculus.
#"what is the derivative at x=n" you can under stand as "what is the gradient when x=n"
#"What is the Integral at x=a to x=b" you can under stand as  "what is the area between the function and the x axis from x=a to x=b".

Conclusion (from the beginning )

derivative = gradient of a tangent = rate of change
Integral = area under the function.

==Differentiation==

===[[Derivative and Gradient]]===
===[[Calculus:Limits|Limits]]===
<math> \lim_{x \to c} f(x) = L,</math>

===[[Calculus:Power Rule|Power Rule]]===
<math>(x^n)' = n*x^{n-1}</math>

===[[Calculus:Derivative of Polynomial Functions|Derivative of Polynomial Functions]]===
=======[[use Notation::Newton]] Derivative of Polynomial Functions=======
#The sum rule <math>(f+g)'=f'+g'</math>
#The Difference Rule <math>(f-g)'=f'-g'</math>
#The Product Rule <math>(f*g)'=f*g'+ g*f'</math>
#The Quotient Rule <math>({f \over g})' = {(gf'-fg') \over g^2} </math>
=======[[use Notation::Leibniz]] Derivative of Polynomial Functions=======
#The sum rule <math>{d (f+g) \over d x} ={d f \over d x} + {d g \over d x}</math>
#The Difference Rule <math>{d (f-g) \over d x}={d f \over d x} - {d g \over d x}</math>
#The Product Rule <math>{d (f g) \over d x}'= f {d g \over d x} + g {d f \over d x}</math>
#The Quotient Rule <math>{d ({f \over g})  \over d x} = {  g {d f \over d x} - f {d g \over d x} \over g^2} </math>

===[[Calculus:Derivative of Trigonometric Functions|Derivative of Trigonometric Functions]]===
#<math>f(sin x)'= cos(x)</math>
#<math>f(cos x)'= -sin(x)</math>
#<math>f(tan x)'= sec^2(x)</math>
#<math>f(cot x)'= -csc^2(x)</math>
#<math>f(csc x)'= -csc(x) cot(x)</math>
#<math>f(sec x)'= sec(x) tan(x)</math>
====[[Calculus:Chain Rule|Chain Rule]]====
#[[use Notation::Newton]] Chain rule <math>[f(g(x))]'=f'(g(x))*g'(x)</math>
#[[use Notation::Leibniz]] Chain rule <math>\frac{d f(g(x))}{d x} = {{d f(g)} \over d g} {d g \over d x}</math>

===[[Derivatives of Logarithmic and Exponential Functions]]===
#Natural log of <math>x</math> : <math>(ln(x))'= {1 \over x} * (x)' </math>
#log of <math>x</math> : <math>(log_a x^z)' = ({ln x^z \over ln  a})'={1 \over ln_a}(ln x^z)'= {z x^{z-1} \over ln       {a*x^z}} </math>

===Higher Derivatives===
#Second derivatives <math>f''(x) = (f'(x))'</math>

======[[use Notation::Newton]] Higher Derivatives======
#function<math> f(x) </math>
#first derivative <math> f'(x) </math>
#Second derivatives <math>f''(x) = (f'(x))'</math>
#Third derivatives <math>f'''(x) = (f''(x))'</math>
#4th derivatives <math>f^{(4)}(x) = (f'''(x))'</math>


======[[[[use Notation::Leibniz]] Higher Derivatives]]======
#function <math>y=f(x)</math>
#first derivative <math> {d y} \over {d x} </math>
#second derivatives <math>  {d^2}y \over {dx^2} </math>
#third derivatives <math>  {d^3}y \over {dx^3} </math>
#4th derivatives <math> {d^4}y \over {dx^4} </math>

===[[Why do we need Higher Derivatives]]===
From the zero derivative to sixth derivative there is a meaning on the graph 
#No derivative function <math> f(x) </math> = Position 
#first derivative <math> f'(x) </math> = Velocity 
#second derivative <math> f''(x) </math> = Acceleration 
#third derivative <math> f'''(x) </math> = [[wikipedia:Jerk (physics)|Jerk]]
#fourth derivative <math>f^{(4)}(x)</math> = [[wikipedia:Snap (physics)|Snap]]
#fifth derivative <math>f^{(5)}(x)</math> = [[wikipedia:Fifth derivative (crackle)|crackle/flounce]]
#sixth derivative <math>f^{(6)}(x)</math> = Pop

===How to finding Local Maxima and Minima by Differentiation===
Our example will be :
<math> f(x)= x^3-12x+1 </math>
And the graph will be the one at the side.
[[File:Screen Shot 2021-07-31 at 9.35.59 PM.png|thumb]]
Then we will need to use the derivatives to find the two point that has 0 slope we will also called it Local maxima and Local minima

<math> f'(x)=3x^2-12 </math>
 
<math> f'(x)=3x^2-12</math>

<math> 3x^2=12</math>

<math> x^2=4 </math>

Then we get x=2 an x=-2 so the, then we put them into the function.

<math>f(x)= 2^3-(12*2)+1=-15</math>

<math>f(x)= -2^3-(12*-2)+1= 17</math>

so the first point will be local maxima it will be at (-2,17)  and then local minima (2,-15).

[[File:Screen Shot 2021-07-31 at 9.55.12 PM.png|thumb]]

===[[Graphing Functions and Their Derivatives]]===
{{:Graphing Functions and Their Derivatives}}

===[[Limits and L'Hospital's Rule]]===
{{:Limits and L'Hospital's Rule}}

==[[Integration]]==
Mathematicians have been playing with the concept of integration for ages, but it was until Newtons time that it was realized that integration and differentiation are inverse operations.

====[[Integral]]====
#Hint <math>F'(x)=f(x)</math> F(x) is not equal to f(x).
#Definite Integral <math>\int_{a}^{b} f(x) \,dx = F(b) - F(a),</math>
#Indefinite Integral <math>\int f(x) \,dx = F(x)+c,</math>

====[[Properties of Indefinite Integral]]====
#sum rule of Indefinite Integral <math>\int [f(x)+g(x)] \,dx = \int f(x) \,dx + \int g(x) \,dx </math>
#The Difference Rule <math>\int [f(x)-g(x)] \,dx = \int f(x) \,dx - \int g(x) \,dx</math>

====[[Integrals With Trigonometric Functions]]====
#<math>\int cosx \,dx = sinx + c </math>
#<math>\int sin x \,dx = -cos x + c </math>
#<math>\int {sec}^2 x \,dx = tan x + c </math>
#<math>\int {csc}^2 x \,dx = -cotx + c </math>
#<math>\int secx \, tanx \,dx = secx + c </math>
#<math>\int cscx \, cotx \,dx = -cscx + c </math>

====[[Integration By Parts]]====

#Performing Integration By Parts #<math>\int f(x) g'(x) \,dx = f(x)g(x) - \int f'(x) g(x) \,dx</math>

==Reference==
#[https://www.youtube.com/watch?v=wcNc0lgCibY&list=PL6LH0ngwf3HsUd_g82b-CSveVjpO8skll  IB Maths AA SL/HL calculus]
#[https://www.youtube.com/watch?v=rBVi_9qAKTU&list=PLybg94GvOJ9ELZEe9s2NXTKr41Yedbw7M  Calculus by Professor Dave Explains]