calculus.org

Calculus Notes

Random notes..

5 Functions

Integral	Function	Derivative
xn+1/(n+1)	xn	nxn	<- Nonsense integral for n=-1
-cos(x)	sin(x)	cos(x)
sin(x)	cos(x)	sin(x)
ecx/c	ecx	cecx
x ln(x)-x	ln(x)	1/x	<- The n=-1 case

The step function

Integral:
Ramp Function:
             /
            /
___________/
-------------------------->x
             ^slope=1
Function:
Step Function:
            ___________
___________|
-------------------------->x
           ^x=0
Derivative:
Delta Function:
            |
            |
____________|___________
-------------------------->x

The area under the function should equal to the function above it. Therefore the area under the delta function is 1

Function composition

Function	Derivative
a * f(x) + b g(x)	a * df/dx + b * df/dx
f(x) * g(x)	f(x) * dg/dx + g(x) * df/dx	Product Rule
f(x) / g(x)	[g(x) * df/dx - f(x) * dg/dx]/g2	Quotient Rule
x = f(y)-1	dx/df = 1 / [dy/dx]
f(g(x))	[df/dy] * [dy/dx] <- y = g(x)	Chain Rule

l’Hospital’s Rule

Where f(x) -> 0 and g(x) -> 0 as x -> a (ie. both function go to zero at some point a)
Then the fraction: \ f(x) / g(x) \ Goes to:\ [df/dx] / [dg/dx]

Fundamental Theorem of Calculus

Integral( Derivative( f(x) )) = f(x)
Derivative( Integral( f(x) )) = f(x) \

As long as they’re continuious… (has a max and min and reaches all values in between)

Mean Value Theorem

If f(x) has a derivative between a and (ie. a <= x <= b)
Then: (f(b) - f(a)) / (b-a) = [df/dx](c) will be true at some c between a and b \ This is “average speed” equation

Taylor series

When you know f(a) and want to find a value close by f(x):
f(x) = f(a) + f’(a)(x-/a/)+ (1/2!)f”(a)(x-a)² + … [1/(n+1)!] * fⁿ⁺¹(a)(x-a)ⁿ⁺¹

In the traveling example:
For some time a you get the location f(a) \ Then you get the speed at that time a and see how much further you’d go by time x and add that to f(a) \ Then you add a correction due to the acceleration \ etc…

Binomial Theorem

Function	Derivative
(1+x)	1 + 1x
(1+x)2	1 + 2x +1x^2
(1+x)3	! + 3x + 3x^2 +1x^3
..	..

Whole number powers give you Pascal’s Triangle. And notice that each one can be differentiated n times

How about if powers aren’t whole?:
(1+x)^p — derivative —> p(1+x)^p-1\ These can be differentiated indefinitely

We can do the Taylor expansion:
(1+x)^p = 1 + px + p(p-1)x²/2! + ..\ And the terms will go on forever

Euler’s equation

Doing the Taylor’s Series for a = 0:
e^x = e⁰ + x + (1/2)x² (1/6)x³ …