Section 4.10: Antiderivatives

Wassup!

Section 4.10 is all about antiderivatives(Precisely how to find antiderivatives).


Before any specifics, the class should know what the antiderivative exactly is.

-Antiderivative(Book's interpretation): F(x) is an antiderivative of f(x) if F'(x)=f(x), on an interval
-Antiderivative(My interpretation): Opposite of the derivative(The class finds the original function, given the derivative of that function)


Given the definition of the antiderivative, a theorem representing the general equation for antiderivatives can be developed: Antiderivative of the given function=F(x)+C, C being the constant

Example:f(x)=3x^4
F(x)=(3x^5)/5+C, which means F(x)=3(x^5)/5+10000,F(x)=3(x^5)/5+573892


Finding the antiderivative(Finding the original function, given the derivative) is just as easy as finding the derivative.
Simply remember the rule: x^n=(x^n+1)/(n+1), when n does not equal -1
REMEMBER that this rule can be applied to any term in a function

Example:f(x)=5x^4+2x^3-3x^2
F(x)=(5x^5)/5+(2x^4)/4-(3x^3)/3=x^5+(2x^4)/4-x^3 + C

Example:f'(x)=5x^4+2x^3-3x^2, f(0)=2(MADE UP, MAY NOT WORK OUT EVENLY)
f'(x)=x^5+(2x^4)/4-x^3+C, C=1+2-1+2=4(WHY? Add the coefficients of the other terms plus what is given[In this case, coefficients are 1,2,-1, and given of 2])

However, the rule: x^n=(x^n+1)/(n+1) has a problem when n=-1, since the antiderivative ends up undefined(Since the denominator=0).

Example: f(x)=x^-1
F(x)=(x^0)/0=Undefined

Luckily, there is another rule that allows us to find the antiderivative of a function when n=-1. The rule: ln x, when n=-1

Example: f(x)=x^-1=1/x
F(x)=ln x + C

Here is a link that can help you understand antiderivatives even more!
http://v5o5jotqkgfu3btr91t7w5fhzedjaoaz8igl.unbsj.ca/~talderso/UWOnotes/UNIT2.pdf

Now, before I leave...
Here are some pictures of some my favorite breakdance crews/breakdancers

BBoy Hong 10












BBoy Wake-Up












Expression Crew








REMEMBER, YOUR NEXT KANE!!!