Tuesday, December 05, 2006

Section 4.10: Antiderivatives


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

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

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

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!

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

BBoy Hong 10

BBoy Wake-Up

Expression Crew



At 5:45 PM, Blogger Joe Polwrek said...

i may be misunderstanding your explanation for your example, but i think the constant, c, is 2. if you plug in 0 for the new function, you get 2 for the constant, not 4. I do not understand why you would add the coefficients together. please correct me if im wrong, which i very well may be (im trying to study for the test)

At 7:40 PM, Blogger Math Maverick said...

Joey's correct - the constant should be 2, not 4. Joey's method of calculating the constant is the correct method...


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