12/4/10

Okay so this week we finished up Chapter 3 and took a test on it. Then on Thursday we started learning antiderivatives, which is like integrating except you're just giving a general solution. Integrating from what I understand is when you are actually finding c, a constant. It's basically finding out what the original function is before you took the derivative of it. So, you're given the derivative and you have to find the original function.
Here are a few examples:
(**By the way, I'm just going to use a "S" for the integral notation because that's the closest thing I can think of to use..)

Ex. 1) S (x+7)dx
*Okay to find the antiderivative of this function you're going to use the formula (a/#+1)x^#+1..this formula goes for all polynomials..(where a is the number in front of the variable and # is the exponent)
*So just like taking a derivative you're going to take the antiderivative of each term separately. So let's start with x. The number in front is understood to be 1, and that goes over the number 2 because when you add 1 to the exponent you get 2. So you should have 1/2x^2
*N0w let's take the antiderivative of 7...and you don't have to use the formula for this because you should know that the derivative of a "#x" gives you that number. So that means the antiderivative of 7 is 7x
*So your answer is 1/2x^2+7x...buttttt, there's something you have to add to that! *Since an antiderivative is a general solution, you don't know for sure if there was a constant included in the original equation. So you just add +c to the equation you found to complete your answer
*So your final answer is 1/2x^2+7x+c

Ex. 2) S (x+1)(3x-2) dx
*First of all, (something I forgot to mention earlier), there is NO product rule, quotient rule, or chain rule in antiderivatives/integrating..So before you take the antiderivative you have to somehow get rid of what's being multiplied or dividing by combining it possibly or breaking up a fraction (if that's what you have..)
*In this case you would have a product rule IF you were taking the derivative of it..but since you're not you first have to foil it out before you can find its antiderivative
*So foiling it out you get 3x^2+x-2
*So your new problem is S (3x^2+x-2)
*Once you use the formula for each separate term you should get this:
3/3x^3+1/2x^2-2x+c
*Simplifying that you get x^3+1/2x^2-2x+c

**Um I'm still having trouble sketching graphs out of thin air. I'm really not sure where to start and at this point I think I'm making up my own ways of doing it..