final post

alright, well some of the things i've gotten comfortable doing this year in calculus is the first and second derivative test. since those were basically the first things we did this year, i learned those the best, and i got kinda lazy throughout the rest of the year, so that's why i was the best at that.

For this post i am going to go into detail on the second derivative test to find all possible points of inflection and intervals of concavity. remember, points of inflection only happen where there is a change of concavity.

Example: f'(x)= 6/(x^(2)+3)

First, you have to take the derivative of that, and you have to use the quotient rule, so the beginning of the problem will look like, [(x^(2)+3)(0)-6(2x)]/(x^(2)+3)^(2), which simplifies to -12x/(x^(2)+3)^2 remember, that was just the first derivative.

Second step is to take the derivative of the first derivative, that would make this step called taking the second derivative.

Once again u need to use the quotient rule, so f''(x)={(x2+3)^2-(12)-[(-12x)2(x^(2)+3)2x} all that over (x^(2)+3)^4 then you get a bunch of stuff, then you simplify, then you cancel, so I am just going to type the end answer of the second derivative. Which is, (3)(6)(x+1)(x-1) all over (x^2+3)^3

The possible points of inflection are found in the numerator of the finished second derivative, in this case, if you look, it would be x=1, and x=-1

so then you set up your points, (-infinity, -1) u (-1, 1) u (1, infinity)

then you plug in. f''(-2)= positive value f''(0)=negative value f''(2)=positive value

then you know that your intervals concave up at (-infinity, -1) u (1,infinity) or x<1,>1

and it is concave down at (-1,1) or -1
and you're points of inflection are x=-1, and x=1

okay, one thing i wish i would have done was do my homework at the beginning of the year, because that would have made learning all the other stuff throughout the year much easier. and my advice to anyone else taking this class is to learn derivatives, integrals, related rates, substitution, limits, and implicit derivatives really well, because that's pretty much all the main focal points of the ap test and calculus in general. and thanks ms robinson for teaching me so much and being such a cool teacher :)