Post #21

Calculus Week #21

So this week in Calculus we came back from the holidays, and we focused on doing some review by doing AP Calculus Practice Examinations.

I just want to say that these were a real eye opener to me because I sucked so hard on these...like... I can't even say how bad it is.

Anyway, some things I've learned that I'll share :-)

If you take the derivative of f^2(x) you treat it as if it was like sin. You do 2f(x) times f'(x) times 1. So its almost as if you have (f(x))^2 and you use the chain rule. Bring two to the front, subtract one. times by the derivative of the inside...well the derivative of f(x) is f'(x) times the derivative of x, which is 1. I never realized that you could apply all of those rules to function notation and it really seemed to help on a lot of problems.

Another thing that is new... say you have

x^2-2
-----
4-x^2 as the limit approaches 2

For this, you end up getting 2/0. For finite limits, if you get a # over 0, it is always infinity. That's a really nice little trick.

Also, all of the problems that deal with velocity and acceleration...like I knew before you had to use derivatives and integrals but I never realized the whole solve for c thing. So anyway, with those I find it easiest to just find all your 3 equations...position, velocity, and acceleration by whatever way possible. Then use what they give you...they will tell you enough information to plug in and solve the equations to be able to get the right thing they want. These have become really easy...

Another thing that's sort of new to me...whenever you have equations solved for x and are doing the area and stuff...like just because you turn the calculator sideways...you still put the one that was on top originally first when you do the integral.

Also, say you have an integral from like 2 to 3x-1... the 3x-1 will change basically like everything. Like if before the integral it had like 1...it would become 2.

On problems that tell you like... x=2t-1 is the substitution used for some integral...how is it changed etc... you use that to find your new bounds and you can eliminate so many answers its amazing. Also, make sure you solve this for t, take its derivative and make sure whatever you have is accounted for in the problem.

If you get confused using fnInt and nDeriv...you can do it another way. Graph the function you want to take the integral of or do a derivative. Then press SECOND, then CALC, and go down to dy/dx for the derivative or Sf(x)dx for integral. If you are doing the dy/dx it will just ask you for a value. You put in the x value that you want to know the derivative at. For the integral, it will ask you your lower bound, you type the number and press enter and then it will ask you for the upper bound...press in the number and press enter. I like this way because it shows you visually where the area you are calculating which is sometimes useful. Also I like this way because you don't have to worry about messing up parenthesizes. A downfall to this method is you will have to write down the decimal version of the number if you want to use it late on whereas if you use fnInt, you can just SECOND ANS. Anyway, just letting you guys know.


Something I still keep forgetting is that say you have the integral of 3x...you can just put it as 3 times the integral of x. This proves very useful...

Something I need to remember is that a secant line is the line between two points...usually the two points they give you.

Anyway, that was a bit of what I was noticing as I studied my notes...I'm off to take the practice online thing...