2nd fundamental theorem of calculus

snakehunter

Can some on pleases explain this too me. I have an AP book, and i am to do a few problems out of it for class, and but cant find it in there ANY WHERE. Any help would be superb!
-Jacob

pig

Mathworld says:

http://mathworld.wolfram.com/SecondF...fCalculus.html

In my book, this is simply called "the fundamental theorem of calculus", and mathworld's first t. of c. is mentioned later than this, so I can understand your confusion.

Ebolamonk3y

Finney's book has this backwards... The second F of C is the Integral Evaluation Theorem... Which is the mathworld's 1st...

Its hand stuff!

Math Is Hard

I've seen it both ways in several books. Luckily, no teacher has ever asked me to state FTC #1 or state FTC #2, they've only asked me to be able to use them.
Did you try any of the problems yet?

JonF

Do you need to understand the proofs? Or just understand the idea?

FTC#2 says
[latex] \int^b_a f(x)dx = F(b) - F(a) [/latex]

where the anti-derivative of f(x) is F(x)

So what you are probably using this Thrm for is evaluating definite integrals (ones with A and B stated).

So what you do is find the anti-derivative of f(x) i.e. F(x), and then evaluate that anti-derivative at a and b, then take the absolute value of their difference.

For example if we wanted to evaluate:

[latex] \int^5_1 3x^2dx [/latex]

we would first find the anti derivative of 3x^2.

Which is x^3.

Then we would evaluate x^3 at 1 and 5 which gives us, 1 and 125. We subtract 1 from 125 and get 124, hence:

[latex] \int^5_1 3x^2dx = 124[/latex]

snakehunter

thanx you guys. I know how to do those with my eyes closed, its just that my AP book has a real habit of doing things w/o thouroughly explaining them. thanx again ill tell my friends about this site!
-Jacob

mathwonk

2nd FTC: If f is any riemann integrable function on the closed bounded interval [a,b], and G is a Lipschitz - continuous function such that for every point x where f is continuous, G is diiferentiable at x with G'(x) = f(x), then the integral of f from a to b, equals G(b)-G(a).

Recall that G is lipschitz continuous on [a,b] if there exists a constant K such that for all points u,v in [a,b] we have |G(v) - G(u)| <= K|v-u|.

JasonRox

Is the first theorem just:

y=x^n
dy/dx=nx^n-1

Is that it?

I wrote general proof for the derivative of a function x^n. Using limits of course, and I ended with nx^n-1, which is what you are suppose to get.

I never seen the proof yet, and it would be great to do it independently.

So, what is the 1st Fundamental Theorem(in most books)?

mathwonk

in most books it says that if f is continuous on [a,b] and G(x) is the integral of f from a to x, then G is differentiable on [a,b] and G'(x) = f(x) for every x in [a,b].

The more general statement is that if f is a Riemann integrable function on [a,b] and G(x) again is the injtegral of f from a to x, then G is Lipschitz continuous, and G is differentiable with G'(x) = f(x) at those points where f is continuous.

Then to derive the 2nd thm from the first you need the generalized mean value theorem, that a function G which is lipschitz continuous and has derivative equal to zero almost everywhere (i.e. except on a set of measure zero) is constant.

in most books the 2nd thm just says that if f is continuous on [a,b] and G is continuous on [a,b] with G'(x) = f(x) for all x in (a,b), then the integral of f from a to b, equals G(b)-G(a).

JasonRox

The Riemann Sum?

You can find the value under a curse with it.