The Fundamental Theorems of Calculus

[misskitty]

Can someone break down these theorems for me please because my book is horrible at explaining them. The examples the book gives shows the initial question but then the answer and none of the steps in between even on the very simple questions. I'm confused.

~Kitty

 

[Galileo]

http://archives.math.utk.edu/visual.calculus/4/ftc.9/

It has proofs in flash.

 

[misskitty]

I understand what it is but what does it mean? What about the other theorems?

~Kitty

 

[Galileo]

It means that differentiation and integration are, in a way, inverse processes. The fundamental theorem only says that in a much more precise way.

And what other theorems are you talking about? There are 2 parts to the fundamental theorem and both are explained (with an step-by-step proof) on the link.

 

[misskitty]

My book says there is a second fundamental theorem of calculus.

~Kitty

 

[Reshma]

The second fundamental theorem of calculus holds for f a continuous function on an open interval I and a any point in I, and states that if F is defined by:
$$F(x) = \int_a^xf(t)dt$$
Then,
$$F'(x) = f(x)$$
at each point in I, where F'(x) is a derivative of F(x).

 

[misskitty]

I didn't understand this one either. My book says even less than that.

~Kitty

 

[mathwonk]

the point is that every continuous function is the derivative of some other function. and more precisely that other function can be defined as a limit of riemann sums (:"riemann integral") of the first function.

intuitively, the moving area function under the graph of f, is an antiderivative of f, if f is continuous.

the other fundamental theorem is really the mean value theorem, which says that two differentiable functions with the same derivative on an interval, differ by a constant on that interval.

it follows that for any antiderivative F of f on the interval [a,b], F(x) differs by a constant from the riemann integral of f from a to x. this trivial corollary of the previous two theorems is somewhat ridiculously called the second fundamental theorem of calculus.

so the main points are how to construct a function witha given derivative, and then how far a function is determined just by knowing its derivative.for instance, any riemann integrable function si continuous almost everywhere, hence its riemann integral is an antiderivative alm,ost everywhere.

but if you are given a function F that has derivative equal to an integrable f almost everywhere, it does not follow that F differs from the riemann integral of f by a constant.

one needs also to assume that F is lipschitz continuous. i.e. the refined mean value theorem does not say that a function F whose derivative is zero almost everywhere is constant, but does say that if the function F is also lipschitz continuous.

thus the two parts of the fundamental theoprem concern
1) the fact that an integral of f is also an antiderivative of f,
and
2) the extent to which that property characterizes the integral of f.

thus as usually stated the 1st theorem says if f is continuous, then the integral is an antiderivative of f everywhere,

and the 2nd theorem says that conversely, if f is continuous on [a,b] and F is an antiderivative of f on [a,b], then F(x) - F(a) equals the integral of f from a to x.

 

[misskitty]

Wow...I need a minute to process that...

~Kitty

 

[mathwonk]

only a minute? it took me 40 years to learn that.

 

[waht]

only a minute? it took me 40 years to learn that.

lol, you must be like 500 years old.

Suppose you unroll a carpet that in not rectangular, but it's edges can form different shapes like a parabola, or anyother function, then as you unroll this carpet the rate of change of the area being swept is exactly its width at any particular instance.

 

[mathwonk]

what you said took me a little less, maybe 10 seconds. its what i said that took longer. maybe you should read it again. but you are right, i am 500 years old.

 

[bomba923]

40 years? Do you mean that you lacked an understanding of the Fundamental Theorem of Calculus for 40 years after you first encountered it?

I don't think this so...but is this what you meant?

 

[misskitty]

Lol, I think its safe to say that it is what he what he meant, but it was in a joking manner.

I understand it I think what confuses me is the proof of this theorem.

~Kitty

 

[arildno]

the other fundamental theorem is really the mean value theorem, which says that two differentiable functions with the same derivative on an interval, differ by a constant on that interval.

This is indeed, a crucial theorem, and unfortunately, it seems that many tend to forget it as one of the two pillars upon which applied calculus is built.

 

[misskitty]

Math, I really don't see how this theorem is really the mean value theorem. Isn't the mean value theorem f(c)(b-a) which is really (1/b-a)(b-a)? Grr (I get furstrated when I get confused)...getting confused...

~Kitty

 

[mathwonk]

well phooey, i have been typing for about 15 minutes and all of a suddeen the browser erased all of my work. anyway i am not kidding. if you understabnd everything i have written and think it ridiculous that ti took me 40 to learn it, more power to you.

but if you think the ftc just says a continuous functiuon is differentiable everywhere and has derivative equal to the original function, and any antiderivative differs from the origiinal continuous function by a constant then you know diddly.

 

[VietDao29]

well phooey, i have been typing for about 15 minutes and all of a suddeen the browser erased all of my work.


anyway i am not kidding. if you understabnd everything i have written and think it ridiculous that ti took me 40 to learn it, more power to you.

but if you think the ftc just says a continuous functiuon is differentiable everywhere and has derivative equal to the original function, and any antiderivative differs from the origiinal continuous function by a constant then you know diddly.

D'oh, what a stupis browser... It prevents me from learning something new, and understand the ftc more deeply...

 

[mathwonk]

i also had a marvellous proof of the riemann hypothesis but that was erased too!

 

[mathwonk]

tive


oh no! it happened again! this si too much. i rewrote the whole integral business explanation, this is frustrating.

i didn't even get the chance to savem, it ahppoened while i was astill wrioting.

this never happened i might add before this new skin was created and forced on us. (dig, dig.)

 

[HallsofIvy]

Basically, the fundamental theorems say that integration and differentiation are (almost) inverses:

If F(x) is an integral of f(x) then f(x) is the derivative of F'(x):
$$\frac{d}{dx}\int f(x)ds= f(x)$$

If f(x) is the derivative of F(x) then F(x) is an anti-derivative of f(x):
$$\int \frac{dF}{dx} dx= F(x)+ constant$$.

That constant is the reason for the "almost" above.

 

[misskitty]

Right. Doesn't adding C just mean that the antiderivative belongs to a family of antiderivatives that are all verticle translations of one another?

~Kitty

 

[misskitty]

By the way Mathwonk, I'm really sorry the browser erased all of your work. Thats horribly crummy.

~Kitty

 

[devious_]

mathwonk, I've developed the habit of selecting everything I write in a post (ctrl+A) every few minutes if I'm writing a lengthy post, and copying it (ctrl+C). That way if the browser crashes or I'm forced to relogin to post I can simply paste with minimal losses.

Maybe you should consider picking up this habit too seeing as your long posts are always excellent.

 

[mathwonk]

with your kind encouragement I have tried typing the post into a wqord document and pasting:

third try:

the two fundamental theorems of calculus are about the relation betwen two ideas: integrals and antiderivatives. Contrary to popular belief they are not the same, but are closely related. or rather for them to be the same in general, one must define antiderivative more carefully than is usually done.

basically the first FTC says to what extent an integral is also an antiderivative, and the second FTC says to what extent an antiderivative is also an integral.

For example, for continuous functions f, antiderivatives and integrals are essentially the same, but why restrict the discussion to continuous functions, when many other functions also have integrals? I.e. to what extent is the integral of a non continuous function also an antiderivative?, and to what extent is the antiderivative of a non continuous function also an integral?


The usual theorems are:

1) If f is continuous on [a,b] then for every x in [a,b] f is also Riemann integrable on [a,x] and the function F defined by F(x) = Riemann integral of f on [a,x], is an antiderivative of f on [a,b]. I.e. F’(x) = f(x) for all x in [a,b].

2) If f is continuous on [a,b], and if G is any diferentiable function on [a,b] such that G’(x) = f(x) for all x in [a,b], then G differs from the previously defined F by a constant, i.e. G(x) -G(a) = F(x), for all x in [a,b].

The proof of the second theorem is obtained by combining the first theorem with the mean value theorem, i.e. given the first theorem, the second one is really the mean value theorem, which tells you the relation between two functions having the same derivative.

But the theorems are more interesting if f is only assumed Riemann integrable but not continuous. We say a statement about [a,b] is true almost everywhere, if for each e>0, the set where it is not true can be covered by a sequence of intervals of total length less than e.

1)A If f is Riemann integrable on [a,b], then f is continuous almost everywhere, hence the function F(x) = integral of f on [a,x], is differentiable with derivative equal to f almost everywhere.
(This is due to Riemann.)

Suprisingly, it is NOT true that any function G which is differentiable with derivative equal to f almost everywhere on [a,b] differs from F by a constant, not even if we assume G is continuous. Namely there exists a (Cantor) function G continuous on [a,b] and with derivative equal to zero almost everywhere, and yet with G not only not constant, but with G increasing weakly monotonically from 0 to 1.

Thus if we take f to be a function which equals zero at every point where G has derivative zero, but f=1 elsewhere, then the integral of f is zero on [a,b], but the antiderivative G of f is not constant, hence does not differ from F by a constant. I.e. the mean value theorem fails for functions like G which are only differentiable almost everywhere, and hence the second FTC, which is essentially the mean value theorem, also fails in this form.

However, we can recover a version of the mean value theorm, hence a second FTC by introducing the concept of Lipschitz continuity. G is Lipschitz continuous on [a,b] if there is some constant K such that for all x,y in [a,b], we always have |G(y)-G(x)| less than or equal to K|y-x|.

1)B If f is Riemann integrable on [a,b], then f is continuous almost everywhere, hence the function F(x) = integral of f on [a,x], is differentiable with derivative equal to f almost everywhere. Moreover the integral function F is Lipschitz continuous on [a,b].

2)B If f is Riemann integrable on [a,b], and G is any Lipschitz continuous function on [a,b] which is differentiable with derivative equal to f at those points where f is continuous, then G does differ from the integral F of f by a constant, i.e. G(x)-G(a) = F(x).


The proof of this version follows from a stronger mean value theorem that shows that a Lipschitz continuous function with derivative zero a.e. is constant. you might try proving this using compactness.

These theorems reach their final form for Lebesgue integrable functions using the still subtler concept of “absolute continuity” in place of Lipschitz continuity.

 

[mathwonk]

the prior version is what took me 40 years to appreciate and understand thoroughly, although of course i had to memorize something about absolute contuity as a grad student, which however I did not understand.

as proof of that, last year I started to tell my class the false theorem that a continuous function with derivative equal to f almost everywhere, must differ from the integral of an integrable f by a constant. fortunately we mathematicians have a 6th sense, called proof, or asking why?, that stops from such behavior, or at least causes us to corect it.

 

[JasonRox]

mathwonk, I've developed the habit of selecting everything I write in a post (ctrl+A) every few minutes if I'm writing a lengthy post, and copying it (ctrl+C). That way if the browser crashes or I'm forced to relogin to post I can simply paste with minimal losses.

Maybe you should consider picking up this habit too seeing as your long posts are always excellent.

I do the same thing. Best advice for anyone.

 

[Galileo]

I suspected some conspiracy trying to keep us from reading mathwonk's wisdom, but mathwonk outsmarted them.

Thanks for the post.

 

[HallsofIvy]

It probabably happened because Mathwonk was "wrioting"! We certainly can't have that!

 

[Galileo]

Probabably so.

 

[mathwonk]

briefly put, if f is only integrable, then every integral of f is also an antiderivative, but not every (continuous) antiderivative of f is an integral of f.