## The Fundamental Theorems of Calculus

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
 PhysOrg.com science news on PhysOrg.com >> Ants and carnivorous plants conspire for mutualistic feeding>> Forecast for Titan: Wild weather could be ahead>> Researchers stitch defects into the world's thinnest semiconductor
 Recognitions: Homework Help Science Advisor http://archives.math.utk.edu/visual.calculus/4/ftc.9/ It has proofs in flash.
 I understand what it is but what does it mean? What about the other theorems? ~Kitty

Recognitions:
Homework Help

## The Fundamental Theorems of Calculus

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.
 My book says there is a second fundamental theorem of calculus. ~Kitty
 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).
 I didn't understand this one either. My book says even less than that. ~Kitty
 Recognitions: Homework Help Science Advisor 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.
 Wow...I need a minute to process that... ~Kitty
 Recognitions: Homework Help Science Advisor only a minute? it took me 40 years to learn that.

 Quote by mathwonk 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.
 Recognitions: Homework Help Science Advisor 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.
 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?
 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

Recognitions:
Gold Member
Homework Help