# 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

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#### misskitty

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

~Kitty

#### Galileo

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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

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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.

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#### misskitty

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

#### mathwonk

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

#### waht

mathwonk said:
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

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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.

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#### 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

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Gold Member
Dearly Missed
mathwonk said:
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

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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 fucntion, and any antiderivative differs from the origiinal continuous function by a constant then you know diddly.

#### VietDao29

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mathwonk said:
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 fucntion, 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

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

#### mathwonk

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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.)

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