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Math Help > Calculus > Calculus Theorems
Once you understand Limits and the definition of "Continuous Function", then you are ready to study five theorems that lead up to the Fundamental Theorem of Calculus.
Here's what you'll find in this section:
(Assume all functions are continuous on [a,b] and if f' is referenced, that f is differentiable on (a,b). ) Intermediate Value Theorem -- if k is between f(a) and f(b), then there exists a c in [a,b] such that f(c)=k. (This proof uses the "completeness property" of the reals) The Bounded Value Theorem also uses the "completeness property" and is used to prove the Extreme Value Theorem -- that a continuous function on a closed interval has a maximum (and a minimum). Relative Extrema Occur Only at Critical Numbers -- if c is an extremum then f'(c)=0 or f'(c) is undefined The Extreme Value theorem and the fact that Relative Extrema Occur Only at Critical Numbers are used to prove Rolle's Theorem -- that if f(a)=f(b) then f'(c)=0 for some c in (a,b) which, in turn, is used to prove the Mean Value Theorem -- that f'(c) = (f(a)-f(b))/(a-b) for some c in (a,b) which is used to prove the Fundamental Theorem of Calculus -- if f is the derivative of F, then the integral from a to b of f(x)dx is F(b)-F(a)
(Assume all functions are continuous on [a,b] and if f' is referenced, that f is differentiable on (a,b). )
(This proof uses the "completeness property" of the reals)
Elementary Calculus: An Approach Using Infinitesimals, by H. Jerome Keisler.
Go back to Calculus Home Limits Definition of Continuous Definition of Interval -- a subset satisfying certain properties of a totally connected set such as the set of real numbers.
Go back to Calculus Home
Limits
Definition of Continuous
Definition of Interval -- a subset satisfying certain properties of a totally connected set such as the set of real numbers.
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