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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). )

  1. 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)

  2. 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).
  3. Relative Extrema Occur Only at Critical Numbers -- if c is an extremum then f'(c)=0 or f'(c) is undefined
  4. 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)
  5. 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)
  6. 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)

Internet references

Elementary Calculus: An Approach Using Infinitesimals, by H. Jerome Keisler.

Related pages in this website

Go back to Calculus Home


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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