Fundamental Theorem of Calculus

Part 1:


   If  ,     then     F'(x) = f(x)
 

PROOF:

This is the average value of f(t), so

  since as h ® 0, x + h ® x , then c® x

EXAMPLE:

,    then
 

Part 2:


If f(x) is continuous on [a,b], then where    F'(x) = f(x)
 

PROOF:

Starting with Part I, we know an antiderivative exists.

Let 

so  F(x) = G(x) + C

      F(b) - F(a) = [ G(b) + C ] - [ G(a) + C ]
                        =  G(b) - G(a)
                   

ALTERNATE PROOF:

F(b) - F(a) = [F(xn ) - F(xn-1 )] + [F(xn-1 ) - F(xn-2 )] + ... + [F(x2 ) - F(x1 )] + [F(x1 ) - F(x0 )]

                   = 

    by the Mean Value Theorem
 

and   F'(c) (xi - xi-1 ) = F(xi ) - F(xi-1 )
 

  Then 

Since F'(ci ) = f(ci ) = f(x), and Dx = dx = xi - xi-1 , then


 

What does the Fundamental Theorem say?

Every continuous function is the derivative of some other function.
Every continuous function has an antiderivative.
The processes of integration and differentiation are inverses of one another.


Mean Value Theorem for Integrals:

Given:  f(x) is continuous on [a,b]

Then:  there exists a c in (a,b) such that

Geometrically, there is a rectangle whose area is precisely equal to the area of the region under the curve.

Average Value of a Function:

From the Mean Value Theorem for Integrals, there exists a c such that

          EXAMPLE:  Find the average value of y = sin x from x = 0 to x = p

Integrals of Even and Odd Functions:

EVEN FUNCTIONS:

ODD FUNCTIONS:

Compare:

  1.          ( a function -- an indefinite integral -- a family of functions --

    2.                                                       C is unknown)
     
  2.      (a function -- an indefinite integral, but a particular function --

    3.                                                      C is known)

         (a number -- a definite integral)


Problems