On-line Math 21

5.2  The Fundamental Theorem of Calculus

Theorem 1 Let f be a continuous function on an open interval containing [a,b] . Set

G(x): = ó õ x a  f(t)dt.

Then, G(x) is differentiable on [a,b] and its derivative is f , that is, G¢(x) = f(x).

Proof:

Define G as in the statement. Then, compute the derivative:

G¢(x) : = lim h® 0  G(x+h)-G(x) h = lim h® 0  ó õ x+h 1  f(t)dt- ó õ x 1  f(t)dt h = lim h® 0  ó õ x+h 1  f(t)dt+ ó õ 1 x  f(t)dt h = lim h® 0  ó õ 1 x  f(t)dt+ ó õ x+h 1  f(t)dt h = lim h® 0  ó õ x+h x  f(t)dt h = lim h® 0  f(c)((x+h)-x) h , by the MVT, integral form. = lim h® 0  f(c) f(x), since c is between x and x+h.

Copyright (c) 2000 by David L. Johnson.