5.3 The Fundamental Theorem of Calculus

Begin by considering an integral as a function –

Let

a) Evaluate g(0), g(2), g(3),g(5) and g(7)

b) On what interval is g increasing?

c) Where does g have a maximum value?

d) Sketch a rough graph of g.

The Fundamental Theorem connects differentiation (grounded in the slope of tangent lines) and integration (grounded in area computation)

The Fundamental Theorem of Calculus - Part I

If f is continuous on [a,b], then the function g defined by

on a<x<b is continuous on [a,b] and differentiable on (a,b)

and g'(x) = f(x).

Proof uses definition of derivative, Extreme Value Theorem, and Squeeze Theorem

Note: If integrate f, then differentiate, the result will be the original function f.

Example

Find the derivative of

#8

#14 Must let u = x² and chain rule

Fundamental Theorem of Calculus - Part 2

If f is continuous on [a,b], then

where F is any antiderivative of f, that is a function such that F '(x) = f(x)

Often in working problems written as

Examples:

(huh? What happened??)

# 26 #30

#32 #34

#36

Assignment: 5.3 page 321; 3, 7-15 odd, 19-37 odd