Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Sections 5.5 & 5.FT The Fundamental Theorems of Calculus

What are the critical numbers (x values)? Where are all the local maximums and minimums? Where are all the global maximums and minimums? Where are the inflection points (x values)? The following is a graph of the derivative of a function f defined on the closed interval [0, 4]. Review Example 0.6, 3.3 Max: 0.6, Min: 0, 4 Max: 0.6, Min: 4 1.7, 3.3

Analyzing the graph of the derivative increasing decreasing maximum rate of decrease still decreasing flattening still decreasing Analyzing the Graph of the Derivative

The Fundamental Theorem of Calculus provides the bridge between two seemingly unrelated branches of calculus: – Differential calculus, which arose from the tangent problem, and – Integral calculus, which arose from the area problem. The Fundamental Theorem gives the precise relationship between these. The Fundamental Theorem of Calculus

Given the rate of change, the definite integral gives total change in a quantity. Suppose F(t) is the quantity. Then F'(t) is the rate of change of that quantity. So the total change in F(t) from t = a to t = b, that is F(b) – F(a), is the definite integral of F′(t) from t=a to t=b. The Fundamental Theorem of Calculus

Suppose a quantity F(t) is given. Then F′(t) is the rate of change of that quantity with respect to t. Compute the total change from t=a to t=b. Change in F = Rate × Time. Compute the total change by computing a Riemann Sum. Break up interval into segments of length ∆ t = (b–a)/n. Consider the first subinterval. Approximate the rate of change in that subinterval by F ′(t 1 ). Then the change in that subinterval is approximately F ′t 1 ) × ∆ t. Continue - adding up all the changes in each subinterval. Total Change in F between a and b is F ′(t 1 ) × ∆ t + F ′(t 2 ) × ∆ t + F ′(t 3 ) × ∆ t F ′(t n ) × ∆ t. As n → ∞, this sum becomes a definite integral. But the total change in F between a and b is also F(b) - F(a). So... The Fundamental Theorem of Calculus: Derivation

The First Fundamental Theorem of Calculus also called The Evaluation Theorem

Example Suppose the following is a graph of the derivative of a function f.

Example

Area Function

Example

So … In other words … Second Fundamental Theorem of Calculus

Examples

Complete Fundamental Theorem of Calculus