Fundamental Theorem of Calculus (FTC)

Author: John J Weber III, PhD Corresponding Textbook Sections:

Prerequisite Knowledge

Calculus I

NOTE: One way to use the information contained within this document is to graphically verify antiderivatives.

Relative Extrema

Definition: A function f(x) has a local (relative) maximum at x=x0 if and only if there is some interval (a,b) that contains x0 such that f(x0)f(x) for all x in (a,b).

Definition: A function f(x) has a local (relative) minimum at x=x0 if and only if there is some interval (a,b) that contains x0 such that f(x0)f(x) for all x in (a,b).

Definition: Critical numbers are location(s), i.e., x-values, of potential local extrema.

First Derivative Test for Local Maxima

First Derivative Test for Local Minima

Intervals of Increase

Intervals of Decrease

Concavity

Practice

Example 01: h(x)=x32x2+3

Example 02: f(x)=x2ex

Example 03: g(x)=x2sin(x) on [2π,π]

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Monday, 18 January 2021 09:33 EDT Last Modified: Monday, 15 August 2022 - 15:08 (EDT)