Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Calculus I
Topic: Antiderivatives
Subtopic: Mean Value Theorem for Integrals and Average Value

Overview

This lesson includes two theorems that are consequences of the Fundamental Theorem of Calculus. The Mean Value Theorem (MVT) for Integrals and its consequence the Average Value (AV) Theorem provide further purpose to integrals from a visual perspective.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 4.5.1 Use graphical symmetry to evaluate a definite integral
• 4.5.2 Use the knowledge about a function being odd or even (and the consequential symmetries) to evaluate a definite integral
• 4.5.3 Understand the hypothesis and conclusion of the mean value theorem for integrals and be able to apply it
• 4.5.4 Understand the hypothesis and conclusion of the average value theorem and be able to apply it

Terminology

Terms you should be able to define: symmetry, odd/even function (and graphical significence)

 The Mean Value Theorem for Integrals If f is continuous on [a,b], then there exists a number c in [a,b] such that int_a^b f(x) dx = (b-a) * f(c). In other words,f" continuous on "[a,b] => EE c in [a,b]∋ int_a^b f(x) dx = (b-a) * f(c)

 The Average Value Theorem of a Function on an Interval If f is integrable on [a,b], then the average value of f on the interval is f_(ave) = 1/(b-a) int_a^b f(x) dx .

Mini-Lectures and Examples

Supplemental Resources (recommended)

Useful video from 3Blue1Brown's The Essence of Calculus: What does area have to do with slope?

Supplemental Resources (optional)

Application of Average Value: Head Injury Criterion (HIC): Severity Index continued at Head Injury Criterion (HIC): HIC Index, example

rev. 2020-11-28