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Definite Integrals in Context

Sarah Miller

Sarah Miller

5 min read

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Study Guide Overview

This study guide covers the definition of the average value of a function over an interval, using the formula involving a definite integral. It introduces the Mean Value Theorem for Integrals, which relates the average value to a constant function with equivalent total change. The guide also includes a geometrical interpretation and worked examples. Key terms like average value and continuous function are explained, and practice questions are provided for reinforcement.

Average Value of a Function

Table of Contents

  1. Introduction
  2. Definition
  3. Mean Value Theorem for Integrals
  4. Geometrical Interpretation
  5. Exam Tip
  6. Worked Example
  7. Practice Questions
  8. Glossary
  9. Summary and Key Takeaways

Introduction

In this section, we will discuss the concept of the average value of a function, its mathematical formulation, and its applications. Understanding this concept is crucial for solving various problems in calculus.

Definition

Key Concept

The average value of a function ff over the interval [a,b][a, b] provides a single number that represents the average output of the function over that interval.

If ff is a continuous function, the average value of ff over the interval [a,b][a, b] is given by:

Average value of f on [a,b]=1b−a∫abf(x),dx\text{Average value of } f \text{ on } [a, b] = \frac{1}{b-a} \int_{a}^{b} f(x) , dx

Mean Value Theorem for Integrals...

Previous Topic - Exponential ModelsNext Topic - Definite Integrals as Accumulated Change

Question 1 of 7

Alright, let's start with something you know! 🚀 What is the formula for finding the average value of a continuous function f(x)f(x) over the interval [a,b][a, b]?

∫abf(x),dx\int_{a}^{b} f(x) , dx

ddxf(x)\frac{d}{dx}f(x)

1b−a∫abf(x),dx\frac{1}{b-a} \int_{a}^{b} f(x) , dx

f(b)−f(a)f(b) - f(a)