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

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

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

Mean Value Theorem for Integrals

The average value of a function will be a number $k$ such that:

$$k \cdot (b - a) = \int_{a}^{b} f(x) , dx$$

This result is referred to as the Mean Value Theorem for Integrals.

Geometrical Interpretation

While we have removed the diagram, the geometric interpretation of the Mean Value Theorem for Integrals can be described textually:

Imagine a curve $y = f(x)$ over the interval $[a, b]$. The average value is the height of a rectangle with the same base $[a, b]$ whose area is equal to the area under the curve $y = f(x)$. This height represents the average value of the function over that interval.

Exam Tip

Worked Example

Let $f$ be the function defined by $f(x) = \sin x$. Calculate the average value of $f$ over the interval $[0, \pi]$.

Solution:

Use the formula for the average value of $f$ on $[a, b]$:

$$\text{Average value of } f \text{ on } [0, \pi] = \frac{1}{\pi - 0} \int_{0}^{\pi} \sin x , dx$$

Calculate the integral:

$$\begin{aligned} \text{Average value} &= \frac{1}{\pi} \int_{0}^{\pi} \sin x , dx \\ &= \frac{1}{\pi} \left[ -\cos x \right]_{0}^{\pi} \\ &= \frac{1}{\pi} \left( -\cos(\pi) - (-\cos(0)) \right) \\ &= \frac{1}{\pi} \left( -(-1) - (-1) \right) \\ &= \frac{1}{\pi} \left( 1 + 1 \right) \\ &= \frac{2}{\pi} \end{aligned}$$

Therefore, the average value is:

$$\text{Average value} = \frac{2}{\pi}$$

Practice Questions

Practice Question

1. Calculate the average value of the function $f(x) = x^2$ over the interval $[1, 3]$.

Practice Question

2. Find the average value of $f(x) = e^x$ over the interval $[0, 1]$.

Glossary

• Average Value: A single number representing the average output of a function over a specific interval.
• Mean Value Theorem for Integrals: A theorem providing a way to find the average value of a function over an interval.
• Continuous Function: A function that is unbroken or uninterrupted over a given interval.

Summary and Key Takeaways

• The average value of a function $f$ over an interval $[a, b]$ is given by $\frac{1}{b-a} \int_{a}^{b} f(x) , dx$.
• The Mean Value Theorem for Integrals helps in finding a constant value that represents the same accumulation of change as the function over the interval.
• The average value of a function is specific to the interval chosen and can vary for different intervals.

Key Takeaways:

1. The formula for the average value of a function is essential for solving related problems.
2. Understanding the geometrical interpretation can help visualize the concept.
3. Always specify the interval when discussing the average value of a function.

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Use these notes to reinforce your understanding of the average value of a function and practice with the provided questions to ensure mastery of the concept.