Average Rate of Change of a Function over a Given Interval

Oct 13, 2025 by windailys.com 65 views

Introduction

The average rate of change of a function over a given interval is a measure of how much the function changes on average over that interval. It is an important concept in calculus and is used to describe the behavior of functions. In this article, we will discuss how to find the average rate of change of a function over a given interval.

What is Average Rate of Change?

The average rate of change of a function f(x) over an interval [a, b] is defined as:

\frac{f(b) - f(a)}{b - a}

This formula calculates the difference in the function values at the endpoints of the interval and divides it by the length of the interval.

Example Problem

Given the function defined in the table below, find the average rate of change of the function over the interval $6 \leq x \leq 18$ .

$x$	$f(x)$
6	49
12	39

Step 1: Identify the Interval

The interval given in the problem is $6 \leq x \leq 18$ . This means that we need to find the average rate of change of the function over this interval.

Step 2: Find the Function Values at the Endpoints

We need to find the function values at the endpoints of the interval, which are $x = 6$ and $x = 18$ . However, the function values are only given for $x = 6$ and $x = 12$ . We can use the given function values to find the function value at $x = 18$ .

Step 3: Find the Function Value at $x = 18$

To find the function value at $x = 18$ , we can use the fact that the function is continuous. This means that we can use the function value at $x = 12$ to estimate the function value at $x = 18$ . However, since we are not given any information about the function's behavior between $x = 12$ and $x = 18$ , we will assume that the function value at $x = 18$ is not given.

Step 4: Calculate the Average Rate of Change

Since we do not have the function value at $x = 18$ , we can only calculate the average rate of change over the interval $6 \leq x \leq 12$ . The average rate of change is given by:

\frac{f(12) - f(6)}{12 - 6}

Substituting the given function values, we get:

\frac{39 - 49}{12 - 6} = \frac{-10}{6} = -\frac{5}{3}

Conclusion

In this article, we discussed how to find the average rate of change of a function over a given interval. We used the formula for average rate of change and applied it to the given function over the interval $6 \leq x \leq 12$ . We found that the average rate of change of the function over this interval is $-\frac{5}{3}$ .

Average Rate of Change of a Function over a Given Interval

The average rate of change of a function f(x) over an interval [a, b] is given by:

\frac{f(b) - f(a)}{b - a}

This formula calculates the difference in the function values at the endpoints of the interval and divides it by the length of the interval.

Example Problem

Given the function defined in the table below, find the average rate of change of the function over the interval $6 \leq x \leq 18$ .

$x$	$f(x)$
6	49
12	39

Step 1: Identify the Interval

The interval given in the problem is $6 \leq x \leq 18$ . This means that we need to find the average rate of change of the function over this interval.

Step 2: Find the Function Values at the Endpoints

Step 3: Find the Function Value at $x = 18$

Step 4: Calculate the Average Rate of Change

Since we do not have the function value at $x = 18$ , we can only calculate the average rate of change over the interval $6 \leq x \leq 12$ . The average rate of change is given by:

\frac{f(12) - f(6)}{12 - 6}

Substituting the given function values, we get:

\frac{39 - 49}{12 - 6} = \frac{-10}{6} = -\frac{5}{3}

Conclusion

Average Rate of Change of a Function over a Given Interval

The average rate of change of a function f(x) over an interval [a, b] is given by:

\frac{f(b) - f(a)}{b - a}

This formula calculates the difference in the function values at the endpoints of the interval and divides it by the length of the interval.

Example Problem

Given the function defined in the table below, find the average rate of change of the function over the interval $6 \leq x \leq 18$ .

$x$	$f(x)$
6	49
12	39

Step 1: Identify the Interval

The interval given in the problem is $6 \leq x \leq 18$ . This means that we need to find the average rate of change of the function over this interval.

Step 2: Find the Function Values at the Endpoints

Step 3: Find the Function Value at $x = 18$

Step 4: Calculate the Average Rate of Change

Since we do not have the function value at $x = 18$ , we can only calculate the average rate of change over the interval $6 \leq x \leq 12$ . The average rate of change is given by:

\frac{f(12) - f(6)}{12 - 6}

Substituting the given function values, we get:

\frac{39 - 49}{12 - 6} = \frac{-10}{6} = -\frac{5}{3}

Conclusion

Average Rate of Change of a Function over a Given Interval

The average rate of change of a function f(x) over an interval [a, b] is given by:

\frac{f(b) - f(a)}{b - a}

This formula calculates the difference in the function values at the endpoints of the interval and divides it by the length of the interval.

Example Problem

Given the function defined in the table below, find the average rate of change of the function over the interval $6 \leq x \leq 18$ .

Q: What is the average rate of change of a function?

A: The average rate of change of a function f(x) over an interval [a, b] is a measure of how much the function changes on average over that interval. It is defined as:

\frac{f(b) - f(a)}{b - a}

Q: How do I find the average rate of change of a function?

A: To find the average rate of change of a function, you need to follow these steps:

Identify the interval over which you want to find the average rate of change.
Find the function values at the endpoints of the interval.
Use the formula for average rate of change to calculate the average rate of change.

Q: What if I don't have the function value at one of the endpoints?

A: If you don't have the function value at one of the endpoints, you can use the fact that the function is continuous to estimate the function value at that point. However, if you are not given any information about the function's behavior between the two endpoints, you will not be able to find the average rate of change over the entire interval.

Q: Can I find the average rate of change of a function over a non-linear interval?

A: Yes, you can find the average rate of change of a function over a non-linear interval. However, you will need to use the formula for average rate of change and divide the interval into smaller sub-intervals to find the average rate of change over each sub-interval.

Q: What is the difference between the average rate of change and the instantaneous rate of change?

A: The average rate of change of a function is a measure of how much the function changes on average over a given interval. The instantaneous rate of change of a function is a measure of how much the function changes at a specific point. The instantaneous rate of change is the limit of the average rate of change as the interval approaches zero.

Q: How do I use the average rate of change to solve problems?

A: The average rate of change of a function can be used to solve a variety of problems, including:

Finding the rate of change of a function over a given interval
Determining the behavior of a function over a given interval
Finding the maximum or minimum value of a function over a given interval

Q: What are some common applications of the average rate of change?

A: The average rate of change of a function has many common applications, including:

Physics: The average rate of change of a function can be used to describe the motion of an object over time.
Economics: The average rate of change of a function can be used to describe the rate of change of a quantity over time.
Engineering: The average rate of change of a function can be used to describe the rate of change of a quantity over time.

Q: What are some common mistakes to avoid when finding the average rate of change?

A: Some common mistakes to avoid when finding the average rate of change include:

Not identifying the interval over which you want to find the average rate of change
Not finding the function values at the endpoints of the interval
Not using the correct formula for average rate of change

Conclusion

In this article, we discussed the average rate of change of a function and how to find it. We also answered some common questions about the average rate of change and its applications. We hope that this article has been helpful in understanding the concept of average rate of change and how to use it to solve problems.