Calculating Average Rates of Change

Oct 13, 2025 by windailys.com 148 views

Introduction

When dealing with functions, understanding their behavior and characteristics is crucial. One such characteristic is the average rate of change, which represents the rate at which the function changes over a given interval. In this article, we will explore the functions $f(x)$ , $g(x)$ , and $h(x)$ and determine the ordering of their average rates of change on the interval $-4 \leq x \leq 5$ .

The Functions $f(x)$ , $g(x)$ , and $h(x)$

The functions $f(x)$ , $g(x)$ , and $h(x)$ are defined as follows:

$f(x) = 2x^2 + 3x - 1$
$g(x) = x^2 + 2x - 3$
$h(x) = 2x^2 - 3x + 1$

Calculating Average Rates of Change

To determine the ordering of the functions according to their average rates of change, we need to calculate the average rate of change for each function on the interval $-4 \leq x \leq 5$ . The average rate of change is given by the formula:

\frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

where $x_1$ and $x_2$ are the endpoints of the interval.

Calculating Average Rate of Change for $f(x)$

To calculate the average rate of change for $f(x)$ , we need to find the values of $f(-4)$ and $f(5)$ .

f(-4) = 2(-4)^2 + 3(-4) - 1 = 32 - 12 - 1 = 19

f(5) = 2(5)^2 + 3(5) - 1 = 50 + 15 - 1 = 64

Now, we can calculate the average rate of change for $f(x)$ :

\frac{\Delta y}{\Delta x} = \frac{f(5) - f(-4)}{5 - (-4)} = \frac{64 - 19}{9} = \frac{45}{9} = 5

Calculating Average Rate of Change for $g(x)$

To calculate the average rate of change for $g(x)$ , we need to find the values of $g(-4)$ and $g(5)$ .

g(-4) = (-4)^2 + 2(-4) - 3 = 16 - 8 - 3 = 5

g(5) = (5)^2 + 2(5) - 3 = 25 + 10 - 3 = 32

Now, we can calculate the average rate of change for $g(x)$ :

\frac{\Delta y}{\Delta x} = \frac{g(5) - g(-4)}{5 - (-4)} = \frac{32 - 5}{9} = \frac{27}{9} = 3

Calculating Average Rate of Change for $h(x)$

To calculate the average rate of change for $h(x)$ , we need to find the values of $h(-4)$ and $h(5)$ .

h(-4) = 2(-4)^2 - 3(-4) + 1 = 32 + 12 + 1 = 45

h(5) = 2(5)^2 - 3(5) + 1 = 50 - 15 + 1 = 36

Now, we can calculate the average rate of change for $h(x)$ :

\frac{\Delta y}{\Delta x} = \frac{h(5) - h(-4)}{5 - (-4)} = \frac{36 - 45}{9} = \frac{-9}{9} = -1

Ordering of Average Rates of Change

Now that we have calculated the average rates of change for each function, we can determine the ordering of the functions according to their average rates of change on the interval $-4 \leq x \leq 5$ . The average rates of change are as follows:

$f(x)$ : 5
$g(x)$ : 3
$h(x)$ : -1

Therefore, the ordering of the functions according to their average rates of change is:

$h(x)$
$g(x)$
$f(x)$

Conclusion

In conclusion, the functions $f(x)$ , $g(x)$ , and $h(x)$ have different average rates of change on the interval $-4 \leq x \leq 5$ . The ordering of the functions according to their average rates of change is $h(x)$ , $g(x)$ , and $f(x)$ . This demonstrates the importance of understanding the behavior and characteristics of functions, including their average rates of change.

References

[1] Calculus: Early Transcendentals, James Stewart, 8th edition
[2] Calculus, Michael Spivak, 4th edition
[3] Functions, Graphs, and Analytic Geometry, Michael Spivak, 4th edition

Discussion

The ordering of the functions according to their average rates of change is a fundamental concept in calculus. It is essential to understand the behavior and characteristics of functions, including their average rates of change, to make informed decisions in various fields, such as science, engineering, and economics.

In this article, we have demonstrated the importance of calculating the average rate of change for functions and determining the ordering of the functions according to their average rates of change. We have also provided examples of how to calculate the average rate of change for different functions.

We hope that this article has provided valuable insights into the functions $f(x)$ , $g(x)$ , and $h(x)$ and their average rates of change on the interval $-4 \leq x \leq 5$ . We encourage readers to explore further and apply the concepts learned in this article to real-world problems.

Introduction

In our previous article, we explored the functions $f(x)$ , $g(x)$ , and $h(x)$ and determined the ordering of their average rates of change on the interval $-4 \leq x \leq 5$ . In this article, we will answer some frequently asked questions related to the functions $f(x)$ , $g(x)$ , and $h(x)$ and their average rates of change.

Q&A

Q: What is the average rate of change?

A: The average rate of change is a measure of how much a function changes over a given interval. It is calculated by finding the difference in the function's values at the endpoints of the interval and dividing by the difference in the x-values.

Q: How do you calculate the average rate of change for a function?

A: To calculate the average rate of change for a function, you need to find the values of the function at the endpoints of the interval and then divide the difference in the function's values by the difference in the x-values.

Q: What is the significance of the average rate of change?

A: The average rate of change is significant because it helps us understand how a function changes over a given interval. It is used in various fields, such as science, engineering, and economics, to make informed decisions.

Q: Can you provide an example of how to calculate the average rate of change for a function?

A: Yes, let's consider the function $f(x) = 2x^2 + 3x - 1$ . To calculate the average rate of change for $f(x)$ on the interval $-4 \leq x \leq 5$ , we need to find the values of $f(-4)$ and $f(5)$ .

f(-4) = 2(-4)^2 + 3(-4) - 1 = 32 - 12 - 1 = 19

f(5) = 2(5)^2 + 3(5) - 1 = 50 + 15 - 1 = 64

Now, we can calculate the average rate of change for $f(x)$ :

\frac{\Delta y}{\Delta x} = \frac{f(5) - f(-4)}{5 - (-4)} = \frac{64 - 19}{9} = \frac{45}{9} = 5

Q: How do you determine the ordering of the functions according to their average rates of change?

A: To determine the ordering of the functions according to their average rates of change, you need to calculate the average rate of change for each function and then compare the values. The function with the smallest average rate of change will be at the bottom of the list, and the function with the largest average rate of change will be at the top.

Q: Can you provide an example of how to determine the ordering of the functions according to their average rates of change?

A: Yes, let's consider the functions $f(x)$ , $g(x)$ , and $h(x)$ . We have already calculated the average rates of change for each function:

$f(x)$ : 5
$g(x)$ : 3
$h(x)$ : -1

Therefore, the ordering of the functions according to their average rates of change is:

$h(x)$
$g(x)$
$f(x)$

Conclusion

In conclusion, the functions $f(x)$ , $g(x)$ , and $h(x)$ have different average rates of change on the interval $-4 \leq x \leq 5$ . The ordering of the functions according to their average rates of change is $h(x)$ , $g(x)$ , and $f(x)$ . We hope that this article has provided valuable insights into the functions $f(x)$ , $g(x)$ , and $h(x)$ and their average rates of change.

References

[1] Calculus: Early Transcendentals, James Stewart, 8th edition
[2] Calculus, Michael Spivak, 4th edition
[3] Functions, Graphs, and Analytic Geometry, Michael Spivak, 4th edition

Discussion

We hope that this article has provided valuable insights into the functions $f(x)$ , $g(x)$ , and $h(x)$ and their average rates of change. We encourage readers to explore further and apply the concepts learned in this article to real-world problems.