Select The Correct Answer.Which Function Has An Average Rate Of Change Of -4 Over The Interval { [-2, 2]$} ? A . ?A. ? A . [ \begin{tabular}{|c|c|c|c|c|c|} \hline X X X & -2 & -1 & 0 & 1 & 2 \ \hline Q ( X ) Q(x) Q ( X ) & -4 & 0 & 0 & -4 & -12

Introduction

In mathematics, 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 essential concept in calculus, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the concept of average rate of change and how to calculate it. We will also use this concept to solve a problem involving a function with a given average rate of change.

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's values at the endpoints of the interval and divides it by the length of the interval. The result is the average rate of change of the function over the interval.

Calculating Average Rate of Change

To calculate the average rate of change of a function, we need to follow these steps:

1. Identify the function and the interval over which we want to calculate the average rate of change.
2. Calculate the function's values at the endpoints of the interval.
3. Subtract the function's value at the starting point from the function's value at the ending point.
4. Divide the result by the length of the interval.

Example Problem

Let's consider the following problem:

Which function has an average rate of change of -4 over the interval {[-2, 2]$}$?

To solve this problem, we need to find the function q(x) that satisfies the given condition. We will use the formula for average rate of change to calculate the function's values at the endpoints of the interval and then find the function that satisfies the given condition.

Step 1: Identify the Function and Interval

The function is q(x), and the interval is {[-2, 2]$}$.

Step 2: Calculate the Function's Values at the Endpoints

We are given the following table of values for the function q(x):

x	q(x)
-2	-4
-1	0
0	0
1	-4
2	-12

We can see that the function's values at the endpoints of the interval are q(-2) = -4 and q(2) = -12.

Step 3: Calculate the Difference in Function Values

We subtract the function's value at the starting point from the function's value at the ending point:

q(2) - q(-2) = -12 - (-4) = -8

Step 4: Calculate the Average Rate of Change

We divide the result by the length of the interval:

$$\frac{-8}{2 - (-2)} = \frac{-8}{4} = -2$$

However, we are given that the average rate of change is -4, not -2. This means that the function q(x) does not satisfy the given condition.

Step 5: Find the Correct Function

We need to find the function q(x) that satisfies the given condition. Let's try to find a function that has an average rate of change of -4 over the interval {[-2, 2]$}$.

We can start by assuming that the function q(x) is a linear function, which means that it can be written in the form q(x) = mx + b, where m is the slope and b is the y-intercept.

We know that the average rate of change of the function is -4, so we can set up the following equation:

$$\frac{q(2) - q(-2)}{2 - (-2)} = -4$$

Simplifying the equation, we get:

$$\frac{q(2) - q(-2)}{4} = -4$$

Multiplying both sides by 4, we get:

q(2) - q(-2) = -16

We know that q(-2) = -4, so we can substitute this value into the equation:

q(2) - (-4) = -16

Simplifying the equation, we get:

q(2) = -20

Now, we need to find the value of q(-2). We know that q(-2) = -4, so we can substitute this value into the equation:

q(2) = -20 q(-2) = -4

We can see that the function q(x) = -4x + 4 satisfies the given condition.

Conclusion

In this article, we explored the concept of average rate of change and how to calculate it. We used this concept to solve a problem involving a function with a given average rate of change. We found that the function q(x) = -4x + 4 satisfies the given condition.

Discussion

The concept of average rate of change is an essential tool in mathematics, and it has numerous applications in various fields. It is used to calculate the rate of change of a function over a given interval, which is essential in understanding the behavior of the function.

In this problem, we used the formula for average rate of change to calculate the function's values at the endpoints of the interval and then found the function that satisfies the given condition. We assumed that the function q(x) is a linear function, which means that it can be written in the form q(x) = mx + b, where m is the slope and b is the y-intercept.

We can see that the function q(x) = -4x + 4 satisfies the given condition, which means that it has an average rate of change of -4 over the interval {[-2, 2]$}$.

References

• Calculus: Early Transcendentals, 8th edition, by James Stewart
• Calculus, 9th edition, by Michael Spivak

Further Reading

• Calculus: A First Course, 8th edition, by Robert Adams
• Calculus: Early Transcendentals, 9th edition, by James Stewart

Glossary

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

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

• Linear function: A linear function is a function that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept.

FAQs

• What is the average rate of change of a function?

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

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

• How do I calculate the average rate of change of a function?

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

1. Identify the function and the interval over which you want to calculate the average rate of change.
2. Calculate the function's values at the endpoints of the interval.
3. Subtract the function's value at the starting point from the function's value at the ending point.
4. Divide the result by the length of the interval.

• What is the formula for average rate of change?

The formula for average rate of change is:

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

• What is a linear function?

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 defined as:

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

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

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

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

1. Identify the function and the interval over which you want to calculate the average rate of change.
2. Calculate the function's values at the endpoints of the interval.
3. Subtract the function's value at the starting point from the function's value at the ending point.
4. Divide the result by the length of the interval.

Q: What is the formula for average rate of change?

A: The formula for average rate of change is:

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

Q: What is a linear function?

A: A linear function is a function that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept.

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

A: To find the average rate of change of a linear function, you can use the formula:

$$\frac{f(b) - f(a)}{b - a} = m$$

where m is the slope of the linear function.

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

A: The average rate of change of a function is the rate of change of the function over a given interval, while the instantaneous rate of change is the rate of change of the function at a specific point.

Q: How do I calculate the instantaneous rate of change of a function?

A: To calculate the instantaneous rate of change of a function, you need to find the derivative of the function and evaluate it at the point of interest.

Q: What is the derivative of a function?

A: The derivative of a function f(x) is a new function that represents the rate of change of the original function. It is denoted by f'(x) and is calculated using the following formula:

$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

Q: How do I use the derivative to find the instantaneous rate of change of a function?

A: To use the derivative to find the instantaneous rate of change of a function, you need to evaluate the derivative at the point of interest. This will give you the rate of change of the function at that specific point.

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

A: The average rate of change of a function is related to the instantaneous rate of change of the function. As the interval over which the average rate of change is calculated gets smaller, the average rate of change approaches the instantaneous rate of change.

Q: How do I use the relationship between average rate of change and instantaneous rate of change to solve problems?

A: To use the relationship between average rate of change and instantaneous rate of change to solve problems, you need to understand how the average rate of change approaches the instantaneous rate of change as the interval gets smaller. This will allow you to use the average rate of change to estimate the instantaneous rate of change.

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

A: Average rate of change and instantaneous rate of change have numerous applications in various fields, including physics, engineering, economics, and computer science. Some common applications include:

• Calculating the rate of change of a function over a given interval
• Finding the maximum or minimum value of a function
• Determining the stability of a system
• Modeling real-world phenomena, such as population growth or chemical reactions

Q: How do I choose between average rate of change and instantaneous rate of change when solving a problem?

A: When choosing between average rate of change and instantaneous rate of change, you need to consider the specific problem and the information given. If you are given a specific interval over which to calculate the rate of change, you should use the average rate of change. If you are interested in the rate of change at a specific point, you should use the instantaneous rate of change.

Q: What are some common mistakes to avoid when working with average rate of change and instantaneous rate of change?

A: Some common mistakes to avoid when working with average rate of change and instantaneous rate of change include:

• Failing to identify the correct interval or point of interest
• Using the wrong formula or method to calculate the rate of change
• Not considering the relationship between average rate of change and instantaneous rate of change
• Not checking the units and dimensions of the rate of change

By understanding the concepts of average rate of change and instantaneous rate of change, you can solve a wide range of problems in mathematics and other fields.