The Average Rate Of Change Of G ( X G(x G ( X ] Between X = 4 X=4 X = 4 And X = 7 X=7 X = 7 Is 5 6 \frac{5}{6} 6 5 . Which Statement Must Be True?A. G ( 7 ) − G ( 4 ) = 5 6 G(7)-g(4)=\frac{5}{6} G ( 7 ) − G ( 4 ) = 6 5 B. 9 ( 7 − 4 ) 7 − 4 = 5 6 \frac{9(7-4)}{7-4}=\frac{5}{6} 7 − 4 9 ( 7 − 4 ) = 6 5 C.

Mar 12, 2025 by ADMIN 312 views

**The Average Rate of Change of a Function**

Understanding the Average Rate of Change

The average rate of change of a function $g(x)$ between two points $x=a$ and $x=b$ is given by the formula:

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

This formula represents the average rate at which the function changes as $x$ moves from $a$ to $b$ .

Given Information

We are given that the average rate of change of $g(x)$ between $x=4$ and $x=7$ is $\frac{5}{6}$ . This means that:

\frac{g(7)-g(4)}{7-4}=\frac{5}{6}

Analyzing the Options

Let's analyze the given options:

A. $g(7)-g(4)=\frac{5}{6}$

This option is incorrect because the average rate of change is given by the formula $\frac{g(b)-g(a)}{b-a}$ , not just $g(b)-g(a)$ .

B. $\frac{9(7-4)}{7-4}=\frac{5}{6}$

This option is incorrect because the left-hand side of the equation is not equal to the right-hand side. The correct calculation is:

\frac{9(7-4)}{7-4}=\frac{9(3)}{3}=9

C. $\frac{g(7)-g(4)}{7-4}=\frac{5}{6}$

This option is correct because it is the definition of the average rate of change of a function.

Conclusion

Based on the analysis of the options, the correct statement that must be true is:

C. $\frac{g(7)-g(4)}{7-4}=\frac{5}{6}$

This statement is true because it is the definition of the average rate of change of a function.

Understanding the Concept of Average Rate of Change

The average rate of change of a function is an important concept in calculus that helps us understand how a function changes as the input variable changes. It is a measure of the average rate at which the function changes over a given interval.

Calculating the Average Rate of Change

To calculate the average rate of change of a function, we use the formula:

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

This formula represents the average rate at which the function changes as $x$ moves from $a$ to $b$ .

Example

Suppose we want to find the average rate of change of the function $g(x)=x^2$ between $x=2$ and $x=5$ . We can use the formula:

\frac{g(5)-g(2)}{5-2}=\frac{5^2-2^2}{5-2}=\frac{25-4}{3}=\frac{21}{3}=7

Therefore, the average rate of change of the function $g(x)=x^2$ between $x=2$ and $x=5$ is $7$ .

Real-World Applications

The concept of average rate of change has many real-world applications. For example, it can be used to calculate the average rate of change of a company's revenue over a given period of time. It can also be used to calculate the average rate of change of a population's growth rate over a given period of time.

Conclusion

In conclusion, the average rate of change of a function is an important concept in calculus that helps us understand how a function changes as the input variable changes. It is a measure of the average rate at which the function changes over a given interval. We can calculate the average rate of change of a function using the formula:

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

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

A: The average rate of change of a function is a measure of the average rate at which the function changes over a given interval. It is calculated using the formula:

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

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:

Identify the function and the interval over which you want to calculate the average rate of change.
Plug the values of the function at the endpoints of the interval into the formula.
Simplify the expression to get the average rate of change.

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

A: The average rate of change is a measure of the average rate at which a function changes over a given interval, while the instantaneous rate of change is a measure of the rate at which a function changes at a specific point. The instantaneous rate of change is calculated using the derivative of the function.

Q: Can I use the average rate of change to predict the future behavior of a function?

A: While the average rate of change can provide some insight into the behavior of a function, it is not a reliable method for predicting the future behavior of a function. The average rate of change is a measure of the average rate of change over a given interval, and it does not take into account any changes that may occur outside of that interval.

Q: How do I use the average rate of change in real-world applications?

A: The average rate of change has many real-world applications, including:

Calculating the average rate of change of a company's revenue over a given period of time.
Calculating the average rate of change of a population's growth rate over a given period of time.
Calculating the average rate of change of a physical system, such as a spring or a pendulum.

Q: Can I use the average rate of change to compare the behavior of different functions?

A: Yes, you can use the average rate of change to compare the behavior of different functions. By calculating the average rate of change of each function over the same interval, you can compare the rates at which they change.

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

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

Failing to identify the correct interval over which to calculate the average rate of change.
Failing to plug the correct values of the function into the formula.
Failing to simplify the expression to get the average rate of change.

Q: Can I use the average rate of change to find the instantaneous rate of change of a function?

A: No, you cannot use the average rate of change to find the instantaneous rate of change of a function. The average rate of change is a measure of the average rate of change over a given interval, while the instantaneous rate of change is a measure of the rate at which a function changes at a specific point. The instantaneous rate of change is calculated using the derivative of the function.

Conclusion

In conclusion, the average rate of change is an important concept in calculus that helps us understand how a function changes over a given interval. It is a measure of the average rate at which the function changes, and it can be used to compare the behavior of different functions. By following the steps outlined in this article, you can calculate the average rate of change of a function and use it to gain insight into its behavior.