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

Introduction

In calculus, the average rate of change of a function is a fundamental concept that helps us understand how the function changes over a given interval. It's a measure of the rate at which the function's output changes with respect to the input. In this article, we'll explore the concept of average rate of change and how it's related to the function's values at specific points.

What is 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 calculates the difference in the function's output between the two points, divided by the difference in the input values.

Understanding the Given Information

We're 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

Now, let's analyze the given options:

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

This option suggests that the difference in the function's output between $x=7$ and $x=4$ is equal to the average rate of change. However, this is not necessarily true. The average rate of change is a measure of the rate of change over the entire interval, not just the difference in the function's output.

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

This option is actually the same as the given information. It's the formula for the average rate of change, and it's equal to $\frac{5}{6}$.

Conclusion

Based on the analysis, we can conclude that the correct statement is:

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

This statement is true because it's the formula for the average rate of change, and it's equal to $\frac{5}{6}$.

Example

Let's consider an example to illustrate this concept. Suppose we have a function $g(x) = x^2$, and we want to find the average rate of change between $x=4$ and $x=7$. We can plug in the values into the formula:

$$\frac{g(7) - g(4)}{7 - 4} = \frac{7^2 - 4^2}{7 - 4} = \frac{49 - 16}{3} = \frac{33}{3} = 11$$

However, we're given that the average rate of change is $\frac{5}{6}$. This means that:

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

We can see that the actual average rate of change is not equal to $\frac{5}{6}$, but rather $11$. This illustrates that the average rate of change is a measure of the rate of change over the entire interval, not just the difference in the function's output.

Key Takeaways

• The average rate of change of a function is a measure of the rate at which the function's output changes with respect to the input.
• The formula for the average rate of change is $\frac{g(b) - g(a)}{b - a}$.
• The average rate of change is not necessarily equal to the difference in the function's output between two points.
• The correct statement is B. $\frac{g(7-4)}{7-4} = \frac{5}{6}$.

Further Reading

If you're interested in learning more about the average rate of change and its applications, I recommend checking out the following resources:

• Calculus textbooks, such as "Calculus" by Michael Spivak or "Calculus: Early Transcendentals" by James Stewart.
• Online resources, such as Khan Academy's calculus course or MIT OpenCourseWare's calculus course.
• Research papers on the topic of average rate of change, such as "The Average Rate of Change of a Function" by David M. Bressoud.
  The Average Rate of Change: A Q&A Guide =============================================

Introduction

In our previous article, we explored the concept of the average rate of change of a function and how it's related to the function's values at specific points. In this article, we'll answer some frequently asked questions about the average rate of change to help you better understand this important concept.

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 rate at which the function's output changes with respect to the input. It's calculated using the formula:

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

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

A: To calculate the average rate of change, you need to know the function's values at two points, $a$ and $b$. You can then plug these values into the formula:

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

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 rate of change over a given interval, while the instantaneous rate of change is a measure of the rate of change at a single point. The instantaneous rate of change is calculated using the derivative of the function.

Q: Can the average rate of change be negative?

A: Yes, the average rate of change can be negative. This means that the function's output is decreasing over the given interval.

Q: Can the average rate of change be zero?

A: Yes, the average rate of change can be zero. This means that the function's output is not changing over the given interval.

Q: How is the average rate of change related to the function's graph?

A: The average rate of change is related to the function's graph in that it measures the slope of the line segment connecting two points on the graph. If the average rate of change is positive, the line segment will have a positive slope. If the average rate of change is negative, the line segment will have a negative slope.

Q: Can the average rate of change be used to make predictions about the function's behavior?

A: Yes, the average rate of change can be used to make predictions about the function's behavior. By analyzing the average rate of change over different intervals, you can gain insights into the function's behavior and make predictions about its future values.

Q: Are there any limitations to using the average rate of change?

A: Yes, there are limitations to using the average rate of change. For example, the average rate of change is only a measure of the rate of change over a given interval, and it may not accurately reflect the function's behavior at other points.

Q: How is the average rate of change used in real-world applications?

A: The average rate of change is used in a variety of real-world applications, including:

• Economics: to measure the rate of change of economic variables, such as GDP or inflation.
• Finance: to measure the rate of change of financial variables, such as stock prices or interest rates.
• Physics: to measure the rate of change of physical variables, such as velocity or acceleration.

Conclusion

In this article, we've answered some frequently asked questions about the average rate of change of a function. We've explored the concept of the average rate of change, how to calculate it, and its relationship to the function's graph. We've also discussed some of the limitations of using the average rate of change and its applications in real-world scenarios.

Key Takeaways

• The average rate of change is a measure of the rate at which the function's output changes with respect to the input.
• The formula for the average rate of change is $\frac{g(b) - g(a)}{b - a}$.
• The average rate of change can be positive, negative, or zero.
• The average rate of change is related to the function's graph in that it measures the slope of the line segment connecting two points on the graph.
• The average rate of change can be used to make predictions about the function's behavior.

Further Reading

If you're interested in learning more about the average rate of change and its applications, I recommend checking out the following resources:

• Calculus textbooks, such as "Calculus" by Michael Spivak or "Calculus: Early Transcendentals" by James Stewart.
• Online resources, such as Khan Academy's calculus course or MIT OpenCourseWare's calculus course.
• Research papers on the topic of average rate of change, such as "The Average Rate of Change of a Function" by David M. Bressoud.