Select The Correct Answer.Consider Function G G G . G ( X ) = 5 X − 1 + 2 G(x)=\frac{5}{x-1}+2 G ( X ) = X − 1 5 ​ + 2 What Is The Average Rate Of Change Of Function G G G Over The Interval − 4 , 3 {-4,3} − 4 , 3 ?A. − 1 2 -\frac{1}{2} − 2 1 ​ B. 1 2 \frac{1}{2} 2 1 ​ C.

Understanding the Concept of Average Rate of Change

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 calculate the average rate of change of a function and apply this concept to a specific function, $g(x)=\frac{5}{x-1}+2$.

What is the Average Rate of Change?

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

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

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

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

To calculate the average rate of change of $g(x)=\frac{5}{x-1}+2$ over the interval $[-4,3]$, we need to follow these steps:

1. Evaluate the function at the endpoints of the interval: We need to find the values of $g(-4)$ and $g(3)$.
2. Calculate the difference in the function values: We will subtract $g(-4)$ from $g(3)$ to get the difference in the function values.
3. Calculate the difference in the input values: We will subtract $-4$ from $3$ to get the difference in the input values.
4. Divide the difference in the function values by the difference in the input values: We will divide the result from step 2 by the result from step 3 to get the average rate of change.

Evaluating the Function at the Endpoints

To evaluate the function at the endpoints, we need to plug in the values of $x$ into the function.

• Evaluating $g(-4)$: We plug in $x=-4$ into the function:

$$g(-4) = \frac{5}{-4-1}+2 = \frac{5}{-5}+2 = -1+2 = 1$$

• Evaluating $g(3)$: We plug in $x=3$ into the function:

$$g(3) = \frac{5}{3-1}+2 = \frac{5}{2}+2 = 2.5+2 = 4.5$$

Calculating the Difference in the Function Values

We subtract $g(-4)$ from $g(3)$ to get the difference in the function values:

$$g(3) - g(-4) = 4.5 - 1 = 3.5$$

Calculating the Difference in the Input Values

We subtract $-4$ from $3$ to get the difference in the input values:

$$3 - (-4) = 3 + 4 = 7$$

Calculating the Average Rate of Change

We divide the difference in the function values by the difference in the input values to get the average rate of change:

$$\frac{g(3) - g(-4)}{3 - (-4)} = \frac{3.5}{7} = \frac{7}{14} = \frac{1}{2}$$

Conclusion

In this article, we discussed how to calculate the average rate of change of a function and applied this concept to a specific function, $g(x)=\frac{5}{x-1}+2$. We evaluated the function at the endpoints of the interval, calculated the difference in the function values, calculated the difference in the input values, and finally divided the difference in the function values by the difference in the input values to get the average rate of change. The average rate of change of $g(x)$ over the interval $[-4,3]$ is $\frac{1}{2}$.

Answer

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

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

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. Evaluate the function at the endpoints of the interval.
2. Calculate the difference in the function values.
3. Calculate the difference in the input values.
4. Divide the difference in the function values by the difference in the input values.

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

A: The formula for calculating the average rate of change is:

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

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

A: The average rate of change can give you an idea of how a function is changing over a given interval, but it is not a reliable method for predicting the future behavior of a function. The average rate of change is a snapshot of the function's behavior at a particular point in time, and it does not take into account any changes that may occur in the future.

Q: How do I apply the concept of average rate of change to real-world problems?

A: The concept of average rate of change can be applied to a wide range of real-world problems, such as:

• Calculating the average rate of change of a company's revenue over a given period of time.
• Determining the average rate of change of a population's growth over a given period of time.
• Calculating the average rate of change of a stock's price over a given period of time.

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 evaluate the function at the endpoints of the interval.
• Failing to calculate the difference in the function values.
• Failing to calculate the difference in the input values.
• Dividing by zero.

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

A: Yes, the average rate of change can be used to compare the behavior of different functions. By calculating the average rate of change of each function over the same interval, you can determine which function is changing more rapidly.

Q: How do I interpret the results of an average rate of change calculation?

A: When interpreting the results of an average rate of change calculation, you should consider the following:

• A positive average rate of change indicates that the function is increasing over the given interval.
• A negative average rate of change indicates that the function is decreasing over the given interval.
• A zero average rate of change indicates that the function is not changing over the given interval.

Conclusion

In this article, we have discussed some frequently asked questions about the average rate of change of a function. We have covered topics such as how to calculate the average rate of change, how to apply the concept to real-world problems, and how to interpret the results of an average rate of change calculation. By understanding the concept of average rate of change, you can gain a deeper understanding of the behavior of functions and make more informed decisions in a wide range of fields.