Given The Function G ( X ) = 1 X G(x) = \frac{1}{x} G ( X ) = X 1 ​ , What Is The Average Rate Of Change Of G ( X G(x G ( X ] With Respect To X X X Between X = 2 X = 2 X = 2 And X = 4 X = 4 X = 4 ?A. − 1 4 -\frac{1}{4} − 4 1 ​ B. − 1 8 -\frac{1}{8} − 8 1 ​ C.

Introduction

In calculus, the average rate of change of a function is a measure of how much the function changes over a given interval. It is an important concept in understanding the behavior of functions and is used in various applications in physics, engineering, and economics. In this article, we will discuss how to calculate the average rate of change of a function using the given function $g(x) = \frac{1}{x}$ as an example.

What is the Average Rate of Change?

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

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

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

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

To calculate the average rate of change of $g(x) = \frac{1}{x}$ between $x = 2$ and $x = 4$, we need to find the values of $g(2)$ and $g(4)$.

Finding $g(2)$ and $g(4)$

To find $g(2)$, we substitute $x = 2$ into the function $g(x) = \frac{1}{x}$:

$$g(2) = \frac{1}{2}$$

To find $g(4)$, we substitute $x = 4$ into the function $g(x) = \frac{1}{x}$:

$$g(4) = \frac{1}{4}$$

Calculating the Average Rate of Change

Now that we have the values of $g(2)$ and $g(4)$, we can calculate the average rate of change of $g(x)$ between $x = 2$ and $x = 4$ using the formula:

$$\frac{\Delta g}{\Delta x} = \frac{g(4) - g(2)}{4 - 2}$$

Substituting the values of $g(2)$ and $g(4)$, we get:

$$\frac{\Delta g}{\Delta x} = \frac{\frac{1}{4} - \frac{1}{2}}{2}$$

Simplifying the expression, we get:

$$\frac{\Delta g}{\Delta x} = \frac{-\frac{1}{4}}{2} = -\frac{1}{8}$$

Conclusion

In this article, we discussed how to calculate the average rate of change of a function using the given function $g(x) = \frac{1}{x}$ as an example. We found the values of $g(2)$ and $g(4)$, and then calculated the average rate of change of $g(x)$ between $x = 2$ and $x = 4$. The result was $-\frac{1}{8}$.

Average Rate of Change Formula

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

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

Step-by-Step Solution

1. Find the values of $f(a)$ and $f(b)$.
2. Substitute the values of $f(a)$ and $f(b)$ into the formula.
3. Simplify the expression to find the average rate of change.

Example Problems

1. Find the average rate of change of $f(x) = 2x^2$ between $x = 1$ and $x = 3$.
2. Find the average rate of change of $g(x) = \frac{1}{x}$ between $x = 2$ and $x = 5$.

Answer Key

1. $\frac{\Delta f}{\Delta x} = \frac{f(3) - f(1)}{3 - 1} = \frac{2(3)^2 - 2(1)^2}{2} = 5$
2. $\frac{\Delta g}{\Delta x} = \frac{g(5) - g(2)}{5 - 2} = \frac{\frac{1}{5} - \frac{1}{2}}{3} = -\frac{1}{30}$
   Q&A: Average Rate of Change =============================

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the average rate of change of a function.

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 over a given interval. It is defined as:

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

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:

1. Find the values of $f(a)$ and $f(b)$.
2. Substitute the values of $f(a)$ and $f(b)$ into the formula.
3. Simplify the expression to find 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 how much the function changes over a given interval, while the instantaneous rate of change is a measure of how much the function changes at a specific point.

Q: Can I use the average rate of change formula for any type of function?

A: Yes, you can use the average rate of change formula for any type of function, including linear, quadratic, polynomial, and rational functions.

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

A: The average rate of change formula can be applied in various real-world problems, such as:

• Calculating the average speed of an object over a given time interval
• Determining the average rate of change of a population over a given time period
• Finding the average rate of change of a stock price over a given time interval

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:

• Not using the correct formula
• Not substituting the correct values into the formula
• Not simplifying the expression correctly

Q: Can I use a calculator to calculate the average rate of change?

A: Yes, you can use a calculator to calculate the average rate of change. However, it is always a good idea to double-check your work by hand to ensure accuracy.

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

A: To graph the average rate of change of a function, you can use a graphing calculator or a computer program. You can also use a table of values to create a graph.

Q: What are some real-world applications of the average rate of change?

A: Some real-world applications of the average rate of change include:

• Calculating the average speed of an object over a given time interval
• Determining the average rate of change of a population over a given time period
• Finding the average rate of change of a stock price over a given time interval

Conclusion

In this article, we answered some frequently asked questions about the average rate of change of a function. We discussed the definition of the average rate of change, how to calculate it, and some common mistakes to avoid. We also explored some real-world applications of the average rate of change and how to graph it.

Additional Resources

For more information on the average rate of change, you can consult the following resources:

• Calculus textbooks
• Online tutorials and videos
• Graphing calculators and computer programs

Practice Problems

1. Find the average rate of change of $f(x) = 2x^2$ between $x = 1$ and $x = 3$.
2. Find the average rate of change of $g(x) = \frac{1}{x}$ between $x = 2$ and $x = 5$.
3. Find the average rate of change of $h(x) = x^3$ between $x = 0$ and $x = 2$.

Answer Key

1. $\frac{\Delta f}{\Delta x} = \frac{f(3) - f(1)}{3 - 1} = \frac{2(3)^2 - 2(1)^2}{2} = 5$
2. $\frac{\Delta g}{\Delta x} = \frac{g(5) - g(2)}{5 - 2} = \frac{\frac{1}{5} - \frac{1}{2}}{3} = -\frac{1}{30}$
3. $\frac{\Delta h}{\Delta x} = \frac{h(2) - h(0)}{2 - 0} = \frac{2^3 - 0^3}{2} = 4$