Given The Function Defined In The Table Below, Find The Average Rate Of Change, In Simplest Form, Of The Function Over The Interval 4 ≤ X ≤ 5 4 \leq X \leq 5 4 ≤ X ≤ 5 . \[ \begin{tabular}{|c|c|} \hline X$ & F ( X ) F(x) F ( X ) \ \hline 2 & 22 \ \hline 3 & 15

Introduction

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 calculus and is used to describe the behavior of functions over different intervals. In this article, we will discuss how to find the average rate of change of a function over a given interval, using a specific function as an example.

The Function

The function we will be using is defined in the table below:

$x$	$f(x)$
2	22
3	15

We are asked to find the average rate of change of this function over the interval $4 \leq x \leq 5$.

Understanding the Concept of Average Rate of Change

The average rate of change of a function over an interval is defined as the difference between the function values at the endpoints of the interval, divided by the length of the interval. Mathematically, this can be represented as:

$$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$$

where $a$ and $b$ are the endpoints of the interval, and $f(a)$ and $f(b)$ are the function values at these points.

Finding the Average Rate of Change

To find the average rate of change of the function over the interval $4 \leq x \leq 5$, we need to first find the function values at the endpoints of the interval. However, we are only given the function values at $x = 2$ and $x = 3$. We cannot directly use these values to find the average rate of change over the interval $4 \leq x \leq 5$.

Extending the Function

To find the average rate of change over the interval $4 \leq x \leq 5$, we need to extend the function to include the points $x = 4$ and $x = 5$. We can do this by assuming that the function is linear between the given points.

Linear Interpolation

We can use linear interpolation to find the function values at $x = 4$ and $x = 5$. Linear interpolation is a method of estimating the value of a function at a point between two known points. The formula for linear interpolation is:

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

where $a$ and $b$ are the endpoints of the interval, and $f(a)$ and $f(b)$ are the function values at these points.

Finding the Function Values at $x = 4$ and $x = 5$

Using linear interpolation, we can find the function values at $x = 4$ and $x = 5$ as follows:

• To find the function value at $x = 4$, we use the formula:

$$f(4) = f(2) + \frac{(4 - 2)}{(3 - 2)}(f(3) - f(2))$$

$$f(4) = 22 + \frac{2}{1}(15 - 22)$$

$$f(4) = 22 + 2(-7)$$

$$f(4) = 22 - 14$$

$$f(4) = 8$$

• To find the function value at $x = 5$, we use the formula:

$$f(5) = f(3) + \frac{(5 - 3)}{(4 - 3)}(f(4) - f(3))$$

$$f(5) = 15 + \frac{2}{1}(8 - 15)$$

$$f(5) = 15 + 2(-7)$$

$$f(5) = 15 - 14$$

$$f(5) = 1$$

Finding the Average Rate of Change

Now that we have the function values at $x = 4$ and $x = 5$, we can find the average rate of change over the interval $4 \leq x \leq 5$ using the formula:

$$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$$

$$\text{Average Rate of Change} = \frac{f(5) - f(4)}{5 - 4}$$

$$\text{Average Rate of Change} = \frac{1 - 8}{1}$$

$$\text{Average Rate of Change} = -7$$

Conclusion

In this article, we discussed how to find the average rate of change of a function over a given interval. We used a specific function as an example and extended the function to include the points $x = 4$ and $x = 5$ using linear interpolation. We then found the average rate of change over the interval $4 \leq x \leq 5$ using the formula for average rate of change. The average rate of change of the function over the interval $4 \leq x \leq 5$ is $-7$.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
  Average Rate of Change: Frequently Asked Questions =====================================================

Introduction

In our previous article, we discussed how to find the average rate of change of a function over a given interval. In this article, we will answer some frequently asked questions about the average rate of change.

Q: What is the average rate of change?

A: 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 calculus and is used to describe the behavior of functions over different intervals.

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

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

1. Find the function values at the endpoints of the interval.
2. Use the formula for average rate of change:

$$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$$

where $a$ and $b$ are the endpoints of the interval, and $f(a)$ and $f(b)$ are the function values at these points.

Q: What if I don't have the function values at the endpoints of the interval?

A: If you don't have the function values at the endpoints of the interval, you can use linear interpolation to estimate the function values. Linear interpolation is a method of estimating the value of a function at a point between two known points.

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: 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 a function changes over a given interval, while the instantaneous rate of change is a measure of how much a function changes at a particular point in time. The instantaneous rate of change is a more precise measure of a function's behavior, but it requires the use of limits and derivatives.

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 comparing the average rates of change of different functions over the same interval, you can get an idea of how each function is changing relative to the others.

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

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

• Economics: The average rate of change can be used to model the behavior of economic systems, such as the rate of inflation or the rate of economic growth.
• Physics: The average rate of change can be used to model the behavior of physical systems, such as the rate of change of velocity or the rate of change of acceleration.
• Biology: The average rate of change can be used to model the behavior of biological systems, such as the rate of change of population growth or the rate of change of disease spread.

Conclusion

In this article, we answered some frequently asked questions about the average rate of change. We discussed how to find the average rate of change of a function, how to use linear interpolation to estimate function values, and how to compare the behavior of different functions. We also discussed some real-world applications of the average rate of change.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Mathematics for Economists, 2nd edition, Carl P. Simon and Lawrence Blume