For The Following Exercise, Find The Average Rate Of Change Of The Function On The Specified Interval.${ K(t) = 5t^2 + \frac{2}{t^3} \text{ On } [-3, 2] }$

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 such as physics, engineering, and economics. In this article, we will discuss how to find the average rate of change of a function on a specified interval.

What is Average Rate of Change?

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

${ \text{Average Rate of Change} = \frac{\text{Change in } f(x)}{\text{Change in } x} }$

This can be written mathematically as:

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

Finding the Average Rate of Change

To find the average rate of change of a function on a specified interval, we need to follow these steps:

1. Find the values of the function at the endpoints of the interval: We need to find the values of the function at the endpoints of the interval, which are a and b.
2. Calculate the change in the function: We need to calculate the change in the function, which is f(b) - f(a).
3. Calculate the change in x: We need to calculate the change in x, which is b - a.
4. Calculate the average rate of change: We need to calculate the average rate of change by dividing the change in the function by the change in x.

Example

Let's consider the function k(t) = 5t^2 + 2/t^3 on the interval [-3, 2]. We need to find the average rate of change of this function on this interval.

Step 1: Find the values of the function at the endpoints of the interval

To find the values of the function at the endpoints of the interval, we need to substitute the values of t into the function.

For t = -3:

${ k(-3) = 5(-3)^2 + \frac{2}{(-3)^3} }$ ${ k(-3) = 5(9) + \frac{2}{-27} }$ ${ k(-3) = 45 - \frac{2}{27} }$ ${ k(-3) = 45 - \frac{2}{27} }$ ${ k(-3) = 45 - 0.074 }$ ${ k(-3) = 44.926 }$

For t = 2:

${ k(2) = 5(2)^2 + \frac{2}{(2)^3} }$ ${ k(2) = 5(4) + \frac{2}{8} }$ ${ k(2) = 20 + \frac{1}{4} }$ ${ k(2) = 20 + 0.25 }$ ${ k(2) = 20.25 }$

Step 2: Calculate the change in the function

To calculate the change in the function, we need to subtract the value of the function at the lower endpoint from the value of the function at the upper endpoint.

${ \text{Change in } k(t) = k(2) - k(-3) }$ ${ \text{Change in } k(t) = 20.25 - 44.926 }$ ${ \text{Change in } k(t) = -24.676 }$

Step 3: Calculate the change in x

To calculate the change in x, we need to subtract the lower endpoint from the upper endpoint.

${ \text{Change in } x = 2 - (-3) }$ ${ \text{Change in } x = 5 }$

Step 4: Calculate the average rate of change

To calculate the average rate of change, we need to divide the change in the function by the change in x.

${ \text{Average Rate of Change} = \frac{\text{Change in } k(t)}{\text{Change in } x} }$ ${ \text{Average Rate of Change} = \frac{-24.676}{5} }$ ${ \text{Average Rate of Change} = -4.9352 }$

Conclusion

In this article, we discussed how to find the average rate of change of a function on a specified interval. We used the function k(t) = 5t^2 + 2/t^3 on the interval [-3, 2] as an example. We found the values of the function at the endpoints of the interval, calculated the change in the function, calculated the change in x, and finally calculated the average rate of change. The average rate of change is an important concept in calculus and is used in various applications.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Calculus, 1st edition, Michael Spivak

Further Reading

• [1] Average Rate of Change, Wolfram MathWorld
• [2] Average Rate of Change, Khan Academy
• [3] Average Rate of Change, MIT OpenCourseWare
  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 on a specified 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 f(x) on an interval [a, b] is a measure of how much the function changes over the interval. It is defined as:

${ \text{Average Rate of Change} = \frac{\text{Change in } f(x)}{\text{Change in } x} }$

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

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

1. Find the values of the function at the endpoints of the interval: You need to find the values of the function at the endpoints of the interval, which are a and b.
2. Calculate the change in the function: You need to calculate the change in the function, which is f(b) - f(a).
3. Calculate the change in x: You need to calculate the change in x, which is b - a.
4. Calculate the average rate of change: You need to calculate the average rate of change by dividing the change in the function by the change in x.

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. The instantaneous rate of change is also known as the derivative of the function.

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

A: No, you cannot use the average rate of change to find the instantaneous rate of change. The average rate of change is an approximation of the instantaneous rate of change, but it is not the same thing.

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:

• Physics: The average rate of change is used to describe the motion of objects, such as the velocity of a car or the acceleration of a ball.
• Engineering: The average rate of change is used to design and optimize systems, such as the flow rate of a pipe or the speed of a conveyor belt.
• Economics: The average rate of change is used to analyze economic data, such as the rate of inflation or the rate of unemployment.

Q: Can I use the average rate of change to find the maximum or minimum value of a function?

A: No, you cannot use the average rate of change to find the maximum or minimum value of a function. The average rate of change is a measure of how much the function changes over a given interval, but it does not provide information about the maximum or minimum value of the function.

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

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

• Not using the correct formula: Make sure to use the correct formula for the average rate of change, which is:

${ \text{Average Rate of Change} = \frac{\text{Change in } f(x)}{\text{Change in } x} }$

• Not calculating the change in the function correctly: Make sure to calculate the change in the function correctly, which is f(b) - f(a).
• Not calculating the change in x correctly: Make sure to calculate the change in x correctly, which is b - a.

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, the difference between the average rate of change and the instantaneous rate of change, and some real-world applications of the average rate of change. We also discussed some common mistakes to avoid when finding the average rate of change.