Find The Average Rate Of Change Over The Given Interval.32. $f(x) = -x^3 + 4; \quad [0, 2]33. $g(x) = X^2 - X + 4; \quad [1, 2]For F ( X F(x F ( X ]:${ f(2) = -(2)^3 + 4 = -8 + 4 = -4 }$[ f(0) = -(0)^3 + 4 =

Introduction

In calculus, the average rate of change is a fundamental concept that helps us understand how a function changes over a given interval. It's a measure of the rate at which the function's output changes in response to changes in the input. In this article, we'll explore how to calculate the average rate of change for two given functions, $f(x)$ and $g(x)$, over the intervals $[0, 2]$ and $[1, 2]$, respectively.

Average Rate of Change Formula

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

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

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

Calculating Average Rate of Change for $f(x)$

Let's apply the formula to calculate the average rate of change for $f(x) = -x^3 + 4$ over the interval $[0, 2]$.

Step 1: Evaluate $f(2)$

To find the average rate of change, we need to evaluate the function at the endpoints of the interval. Let's start with $f(2)$.

$$f(2) = -(2)^3 + 4 = -8 + 4 = -4$$

Step 2: Evaluate $f(0)$

Next, we need to evaluate the function at the other endpoint of the interval, which is $x = 0$.

$$f(0) = -(0)^3 + 4 = 4$$

Step 3: Calculate the Average Rate of Change

Now that we have the values of $f(2)$ and $f(0)$, we can plug them into the formula to calculate the average rate of change.

$$\frac{f(2) - f(0)}{2 - 0} = \frac{-4 - 4}{2} = \frac{-8}{2} = -4$$

Therefore, the average rate of change of $f(x)$ over the interval $[0, 2]$ is $-4$.

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

Now, let's apply the same formula to calculate the average rate of change for $g(x) = x^2 - x + 4$ over the interval $[1, 2]$.

Step 1: Evaluate $g(2)$

To find the average rate of change, we need to evaluate the function at the endpoints of the interval. Let's start with $g(2)$.

$$g(2) = (2)^2 - 2 + 4 = 4 - 2 + 4 = 6$$

Step 2: Evaluate $g(1)$

Next, we need to evaluate the function at the other endpoint of the interval, which is $x = 1$.

$$g(1) = (1)^2 - 1 + 4 = 1 - 1 + 4 = 4$$

Step 3: Calculate the Average Rate of Change

Now that we have the values of $g(2)$ and $g(1)$, we can plug them into the formula to calculate the average rate of change.

$$\frac{g(2) - g(1)}{2 - 1} = \frac{6 - 4}{1} = \frac{2}{1} = 2$$

Therefore, the average rate of change of $g(x)$ over the interval $[1, 2]$ is $2$.

Conclusion

In this article, we've explored how to calculate the average rate of change for two given functions, $f(x)$ and $g(x)$, over the intervals $[0, 2]$ and $[1, 2]$, respectively. We've applied the formula for average rate of change and evaluated the functions at the endpoints of the intervals to find the average rate of change. The results show that the average rate of change of $f(x)$ over the interval $[0, 2]$ is $-4$, while the average rate of change of $g(x)$ over the interval $[1, 2]$ is $2$. This demonstrates the importance of understanding the average rate of change in calculus and its applications in various fields.

Discussion

• What is the average rate of change, and how is it calculated?
• How does the average rate of change relate to the concept of slope?
• Can you think of any real-world applications of the average rate of change?

References

• [1] Calculus: Early Transcendentals, James Stewart, 8th edition.
• [2] Calculus, Michael Spivak, 4th edition.

Additional Resources

• Khan Academy: Average Rate of Change
• MIT OpenCourseWare: Calculus I
• Wolfram Alpha: Average Rate of Change Calculator
  Average Rate of Change Q&A =============================

Frequently Asked Questions

Q: What is the average rate of change, and how is it calculated?

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

Q: How does the average rate of change relate to the concept of slope?

A: The average rate of change is closely related to the concept of slope. In fact, the average rate of change is equal to the slope of the secant line that passes through the points on the graph of the function at the endpoints of the interval.

Q: Can you think of any real-world applications of the average rate of change?

A: Yes, the average rate of change has many real-world applications. For example, it can be used to calculate the average speed of an object over a given time interval, the average rate of change of a population over time, or the average rate of change of a company's revenue over a given period.

Q: How do I calculate the average rate of change for a function that is not a linear function?

A: To calculate the average rate of change for a non-linear function, you can use the formula:

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

where $f(x)$ is the function, and $a$ and $b$ are the endpoints of the interval.

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

A: No, the average rate of change is not the same as the instantaneous rate of change. The instantaneous rate of change is a measure of the rate at which a function changes at a single point, while the average rate of change is a measure of the rate at which a function changes over a given interval.

Q: How do I use the average rate of change to solve problems in calculus?

A: The average rate of change is a fundamental concept in calculus, and it's used to solve a wide range of problems. For example, you can use the average rate of change to find the maximum or minimum value of a function, to determine the intervals on which a function is increasing or decreasing, or to calculate the area under a curve.

Q: Can I use the average rate of change to solve problems in physics?

A: Yes, the average rate of change is used extensively in physics to solve problems involving motion, velocity, and acceleration. For example, you can use the average rate of change to calculate the average speed of an object over a given time interval, or to determine the acceleration of an object over a given time interval.

Q: How do I apply the average rate of change to solve problems in economics?

A: The average rate of change is used in economics to solve problems involving growth rates, inflation rates, and interest rates. For example, you can use the average rate of change to calculate the average growth rate of a company's revenue over a given period, or to determine the average inflation rate over a given period.

Conclusion

The average rate of change is a fundamental concept in calculus that has many real-world applications. It's used to calculate the rate at which a function changes over a given interval, and it's a key tool for solving problems in physics, economics, and other fields. By understanding the 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 contexts.

Additional Resources

• Khan Academy: Average Rate of Change
• MIT OpenCourseWare: Calculus I
• Wolfram Alpha: Average Rate of Change Calculator
• Calculus: Early Transcendentals, James Stewart, 8th edition.
• Calculus, Michael Spivak, 4th edition.