Given F ( X ) = 17 − X 2 F(x) = 17 - X^2 F ( X ) = 17 − X 2 , What Is The Average Rate Of Change In F ( X F(x F ( X ] Over The Interval 1 , 5 {1, 5} 1 , 5 ?A. − 6 -6 − 6 B. − 1 2 -\frac{1}{2} − 2 1 ​ C. 1 4 \frac{1}{4} 4 1 ​ D. 1 1 1

Introduction

In mathematics, 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 calculate the average rate of change of the function $f(x) = 17 - x^2$ over the interval $[1, 5]$.

Understanding the Concept of Average Rate of Change

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

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

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

Calculating the Average Rate of Change of $f(x) = 17 - x^2$

To calculate the average rate of change of $f(x) = 17 - x^2$ over the interval $[1, 5]$, we need to find the values of $f(1)$ and $f(5)$.

Finding $f(1)$

To find $f(1)$, we substitute $x = 1$ into the function:

$$f(1) = 17 - (1)^2 = 17 - 1 = 16$$

Finding $f(5)$

To find $f(5)$, we substitute $x = 5$ into the function:

$$f(5) = 17 - (5)^2 = 17 - 25 = -8$$

Calculating the Average Rate of Change

Now that we have found $f(1)$ and $f(5)$, we can calculate the average rate of change of $f(x) = 17 - x^2$ over the interval $[1, 5]$:

$$\frac{f(5) - f(1)}{5 - 1} = \frac{-8 - 16}{4} = \frac{-24}{4} = -6$$

Conclusion

In this article, we calculated the average rate of change of the function $f(x) = 17 - x^2$ over the interval $[1, 5]$. We found that the average rate of change is $-6$. This means that on average, the function decreases by $6$ units over the interval $[1, 5]$.

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 can be used to calculate the average rate of change of any function over a given interval.

Example Problems

• Find the average rate of change of the function $f(x) = 2x^2 - 3x + 1$ over the interval $[0, 2]$.
• Find the average rate of change of the function $f(x) = x^3 - 2x^2 + x - 1$ over the interval $[1, 3]$.

Solutions

• To find the average rate of change of the function $f(x) = 2x^2 - 3x + 1$ over the interval $[0, 2]$, we need to find the values of $f(0)$ and $f(2)$.
• To find the average rate of change of the function $f(x) = x^3 - 2x^2 + x - 1$ over the interval $[1, 3]$, we need to find the values of $f(1)$ and $f(3)$.

Step-by-Step Solutions

• To find the average rate of change of the function $f(x) = 2x^2 - 3x + 1$ over the interval $[0, 2]$, we need to follow these steps:
  1. Find $f(0)$ by substituting $x = 0$ into the function.
  2. Find $f(2)$ by substituting $x = 2$ into the function.
  3. Calculate the average rate of change using the formula: $\frac{f(2) - f(0)}{2 - 0}$.
• To find the average rate of change of the function $f(x) = x^3 - 2x^2 + x - 1$ over the interval $[1, 3]$, we need to follow these steps:
  1. Find $f(1)$ by substituting $x = 1$ into the function.
  2. Find $f(3)$ by substituting $x = 3$ into the function.
  3. Calculate the average rate of change using the formula: $\frac{f(3) - f(1)}{3 - 1}$.

Final Answers

• The average rate of change of the function $f(x) = 2x^2 - 3x + 1$ over the interval $[0, 2]$ is $\frac{f(2) - f(0)}{2 - 0}$.
• The average rate of change of the function $f(x) = x^3 - 2x^2 + x - 1$ over the interval $[1, 3]$ is $\frac{f(3) - f(1)}{3 - 1}$.

Conclusion

$$$$$$$$$$

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 length of the interval.

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 the function at the endpoints of the interval.
2. Calculate the difference in the function values at the endpoints.
3. Divide the difference by the length of the interval.

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

A: The formula for the average rate of change of a function is:

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

Q: How do I apply the formula for the average rate of change of a function?

A: To apply the formula, you need to substitute the values of the function at the endpoints of the interval into the formula and simplify.

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

A: The average rate of change of a function is an important concept in calculus and is used to describe the behavior of functions. It helps us understand how the function changes over a given interval and can be used to make predictions about the function's behavior.

Q: Can the average rate of change of a function be positive or negative?

A: Yes, the average rate of change of a function can be positive or negative. If the function increases over the interval, the average rate of change will be positive. If the function decreases over the interval, the average rate of change will be negative.

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

A: To determine the sign of the average rate of change of a function, you need to examine the behavior of the function over the interval. If the function increases over the interval, the average rate of change will be positive. If the function decreases over the interval, the average rate of change will be negative.

Q: Can the average rate of change of a function be zero?

A: Yes, the average rate of change of a function can be zero. This occurs when the function does not change over the interval.

Q: How do I determine if the average rate of change of a function is zero?

A: To determine if the average rate of change of a function is zero, you need to examine the behavior of the function over the interval. If the function does not change over the interval, the average rate of change will be zero.

Q: Can the average rate of change of a function be undefined?

A: Yes, the average rate of change of a function can be undefined. This occurs when the function is not defined at one or both of the endpoints of the interval.

Q: How do I determine if the average rate of change of a function is undefined?

A: To determine if the average rate of change of a function is undefined, you need to examine the behavior of the function over the interval. If the function is not defined at one or both of the endpoints of the interval, the average rate of change will be undefined.

Q: What are some common applications of the average rate of change of a function?

A: The average rate of change of a function has many applications in various fields, including:

• Physics: The average rate of change of a function is used to describe the motion of objects.
• Engineering: The average rate of change of a function is used to design and optimize systems.
• Economics: The average rate of change of a function is used to model economic systems and make predictions about economic trends.

Conclusion

In this article, we answered some common questions about the average rate of change of a function. We discussed the formula for the average rate of change, how to apply it, and the significance of the average rate of change. We also covered some common applications of the average rate of change of a function.