Evaluate the function at the given points

Oct 13, 2025 by magicalceylontours.com 28 views

Introduction

In mathematics, the average rate of change of a function is a measure of how much the function's output changes when its input changes by a certain amount. This concept is crucial in various fields, including physics, engineering, and economics. In this article, we will explore the average rate of change of a given function, $f(x)=\sqrt{x-4}$ , from 8 to 14.

What is the Average Rate of Change?

The average rate of change of a function $f(x)$ from $x=a$ to $x=b$ is given by the formula:

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

This formula calculates the difference in the function's output between two points, $a$ and $b$ , and divides it by the difference in the input values.

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

To find the average rate of change of $f(x)=\sqrt{x-4}$ from 8 to 14, we need to follow these steps:

Evaluate the function at the given points: We need to find the values of $f(8)$ and $f(14)$ .
Calculate the difference in the function's output: We will subtract $f(8)$ from $f(14)$ to find the difference in the function's output.
Calculate the difference in the input values: We will subtract 8 from 14 to find the difference in the input values.
Apply the formula for the average rate of change: We will use the formula $\frac{f(b)-f(a)}{b-a}$ to calculate the average rate of change.

Step 1: Evaluate the function at the given points

To evaluate the function at the given points, we need to substitute the values of $x$ into the function.

f(8)=\sqrt{8-4}=\sqrt{4}=2

f(14)=\sqrt{14-4}=\sqrt{10}\approx 3.16

Step 2: Calculate the difference in the function's output

We will subtract $f(8)$ from $f(14)$ to find the difference in the function's output.

f(14)-f(8)\approx 3.16-2=1.16

Step 3: Calculate the difference in the input values

We will subtract 8 from 14 to find the difference in the input values.

14-8=6

Step 4: Apply the formula for the average rate of change

We will use the formula $\frac{f(b)-f(a)}{b-a}$ to calculate the average rate of change.

\frac{f(14)-f(8)}{14-8}\approx \frac{1.16}{6}\approx 0.19

Conclusion

In conclusion, the average rate of change of $f(x)=\sqrt{x-4}$ from 8 to 14 is approximately 0.19. This means that for every unit increase in the input value, the function's output increases by approximately 0.19 units.

Answer

The correct answer is:

C. 0.19

Discussion

The average rate of change of a function is an important concept in mathematics, and it has various applications in real-world problems. In this article, we have explored the average rate of change of a given function, $f(x)=\sqrt{x-4}$ , from 8 to 14. We have used the formula for the average rate of change and calculated the value of the average rate of change. The result is approximately 0.19, which means that for every unit increase in the input value, the function's output increases by approximately 0.19 units.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Calculus, 2nd edition, James Stewart
[3] Mathematics for the Nonmathematician, Morris Kline

Additional Resources

[1] Khan Academy: Average Rate of Change
[2] MIT OpenCourseWare: Calculus
[3] Wolfram Alpha: Average Rate of Change
Q&A: Average Rate of Change =============================

Frequently Asked Questions

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's output changes when its input changes by a certain amount. It is calculated using the formula:

\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:

Evaluate the function at the given points.
Calculate the difference in the function's output.
Calculate the difference in the input values.
Apply the formula for 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's output changes over a certain interval, while the instantaneous rate of change is a measure of how much the function's output changes at a specific point.

Q: Can the average rate of change be negative?

A: Yes, the average rate of change can be negative. This means that the function's output is decreasing over the given interval.

Q: Can the average rate of change be zero?

A: Yes, the average rate of change can be zero. This means that the function's output is not changing over the given interval.

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

A: The average rate of change has various applications in real-world problems, such as:

Physics: to calculate the velocity of an object
Engineering: to calculate the rate of change of a system's output
Economics: to calculate the rate of change of a country's GDP

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 evaluating the function at the given points
Not calculating the difference in the function's output
Not calculating the difference in the input values
Not applying the formula for the average rate of change correctly

Q: How do I round the average rate of change to the nearest hundredth?

A: To round the average rate of change to the nearest hundredth, you need to follow these steps:

Calculate the average rate of change using the formula.
Round the result to the nearest hundredth.

Conclusion

In conclusion, the average rate of change is an important concept in mathematics, and it has various applications in real-world problems. By understanding how to calculate the average rate of change and avoiding common mistakes, you can use this concept to solve problems in physics, engineering, economics, and other fields.

Additional Resources

[1] Khan Academy: Average Rate of Change
[2] MIT OpenCourseWare: Calculus
[3] Wolfram Alpha: Average Rate of Change

Discussion

The average rate of change is a fundamental concept in mathematics, and it has various applications in real-world problems. By understanding how to calculate the average rate of change and avoiding common mistakes, you can use this concept to solve problems in physics, engineering, economics, and other fields.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Calculus, 2nd edition, James Stewart
[3] Mathematics for the Nonmathematician, Morris Kline