Choose The Square Root Function That Has A Smaller Estimated Average Rate Of Change Over The Interval \[0,9\] Than The Cube Root Function.A. $f(x) = \sqrt{x} - 1$B. $f(z) = \sqrt{3z} - 1$C. $f(z) = \frac{1}{3} \sqrt{x} -

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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 estimate the rate at which a function changes over a specific interval. In this article, we will discuss how to choose the square root function that has a smaller estimated average rate of change over the interval [0,9] than the cube root function.

Understanding the Average Rate of Change

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

f(b)βˆ’f(a)bβˆ’a\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.

The Square Root Function

The square root function is given by the formula:

f(x)=xf(x) = \sqrt{x}

However, in this problem, we are given two different square root functions:

A. f(x)=xβˆ’1f(x) = \sqrt{x} - 1 B. f(z)=3zβˆ’1f(z) = \sqrt{3z} - 1 C. f(z)=13xβˆ’1f(z) = \frac{1}{3} \sqrt{x} - 1

We need to choose the function that has a smaller estimated average rate of change over the interval [0,9] than the cube root function.

The Cube Root Function

The cube root function is given by the formula:

f(x)=x3f(x) = \sqrt[3]{x}

However, in this problem, we are not given the cube root function explicitly. We are only given the square root functions and need to compare them to the cube root function.

Comparing the Square Root Functions

To compare the square root functions, we need to calculate their average rates of change over the interval [0,9]. We can do this by plugging in the values of x into the functions and calculating the differences in the function values at the endpoints of the interval.

For function A, we have:

f(0)=0βˆ’1=βˆ’1f(0) = \sqrt{0} - 1 = -1

f(9)=9βˆ’1=2f(9) = \sqrt{9} - 1 = 2

The average rate of change of function A is:

f(9)βˆ’f(0)9βˆ’0=2βˆ’(βˆ’1)9=39=13\frac{f(9) - f(0)}{9 - 0} = \frac{2 - (-1)}{9} = \frac{3}{9} = \frac{1}{3}

For function B, we have:

f(0)=3(0)βˆ’1=βˆ’1f(0) = \sqrt{3(0)} - 1 = -1

f(9)=3(9)βˆ’1=4f(9) = \sqrt{3(9)} - 1 = 4

The average rate of change of function B is:

f(9)βˆ’f(0)9βˆ’0=4βˆ’(βˆ’1)9=59\frac{f(9) - f(0)}{9 - 0} = \frac{4 - (-1)}{9} = \frac{5}{9}

For function C, we have:

f(0)=130βˆ’1=βˆ’1f(0) = \frac{1}{3} \sqrt{0} - 1 = -1

f(9)=139βˆ’1=23f(9) = \frac{1}{3} \sqrt{9} - 1 = \frac{2}{3}

The average rate of change of function C is:

f(9)βˆ’f(0)9βˆ’0=23βˆ’(βˆ’1)9=539=527\frac{f(9) - f(0)}{9 - 0} = \frac{\frac{2}{3} - (-1)}{9} = \frac{\frac{5}{3}}{9} = \frac{5}{27}

Comparing the Average Rates of Change

Now that we have calculated the average rates of change of the square root functions, we can compare them to the cube root function. However, we are not given the cube root function explicitly. We can assume that the cube root function has an average rate of change that is greater than the square root functions.

Conclusion

Based on the calculations above, we can see that function A has the largest average rate of change, followed by function B, and then function C. Therefore, function C has the smallest average rate of change over the interval [0,9] than the cube root function.

Answer

The correct answer is:

C. f(z)=13xβˆ’1f(z) = \frac{1}{3} \sqrt{x} - 1

Introduction

In our previous article, we discussed how to choose the square root function that has a smaller estimated average rate of change over the interval [0,9] than the cube root function. We calculated the average rates of change of three different square root functions and compared them to the cube root function. In this article, we will answer some frequently asked questions about the topic.

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

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

f(b)βˆ’f(a)bβˆ’a\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.

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 plug in the values of x into the function and calculate the differences in the function values at the endpoints of the interval. Then, you divide the difference by the length of the interval.

Q: What is the difference between the average rate of change and the instantaneous rate of change?

A: The average rate of change of a function is a measure of how much the function changes on average over a specific interval. The instantaneous rate of change of a function is a measure of how much the function changes at a specific point. The instantaneous rate of change is calculated using the derivative of the function.

Q: How do I choose the square root function with a smaller estimated average rate of change?

A: To choose the square root function with a smaller estimated average rate of change, you need to calculate the average rates of change of the different square root functions and compare them to the cube root function. The function with the smallest average rate of change is the one you want to choose.

Q: What is the significance of the cube root function in this problem?

A: The cube root function is used as a reference point in this problem. We are comparing the average rates of change of the square root functions to the cube root function to determine which square root function has a smaller estimated average rate of change.

Q: Can I use this method to compare other functions?

A: Yes, you can use this method to compare other functions. The method involves calculating the average rates of change of the functions and comparing them to each other.

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

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

  • Modeling population growth
  • Predicting stock prices
  • Calculating the rate of change of a chemical reaction
  • Determining the rate of change of a physical system

Conclusion

In this article, we answered some frequently asked questions about choosing the square root function with a smaller estimated average rate of change. We discussed the concept of the average rate of change, how to calculate it, and its significance in real-world applications. We also provided some tips on how to choose the square root function with a smaller estimated average rate of change.

Additional Resources

For more information on the average rate of change and its applications, you can refer to the following resources:

  • Calculus textbooks
  • Online resources such as Khan Academy and MIT OpenCourseWare
  • Research papers on the topic of average rate of change

Final Thoughts

The average rate of change is an important concept in mathematics and has many real-world applications. By understanding how to calculate and compare the average rates of change of different functions, you can make informed decisions in a variety of fields.