Which Exponential Function Has The Greatest Average Rate Of Change Over The Interval 0 , 2 {0,2} 0 , 2 ?A. ${ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \ \hline g(x) & 8 & 10 & \frac{25}{2} & \frac{125}{8} \ \hline \end{array} }$B.
Which Exponential Function Has the Greatest Average Rate of Change Over the Interval [0,2]?
In this article, we will explore the concept of average rate of change and how it applies to exponential functions. We will examine two different exponential functions, A and B, and determine which one has the greatest average rate of change over the interval [0,2]. To do this, we will first need to understand what average rate of change is and how it is calculated.
What is Average Rate of Change?
The average rate of change of a function over a given interval is a measure of how much the function changes over that interval. It is calculated by finding the difference in the function's values at the endpoints of the interval and dividing by the length of the interval.
Calculating Average Rate of Change
The formula for calculating the average rate of change of a function f(x) over the interval [a,b] is:
Exponential Function A
The exponential function A is given by the table:
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| g(x) | 8 | 10 | 25/2 | 125/8 |
To calculate the average rate of change of function A over the interval [0,2], we will use the formula above.
Calculating Average Rate of Change for Function A
First, we need to find the values of g(0) and g(2).
g(0) = 8 g(2) = 25/2
Next, we will substitute these values into the formula for average rate of change.
Exponential Function B
The exponential function B is given by the table:
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| g(x) | 8 | 12 | 18 | 27 |
To calculate the average rate of change of function B over the interval [0,2], we will use the formula above.
Calculating Average Rate of Change for Function B
First, we need to find the values of g(0) and g(2).
g(0) = 8 g(2) = 18
Next, we will substitute these values into the formula for average rate of change.
Comparing the Average Rates of Change
Now that we have calculated the average rates of change for both functions A and B, we can compare them to determine which one has the greatest average rate of change over the interval [0,2].
Function A has an average rate of change of 9/4, while function B has an average rate of change of 5. Since 9/4 is greater than 5, function A has the greatest average rate of change over the interval [0,2].
In conclusion, we have determined that function A has the greatest average rate of change over the interval [0,2]. This is because its average rate of change is 9/4, which is greater than the average rate of change of function B, which is 5. We hope that this article has provided a clear understanding of how to calculate and compare average rates of change for exponential functions.
- [1] Calculus: Early Transcendentals, James Stewart, 8th edition
- [2] Exponential Functions, Math Open Reference
- [3] Average Rate of Change, Khan Academy
- [1] Exponential Functions, Wolfram MathWorld
- [2] Average Rate of Change, Mathway
- [3] Calculus, MIT OpenCourseWare
Q&A: Exponential Functions and Average Rate of Change =====================================================
In our previous article, we explored the concept of average rate of change and how it applies to exponential functions. We examined two different exponential functions, A and B, and determined which one had the greatest average rate of change over the interval [0,2]. In this article, we will answer some frequently asked questions about exponential functions and average rate of change.
Q: What is an exponential function?
A: An exponential function is a function of the form f(x) = ab^x, where a and b are constants and b is a positive number not equal to 1.
Q: How do I calculate the average rate of change of an exponential function?
A: To calculate the average rate of change of an exponential function, you need to find the difference in the function's values at the endpoints of the interval and divide by the length of the interval. The formula for calculating the average rate of change of a function f(x) over the interval [a,b] is:
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.
Q: How do I determine which exponential function has the greatest average rate of change over a given interval?
A: To determine which exponential function has the greatest average rate of change over a given interval, you need to calculate the average rate of change of each function over that interval and compare the results.
Q: Can I use the average rate of change to determine the maximum or minimum value of an exponential function?
A: No, the average rate of change is not a reliable method for determining the maximum or minimum value of an exponential function. The maximum or minimum value of an exponential function can be determined using other methods, such as finding the critical points of the function.
Q: How do I apply the concept of average rate of change to real-world problems?
A: The concept of average rate of change can be applied to a wide range of real-world problems, such as:
- Modeling population growth or decline
- Analyzing the rate of change of a physical system
- Determining the rate of change of a financial investment
- Calculating the average rate of change of a chemical reaction
Q: What are some common mistakes to avoid when calculating the average rate of change of an exponential function?
A: Some common mistakes to avoid when calculating the average rate of change of an exponential function include:
- Failing to use the correct formula for calculating the average rate of change
- Not using the correct values for the function at the endpoints of the interval
- Not checking for errors in the calculation
In conclusion, we have answered some frequently asked questions about exponential functions and average rate of change. We hope that this article has provided a clear understanding of the concept of average rate of change and how it applies to exponential functions.
- [1] Calculus: Early Transcendentals, James Stewart, 8th edition
- [2] Exponential Functions, Math Open Reference
- [3] Average Rate of Change, Khan Academy
- [1] Exponential Functions, Wolfram MathWorld
- [2] Average Rate of Change, Mathway
- [3] Calculus, MIT OpenCourseWare