Select The Correct Answer.Which Exponential Function Has The Greatest Average Rate Of Change Over The Interval \[0,2\]?A. $j(x) = 3(1.6)^x$B. An Exponential Function, $f$, With A $y$-intercept Of 1.5 And A Common

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Introduction

In mathematics, exponential functions are a fundamental concept that describe growth and decay phenomena in various fields, including finance, biology, and physics. When dealing with exponential functions, it's essential to understand the concept of average rate of change, which measures the rate at which the function changes over a given interval. In this article, we will compare two exponential functions, A and B, to determine which one has the greatest average rate of change over the interval [0,2].

Understanding Exponential Functions

An exponential function is a mathematical function of the form f(x) = ab^x, where a is the initial value (or y-intercept), b is the base, and x is the variable. The base, b, determines the rate of growth or decay of the function. If b > 1, the function grows exponentially, while if 0 < b < 1, the function decays exponentially.

Function A: 3(1.6)^x

Function A is an exponential function with a y-intercept of 3 and a base of 1.6. The function can be written as:

f(x) = 3(1.6)^x

To find the average rate of change of this function over the interval [0,2], we need to calculate the difference quotient:

Average rate of change = (f(2) - f(0)) / (2 - 0)

First, let's calculate f(2) and f(0):

f(2) = 3(1.6)^2 = 3(2.56) = 7.68 f(0) = 3(1.6)^0 = 3(1) = 3

Now, we can calculate the average rate of change:

Average rate of change = (7.68 - 3) / 2 = 4.68 / 2 = 2.34

Function B: An Exponential Function with a y-Intercept of 1.5

Function B is an exponential function with a y-intercept of 1.5 and a base of 2. The function can be written as:

f(x) = 1.5(2)^x

To find the average rate of change of this function over the interval [0,2], we need to calculate the difference quotient:

Average rate of change = (f(2) - f(0)) / (2 - 0)

First, let's calculate f(2) and f(0):

f(2) = 1.5(2)^2 = 1.5(4) = 6 f(0) = 1.5(2)^0 = 1.5(1) = 1.5

Now, we can calculate the average rate of change:

Average rate of change = (6 - 1.5) / 2 = 4.5 / 2 = 2.25

Comparing the Average Rates of Change

Now that we have calculated the average rates of change for both functions, 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 2.34, while Function B has an average rate of change of 2.25. Therefore, Function A has the greatest average rate of change over the interval [0,2].

Conclusion

In conclusion, when comparing two exponential functions, A and B, over the interval [0,2], we found that Function A has the greatest average rate of change. This is because the base of Function A (1.6) is greater than the base of Function B (2), resulting in a faster rate of growth. Understanding the concept of average rate of change and the properties of exponential functions is essential in various fields, including finance, biology, and physics.

References

Additional Resources

Introduction

Exponential functions are a fundamental concept in mathematics that describe growth and decay phenomena in various fields, including finance, biology, and physics. In our previous article, we compared two exponential functions, A and B, to determine which one has the greatest average rate of change over the interval [0,2]. In this article, we will answer some frequently asked questions about exponential functions to help you better understand this concept.

Q: What is an exponential function?

A: An exponential function is a mathematical function of the form f(x) = ab^x, where a is the initial value (or y-intercept), b is the base, and x is the variable.

Q: What is the base of an exponential function?

A: The base of an exponential function is the number that is raised to the power of x. In the function f(x) = ab^x, b is the base.

Q: What is the y-intercept of an exponential function?

A: The y-intercept of an exponential function is the value of the function when x = 0. In the function f(x) = ab^x, the y-intercept is a.

Q: How do I determine if an exponential function is growing or decaying?

A: If the base of the exponential function is greater than 1, the function is growing exponentially. If the base is between 0 and 1, the function is decaying exponentially.

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 calculate the difference quotient:

Average rate of change = (f(2) - f(0)) / (2 - 0)

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

A: The average rate of change of an exponential function measures the rate at which the function changes over a given interval. It is an important concept in various fields, including finance, biology, and physics.

Q: Can you provide an example of an exponential function?

A: Yes, an example of an exponential function is f(x) = 2^x. This function has a y-intercept of 1 and a base of 2.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the function.

Q: What are some real-world applications of exponential functions?

A: Exponential functions have many real-world applications, including:

  • Finance: Exponential functions are used to model population growth, compound interest, and inflation.
  • Biology: Exponential functions are used to model population growth, disease spread, and chemical reactions.
  • Physics: Exponential functions are used to model radioactive decay, electrical circuits, and population growth.

Conclusion

In conclusion, exponential functions are a fundamental concept in mathematics that describe growth and decay phenomena in various fields. By understanding the properties of exponential functions, you can better analyze and model real-world phenomena. We hope this Q&A guide has helped you better understand exponential functions.

References

Additional Resources

Note: The references and additional resources provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.