What Is The Multiplicative Rate Of Change For The Exponential Function $f(x) = 2\left(\frac{5}{2}\right)^{-x}$?A. 0.4 B. 0.6 C. 1.5 D. 2.5

Introduction

In mathematics, the multiplicative rate of change is a concept used to describe how a function changes as its input changes. For exponential functions, this rate of change is constant and can be calculated using the formula for the derivative of an exponential function. In this article, we will explore the concept of the multiplicative rate of change for the exponential function $f(x) = 2\left(\frac{5}{2}\right)^{-x}$.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that can be written in the form $f(x) = ab^x$, where $a$ and $b$ are constants. The base $b$ is the key to understanding the behavior of the function. If $b$ is greater than 1, the function will increase exponentially as $x$ increases. If $b$ is between 0 and 1, the function will decrease exponentially as $x$ increases.

The Multiplicative Rate of Change

The multiplicative rate of change is a measure of how much the function changes as its input changes. For exponential functions, this rate of change is constant and can be calculated using the formula for the derivative of an exponential function. The derivative of an exponential function $f(x) = ab^x$ is given by $f'(x) = ab^x \ln(b)$, where $\ln(b)$ is the natural logarithm of the base $b$.

Calculating the Multiplicative Rate of Change

To calculate the multiplicative rate of change for the exponential function $f(x) = 2\left(\frac{5}{2}\right)^{-x}$, we need to find the derivative of the function. Using the formula for the derivative of an exponential function, we get:

$$f'(x) = 2\left(\frac{5}{2}\right)^{-x} \ln\left(\frac{5}{2}\right)$$

To find the multiplicative rate of change, we need to evaluate the derivative at a specific value of $x$. Let's evaluate the derivative at $x = 0$.

$$f'(0) = 2\left(\frac{5}{2}\right)^0 \ln\left(\frac{5}{2}\right)$$

Simplifying the expression, we get:

$$f'(0) = 2 \ln\left(\frac{5}{2}\right)$$

Evaluating the Natural Logarithm

The natural logarithm of a number is the logarithm to the base $e$, where $e$ is a mathematical constant approximately equal to 2.71828. To evaluate the natural logarithm of $\frac{5}{2}$, we can use a calculator or a mathematical software package.

$$\ln\left(\frac{5}{2}\right) \approx 0.8047$$

Calculating the Multiplicative Rate of Change

Now that we have evaluated the natural logarithm of $\frac{5}{2}$, we can calculate the multiplicative rate of change.

$$f'(0) = 2 \ln\left(\frac{5}{2}\right) \approx 2 \times 0.8047 \approx 1.6094$$

Conclusion

In conclusion, the multiplicative rate of change for the exponential function $f(x) = 2\left(\frac{5}{2}\right)^{-x}$ is approximately 1.6094. This means that for every unit increase in $x$, the function will increase by a factor of approximately 1.6094.

Comparison with Answer Choices

Now that we have calculated the multiplicative rate of change, we can compare it with the answer choices.

• A. 0.4: This is not equal to the multiplicative rate of change.
• B. 0.6: This is not equal to the multiplicative rate of change.
• C. 1.5: This is not equal to the multiplicative rate of change.
• D. 2.5: This is not equal to the multiplicative rate of change.

Final Answer

The final answer is $\boxed{1.6094}$.

References

• [1] Calculus: Early Transcendentals, James Stewart, 8th edition.
• [2] Exponential Functions, Math Open Reference.
• [3] Natural Logarithm, Wolfram MathWorld.

Additional Resources

• [1] Exponential Functions, Khan Academy.
• [2] Derivatives of Exponential Functions, Mathway.
• [3] Natural Logarithm, Symbolab.
  Q&A: Multiplicative Rate of Change for Exponential Functions ===========================================================

Introduction

In our previous article, we explored the concept of the multiplicative rate of change for exponential functions. We calculated the multiplicative rate of change for the function $f(x) = 2\left(\frac{5}{2}\right)^{-x}$ and found that it is approximately 1.6094. In this article, we will answer some frequently asked questions about the multiplicative rate of change for exponential functions.

Q: What is the multiplicative rate of change?

A: The multiplicative rate of change is a measure of how much a function changes as its input changes. For exponential functions, this rate of change is constant and can be calculated using the formula for the derivative of an exponential function.

Q: How do I calculate the multiplicative rate of change for an exponential function?

A: To calculate the multiplicative rate of change for an exponential function, you need to find the derivative of the function using the formula for the derivative of an exponential function. Then, evaluate the derivative at a specific value of x.

Q: What is the formula for the derivative of an exponential function?

A: The formula for the derivative of an exponential function $f(x) = ab^x$ is given by $f'(x) = ab^x \ln(b)$, where $\ln(b)$ is the natural logarithm of the base b.

Q: How do I evaluate the natural logarithm of a number?

A: You can evaluate the natural logarithm of a number using a calculator or a mathematical software package.

Q: What is the multiplicative rate of change for the function $f(x) = 2^x$?

A: To calculate the multiplicative rate of change for the function $f(x) = 2^x$, we need to find the derivative of the function using the formula for the derivative of an exponential function. Then, evaluate the derivative at a specific value of x.

$$f'(x) = 2^x \ln(2)$$

Evaluating the derivative at x = 0, we get:

$$f'(0) = 2^0 \ln(2) = \ln(2)$$

The multiplicative rate of change for the function $f(x) = 2^x$ is approximately 0.6931.

Q: What is the multiplicative rate of change for the function $f(x) = 3^{-x}$?

A: To calculate the multiplicative rate of change for the function $f(x) = 3^{-x}$, we need to find the derivative of the function using the formula for the derivative of an exponential function. Then, evaluate the derivative at a specific value of x.

$$f'(x) = -3^{-x} \ln(3)$$

Evaluating the derivative at x = 0, we get:

$$f'(0) = -3^0 \ln(3) = -\ln(3)$$

The multiplicative rate of change for the function $f(x) = 3^{-x}$ is approximately -1.0986.

Q: Can I use the multiplicative rate of change to predict the behavior of an exponential function?

A: Yes, you can use the multiplicative rate of change to predict the behavior of an exponential function. If the multiplicative rate of change is greater than 1, the function will increase exponentially as x increases. If the multiplicative rate of change is between 0 and 1, the function will decrease exponentially as x increases.

Conclusion

In conclusion, the multiplicative rate of change is a measure of how much a function changes as its input changes. For exponential functions, this rate of change is constant and can be calculated using the formula for the derivative of an exponential function. We answered some frequently asked questions about the multiplicative rate of change for exponential functions and provided examples of how to calculate the multiplicative rate of change for different functions.

References

• [1] Calculus: Early Transcendentals, James Stewart, 8th edition.
• [2] Exponential Functions, Math Open Reference.
• [3] Natural Logarithm, Wolfram MathWorld.

Additional Resources

• [1] Exponential Functions, Khan Academy.
• [2] Derivatives of Exponential Functions, Mathway.
• [3] Natural Logarithm, Symbolab.