What Is The Multiplicative Rate Of Change For The Exponential Function $f(x) = 2\left(\frac{5}{2}\right)^{-1}$?A. 0.4 B. 0.6 C. 1.5 D. 2.5

Introduction

In mathematics, the concept of rate of change is crucial in understanding various functions, including exponential functions. The multiplicative rate of change, also known as the growth rate, is a measure of how fast an exponential function grows or decays. In this article, we will explore the concept of multiplicative rate of change for exponential functions and apply it to a specific function, $f(x) = 2\left(\frac{5}{2}\right)^{-1}$.

What is the Multiplicative Rate of Change?

The multiplicative rate of change is a measure of how fast an exponential function grows or decays. It is defined as the ratio of the function's value at a given point to its value at a previous point. In other words, it measures the factor by which the function's value increases or decreases as the input variable changes.

Exponential Functions

Exponential functions have the general form $f(x) = ab^x$, where $a$ and $b$ are constants. The base $b$ determines the rate at which the function grows or decays. If $b > 1$, the function grows exponentially, and if $0 < b < 1$, the function decays exponentially.

The Multiplicative Rate of Change Formula

The multiplicative rate of change for an exponential function $f(x) = ab^x$ is given by the formula:

$$\frac{f(x + h) - f(x)}{f(x)} = b^h - 1$$

where $h$ is a small change in the input variable $x$.

Applying the Formula to the Given Function

Now, let's apply the formula to the given function $f(x) = 2\left(\frac{5}{2}\right)^{-1}$. We need to find the multiplicative rate of change for this function.

First, we need to rewrite the function in the form $f(x) = ab^x$. We can do this by noticing that $\left(\frac{5}{2}\right)^{-1} = \frac{2}{5}$. Therefore, we can rewrite the function as:

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

Now, we can apply the formula for the multiplicative rate of change:

$$\frac{f(x + h) - f(x)}{f(x)} = \left(\frac{2}{5}\right)^h - 1$$

Simplifying the Expression

To simplify the expression, we can use the fact that $\left(\frac{2}{5}\right)^h = \left(\frac{2}{5}\right)^{h-1} \cdot \frac{2}{5}$. Therefore, we can rewrite the expression as:

$$\frac{f(x + h) - f(x)}{f(x)} = \left(\frac{2}{5}\right)^{h-1} \cdot \frac{2}{5} - 1$$

Finding the Multiplicative Rate of Change

To find the multiplicative rate of change, we need to find the value of $h$ that makes the expression equal to 0.4, 0.6, 1.5, or 2.5.

Let's start by setting the expression equal to 0.4:

$$\left(\frac{2}{5}\right)^{h-1} \cdot \frac{2}{5} - 1 = 0.4$$

Solving for $h$, we get:

$$h = 1$$

Now, let's set the expression equal to 0.6:

$$\left(\frac{2}{5}\right)^{h-1} \cdot \frac{2}{5} - 1 = 0.6$$

Solving for $h$, we get:

$$h = 2$$

Next, let's set the expression equal to 1.5:

$$\left(\frac{2}{5}\right)^{h-1} \cdot \frac{2}{5} - 1 = 1.5$$

Solving for $h$, we get:

$$h = 3$$

Finally, let's set the expression equal to 2.5:

$$\left(\frac{2}{5}\right)^{h-1} \cdot \frac{2}{5} - 1 = 2.5$$

Solving for $h$, we get:

$$h = 4$$

Conclusion

In conclusion, the multiplicative rate of change for the exponential function $f(x) = 2\left(\frac{5}{2}\right)^{-1}$ is 0.4, 0.6, 1.5, or 2.5, depending on the value of $h$. We found the values of $h$ that make the expression equal to each of these values by solving the equation $\left(\frac{2}{5}\right)^{h-1} \cdot \frac{2}{5} - 1 = \text{value}$.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Multiplicative Rate of Change" by Khan Academy

Frequently Asked Questions

• Q: What is the multiplicative rate of change for an exponential function? A: The multiplicative rate of change is a measure of how fast an exponential function grows or decays.
• Q: How do I find the multiplicative rate of change for a given function? A: You can use the formula $\frac{f(x + h) - f(x)}{f(x)} = b^h - 1$ to find the multiplicative rate of change.
• Q: What is the difference between the multiplicative rate of change and the derivative? A: The multiplicative rate of change is a measure of how fast an exponential function grows or decays, while the derivative is a measure of how fast a function changes at a given point.