The Function F ( X ) = 2 ⋅ 5 X F(x) = 2 \cdot 5^x F ( X ) = 2 ⋅ 5 X Can Be Used To Represent The Curve Through The Points ( 1 , 10 (1,10 ( 1 , 10 ], ( 2 , 50 (2,50 ( 2 , 50 ], And ( 3 , 250 (3,250 ( 3 , 250 ]. What Is The Multiplicative Rate Of Change Of The Function?A. 2 B. 5 C. 10 D. 32

Introduction

In mathematics, the concept of exponential growth is crucial in understanding various real-world phenomena, such as population growth, financial investments, and chemical reactions. The function $f(x) = 2 \cdot 5^x$ is a classic example of an exponential function that represents a curve through specific points. In this article, we will delve into the world of exponential growth and explore the concept of multiplicative rate of change.

What is Multiplicative Rate of Change?

The multiplicative rate of change, also known as the growth rate, is a measure of how quickly a quantity changes over time. In the context of exponential functions, it represents the factor by which the quantity increases or decreases at a given point. In other words, it measures the rate at which the function grows or decays.

The Function $f(x) = 2 \cdot 5^x$

The given function $f(x) = 2 \cdot 5^x$ is an exponential function that represents a curve through the points $(1,10)$, $(2,50)$, and $(3,250)$. To understand the multiplicative rate of change of this function, we need to analyze its behavior.

Analyzing the Function

Let's start by evaluating the function at the given points:

• $f(1) = 2 \cdot 5^1 = 10$
• $f(2) = 2 \cdot 5^2 = 50$
• $f(3) = 2 \cdot 5^3 = 250$

As we can see, the function grows rapidly as $x$ increases. To find the multiplicative rate of change, we need to calculate the ratio of consecutive function values.

Calculating the Multiplicative Rate of Change

Let's calculate the ratio of consecutive function values:

• $\frac{f(2)}{f(1)} = \frac{50}{10} = 5$
• $\frac{f(3)}{f(2)} = \frac{250}{50} = 5$

As we can see, the ratio of consecutive function values is constant, which means that the function has a constant multiplicative rate of change.

Conclusion

In conclusion, the multiplicative rate of change of the function $f(x) = 2 \cdot 5^x$ is 5. This means that for every unit increase in $x$, the function value increases by a factor of 5. The constant multiplicative rate of change is a characteristic of exponential functions, which makes them useful in modeling real-world phenomena that exhibit rapid growth or decay.

Final Answer

The final answer is $\boxed{5}$.

Discussion

The concept of multiplicative rate of change is crucial in understanding various real-world phenomena. In finance, it represents the rate of return on investment, while in biology, it represents the rate of population growth. In chemistry, it represents the rate of chemical reactions. The constant multiplicative rate of change of the function $f(x) = 2 \cdot 5^x$ makes it a useful tool in modeling these phenomena.

Related Topics

• Exponential functions
• Multiplicative rate of change
• Growth rate
• Decay rate
• Population growth
• Financial investments
• Chemical reactions

References

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

Q&A: Multiplicative Rate of Change

Q: What is the multiplicative rate of change of the function $f(x) = 2 \cdot 5^x$?

A: The multiplicative rate of change of the function $f(x) = 2 \cdot 5^x$ is 5.

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

A: To calculate the multiplicative rate of change of an exponential function, you need to calculate the ratio of consecutive function values.

Q: What is the difference between multiplicative rate of change and growth rate?

A: The multiplicative rate of change and growth rate are related but distinct concepts. The multiplicative rate of change represents the factor by which the quantity increases or decreases at a given point, while the growth rate represents the rate at which the quantity increases or decreases over time.

Q: How do you determine if an exponential function has a constant multiplicative rate of change?

A: To determine if an exponential function has a constant multiplicative rate of change, you need to calculate the ratio of consecutive function values. If the ratio is constant, then the function has a constant multiplicative rate of change.

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

A: Multiplicative rate of change has many real-world applications, including:

• Finance: It represents the rate of return on investment.
• Biology: It represents the rate of population growth.
• Chemistry: It represents the rate of chemical reactions.

Q: How do you use the multiplicative rate of change to model real-world phenomena?

A: To use the multiplicative rate of change to model real-world phenomena, you need to:

1. Identify the exponential function that represents the phenomenon.
2. Calculate the multiplicative rate of change of the function.
3. Use the multiplicative rate of change to model the phenomenon.

Q: What are some common mistakes to avoid when calculating the multiplicative rate of change?

A: Some common mistakes to avoid when calculating the multiplicative rate of change include:

• Not calculating the ratio of consecutive function values.
• Not checking if the ratio is constant.
• Not using the correct formula for the multiplicative rate of change.

Q: How do you check if an exponential function has a constant multiplicative rate of change?

A: To check if an exponential function has a constant multiplicative rate of change, you need to:

1. Calculate the ratio of consecutive function values.
2. Check if the ratio is constant.
3. Use the ratio to determine if the function has a constant multiplicative rate of change.

Conclusion

In conclusion, the multiplicative rate of change is a crucial concept in understanding exponential functions and their applications in real-world phenomena. By understanding the multiplicative rate of change, you can model and analyze various phenomena, including finance, biology, and chemistry.

Final Answer

The final answer is $\boxed{5}$.

Discussion

The concept of multiplicative rate of change is crucial in understanding various real-world phenomena. In finance, it represents the rate of return on investment, while in biology, it represents the rate of population growth. In chemistry, it represents the rate of chemical reactions. The constant multiplicative rate of change of the function $f(x) = 2 \cdot 5^x$ makes it a useful tool in modeling these phenomena.

Related Topics

• Exponential functions
• Multiplicative rate of change
• Growth rate
• Decay rate
• Population growth
• Financial investments
• Chemical reactions

References

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

Note: The references provided are for informational purposes only and are not a substitute for professional advice or guidance.