Which Function Increases At The Fastest Rate Between X = 0 X=0 X = 0 And X = 8 X=8 X = 8 ?Linear Function$[ \begin{array}{|c|c|} \hline \multicolumn{2}{|c|}{f(x)=2x+2} \ \hline X & F(x) \ \hline 0 & 2 \ \hline 2 & 6 \ \hline 4 & 10 \ \hline 6 &

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Introduction

When comparing the rates of change of two or more functions, it's essential to understand the concept of the derivative. The derivative of a function represents the rate at which the function changes as its input changes. In this article, we will explore the rates of change of two linear functions, $f(x)=2x+2$ and $g(x)=3x-1$, between $x=0$ and $x=8$. We will determine which function increases at the fastest rate during this interval.

Linear Function 1: $f(x)=2x+2$

The first linear function is $f(x)=2x+2$. This function has a slope of 2, which means that for every unit increase in $x$, the value of $f(x)$ increases by 2 units. To understand the rate of change of this function, we can calculate its derivative.

Derivative of $f(x)=2x+2$

The derivative of $f(x)=2x+2$ is given by:

$$f'(x)=\frac{d}{dx}(2x+2)=2$$

This means that the rate of change of $f(x)$ is constant and equal to 2.

Rate of Change of $f(x)=2x+2$ between $x=0$ and $x=8$

To determine the rate of change of $f(x)$ between $x=0$ and $x=8$, we can calculate the difference quotient:

$$\frac{f(8)-f(0)}{8-0}=\frac{(2(8)+2)-(2(0)+2)}{8-0}=\frac{18-2}{8}=\frac{16}{8}=2$$

This means that the rate of change of $f(x)$ between $x=0$ and $x=8$ is also 2.

Linear Function 2: $g(x)=3x-1$

The second linear function is $g(x)=3x-1$. This function has a slope of 3, which means that for every unit increase in $x$, the value of $g(x)$ increases by 3 units. To understand the rate of change of this function, we can calculate its derivative.

Derivative of $g(x)=3x-1$

The derivative of $g(x)=3x-1$ is given by:

$$g'(x)=\frac{d}{dx}(3x-1)=3$$

This means that the rate of change of $g(x)$ is constant and equal to 3.

Rate of Change of $g(x)=3x-1$ between $x=0$ and $x=8$

To determine the rate of change of $g(x)$ between $x=0$ and $x=8$, we can calculate the difference quotient:

$$\frac{g(8)-g(0)}{8-0}=\frac{(3(8)-1)-(3(0)-1)}{8-0}=\frac{24-1}{8}=\frac{23}{8}=2.875$$

This means that the rate of change of $g(x)$ between $x=0$ and $x=8$ is approximately 2.875.

Comparison of Rates of Change

Now that we have calculated the rates of change of both functions, we can compare them to determine which function increases at the fastest rate between $x=0$ and $x=8$.

Function	Rate of Change
$f(x)=2x+2$	2
$g(x)=3x-1$	2.875

Based on the calculations, we can see that the rate of change of $g(x)=3x-1$ is greater than the rate of change of $f(x)=2x+2$ between $x=0$ and $x=8$. Therefore, we can conclude that the function $g(x)=3x-1$ increases at the fastest rate between $x=0$ and $x=8$.

Conclusion

In this article, we compared the rates of change of two linear functions, $f(x)=2x+2$ and $g(x)=3x-1$, between $x=0$ and $x=8$. We calculated the derivatives of both functions and determined their rates of change. Based on the calculations, we found that the function $g(x)=3x-1$ increases at the fastest rate between $x=0$ and $x=8$. This demonstrates the importance of understanding the concept of the derivative and its application in determining the rates of change of functions.

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Introduction

In our previous article, we compared the rates of change of two linear functions, $f(x)=2x+2$ and $g(x)=3x-1$, between $x=0$ and $x=8$. We determined that the function $g(x)=3x-1$ increases at the fastest rate between $x=0$ and $x=8$. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the derivative of a function?

A: The derivative of a function represents the rate at which the function changes as its input changes. It is a measure of how fast the function is changing at a given point.

Q: How do you calculate the derivative of a function?

A: To calculate the derivative of a function, you can use the power rule, which states that if $f(x)=x^n$, then $f'(x)=nx^{n-1}$. You can also use the sum rule, which states that if $f(x)=g(x)+h(x)$, then $f'(x)=g'(x)+h'(x)$.

Q: What is the difference quotient?

A: The difference quotient is a formula used to calculate the rate of change of a function between two points. It is given by:

$$\frac{f(x_2)-f(x_1)}{x_2-x_1}$$

Q: How do you determine which function increases at the fastest rate?

A: To determine which function increases at the fastest rate, you can compare the rates of change of the two functions. The function with the greatest rate of change is the one that increases at the fastest rate.

Q: Can you give an example of a function that increases at a slower rate?

A: Yes, consider the function $f(x)=x$. This function has a slope of 1, which means that it increases at a slower rate than the function $g(x)=3x-1$, which has a slope of 3.

Q: What is the significance of understanding the concept of the derivative?

A: Understanding the concept of the derivative is crucial in many fields, including physics, engineering, and economics. It allows us to model and analyze the behavior of complex systems, and to make predictions about future outcomes.

Q: Can you give an example of a real-world application of the concept of the derivative?

A: Yes, consider the example of a car accelerating from 0 to 60 mph. The derivative of the car's velocity with respect to time represents the acceleration of the car. By understanding the concept of the derivative, we can model and analyze the behavior of the car's acceleration, and make predictions about its future behavior.

Conclusion

In this article, we answered some frequently asked questions related to the topic of which function increases at the fastest rate between $x=0$ and $x=8$. We hope that this article has provided you with a better understanding of the concept of the derivative and its application in determining the rates of change of functions.

Additional Resources

For more information on the concept of the derivative and its application in determining the rates of change of functions, we recommend the following resources:

• Textbook: Calculus by Michael Spivak
• Online Course: Calculus by MIT OpenCourseWare
• Video Lecture: Derivatives by Khan Academy

We hope that these resources will be helpful in your studies.