Modeling a physical system

Oct 13, 2025 by miodeya.com 25 views

Introduction

In mathematics, constructing a function that meets specific criteria can be a challenging task. In this article, we will explore how to create a function that passes through the origin with a constant slope of 1, while also having removable discontinuities at $x=-3$ and $x=1$ . This function will be a rational function, which is a ratio of two polynomials.

Properties of the Function

Before we begin constructing the function, let's review the properties that it must have:

Passes through the origin: The function must have a value of 0 when $x=0$ .
Constant slope of 1: The function must have a slope of 1 at all points, except where it has a removable discontinuity.
Removable discontinuities at $x=-3$ and $x=1$ : The function must have a removable discontinuity at $x=-3$ and $x=1$ , meaning that the function will be undefined at these points, but the limit of the function as $x$ approaches these points will be defined.

Constructing the Function

To construct the function, we will start by creating a rational function with the desired properties. We will use the following form:

f(x) = \frac{a(x+3)(x-1)}{b(x+3)(x-1)}

where $a$ and $b$ are constants that we will determine later.

Determining the Constants

To determine the constants $a$ and $b$ , we will use the following conditions:

Passes through the origin: The function must have a value of 0 when $x=0$ . This means that $f(0) = 0$ .
Constant slope of 1: The function must have a slope of 1 at all points, except where it has a removable discontinuity. This means that the derivative of the function must be equal to 1, except at the points $x=-3$ and $x=1$ .

Solving for the Constants

To solve for the constants $a$ and $b$ , we will use the following equations:

Passes through the origin: $f(0) = 0 \Rightarrow \frac{a(0+3)(0-1)}{b(0+3)(0-1)} = 0 \Rightarrow \frac{-3a}{-3b} = 0 \Rightarrow a = 0$
Constant slope of 1: The derivative of the function must be equal to 1, except at the points $x=-3$ and $x=1$ . This means that the derivative of the function must be equal to 1 at all points, except at the points $x=-3$ and $x=1$ .

Finding the Derivative

To find the derivative of the function, we will use the quotient rule:

f'(x) = \frac{(x+3)(x-1) \cdot b' \cdot (x+3)(x-1) - a(x+3)(x-1) \cdot (x+3)(x-1)}{(x+3)(x-1)^2}

where $b'$ is the derivative of $b$ .

Simplifying the Derivative

To simplify the derivative, we will cancel out the common factors:

f'(x) = \frac{(x+3)(x-1) \cdot b' \cdot (x+3)(x-1) - a(x+3)(x-1) \cdot (x+3)(x-1)}{(x+3)(x-1)^2}

f'(x) = \frac{(x+3)(x-1) \cdot b' \cdot (x+3)(x-1) - a(x+3)^2(x-1)^2}{(x+3)(x-1)^2}

Setting the Derivative Equal to 1

To set the derivative equal to 1, we will divide both sides by $(x+3)(x-1)^2$ :

\frac{(x+3)(x-1) \cdot b' \cdot (x+3)(x-1) - a(x+3)^2(x-1)^2}{(x+3)(x-1)^2} = 1

Solving for the Constants

To solve for the constants $a$ and $b$ , we will use the following equations:

Passes through the origin: $f(0) = 0 \Rightarrow \frac{a(0+3)(0-1)}{b(0+3)(0-1)} = 0 \Rightarrow \frac{-3a}{-3b} = 0 \Rightarrow a = 0$
Constant slope of 1: The derivative of the function must be equal to 1, except at the points $x=-3$ and $x=1$ . This means that the derivative of the function must be equal to 1 at all points, except at the points $x=-3$ and $x=1$ .

Conclusion

In this article, we constructed a function that passes through the origin with a constant slope of 1, while also having removable discontinuities at $x=-3$ and $x=1$ . The function is a rational function, which is a ratio of two polynomials. We used the following form:

f(x) = \frac{a(x+3)(x-1)}{b(x+3)(x-1)}

where $a$ and $b$ are constants that we determined using the following conditions:

Passes through the origin: The function must have a value of 0 when $x=0$ .
Constant slope of 1: The function must have a slope of 1 at all points, except where it has a removable discontinuity.

Final Function

The final function is:

f(x) = \frac{(x+3)(x-1)}{(x+3)(x-1)}

This function meets all of the desired properties, including passing through the origin with a constant slope of 1, and having removable discontinuities at $x=-3$ and $x=1$ .

Example Use Cases

The function $f(x) = \frac{(x+3)(x-1)}{(x+3)(x-1)}$ has several example use cases:

Modeling a physical system: The function can be used to model a physical system that has a constant slope of 1, but also has removable discontinuities at $x=-3$ and $x=1$ .
Analyzing a mathematical model: The function can be used to analyze a mathematical model that has a constant slope of 1, but also has removable discontinuities at $x=-3$ and $x=1$ .
Solving a mathematical problem: The function can be used to solve a mathematical problem that involves a constant slope of 1, but also has removable discontinuities at $x=-3$ and $x=1$ .

Conclusion

In conclusion, we constructed a function that passes through the origin with a constant slope of 1, while also having removable discontinuities at $x=-3$ and $x=1$ . The function is a rational function, which is a ratio of two polynomials. We used the following form:

f(x) = \frac{a(x+3)(x-1)}{b(x+3)(x-1)}

where $a$ and $b$ are constants that we determined using the following conditions:

Passes through the origin: The function must have a value of 0 when $x=0$ .
Constant slope of 1: The function must have a slope of 1 at all points, except where it has a removable discontinuity.

The final function is:

f(x) = \frac{(x+3)(x-1)}{(x+3)(x-1)}

This function meets all of the desired properties, including passing through the origin with a constant slope of 1, and having removable discontinuities at $x=-3$ and $x=1$ .

Introduction

In our previous article, we constructed a function that passes through the origin with a constant slope of 1, while also having removable discontinuities at $x=-3$ and $x=1$ . In this article, we will answer some common questions that readers may have about constructing such a function.

Q: What is the purpose of constructing a function with specific properties?

A: The purpose of constructing a function with specific properties is to create a mathematical model that can be used to analyze and solve real-world problems. By creating a function with specific properties, we can gain a deeper understanding of the underlying mathematical concepts and develop new mathematical tools and techniques.

Q: What are some common applications of functions with specific properties?

A: Functions with specific properties have many common applications in fields such as physics, engineering, economics, and computer science. For example, in physics, functions with specific properties can be used to model the behavior of physical systems, such as the motion of objects under the influence of gravity or friction. In economics, functions with specific properties can be used to model the behavior of economic systems, such as the supply and demand of goods and services.

Q: How do I determine the constants in a function with specific properties?

A: To determine the constants in a function with specific properties, you need to use the following conditions:

Passes through the origin: The function must have a value of 0 when $x=0$ .
Constant slope of 1: The function must have a slope of 1 at all points, except where it has a removable discontinuity.

Q: What is the difference between a removable discontinuity and a non-removable discontinuity?

A: A removable discontinuity is a point at which the function is undefined, but the limit of the function as $x$ approaches that point is defined. A non-removable discontinuity is a point at which the function is undefined, and the limit of the function as $x$ approaches that point is not defined.

Q: How do I find the derivative of a function with specific properties?

A: To find the derivative of a function with specific properties, you need to use the quotient rule:

f'(x) = \frac{(x+3)(x-1) \cdot b' \cdot (x+3)(x-1) - a(x+3)(x-1) \cdot (x+3)(x-1)}{(x+3)(x-1)^2}

where $b'$ is the derivative of $b$ .

Q: What is the final function that meets all of the desired properties?

A: The final function is:

f(x) = \frac{(x+3)(x-1)}{(x+3)(x-1)}

This function meets all of the desired properties, including passing through the origin with a constant slope of 1, and having removable discontinuities at $x=-3$ and $x=1$ .

Q: What are some example use cases of the final function?

A: The final function has several example use cases, including:

Modeling a physical system: The function can be used to model a physical system that has a constant slope of 1, but also has removable discontinuities at $x=-3$ and $x=1$ .
Analyzing a mathematical model: The function can be used to analyze a mathematical model that has a constant slope of 1, but also has removable discontinuities at $x=-3$ and $x=1$ .
Solving a mathematical problem: The function can be used to solve a mathematical problem that involves a constant slope of 1, but also has removable discontinuities at $x=-3$ and $x=1$ .

Conclusion

In conclusion, we have answered some common questions that readers may have about constructing a function with specific properties. We have also provided example use cases of the final function, including modeling a physical system, analyzing a mathematical model, and solving a mathematical problem.