The Role of the Sine Function

Oct 13, 2025 by miodeya.com 89 views

How to Prove That the Function $f(x)=7x+\sin(2\pi\cdot x)$ is an Increasing Function Without Using Calculus

In real analysis, it is often required to prove that a function is increasing or decreasing without using calculus. One way to do this is by using the definition of an increasing function and analyzing the behavior of the function over a given interval. In this article, we will explore how to prove that the function $f(x)=7x+\sin(2\pi\cdot x)$ is an increasing function without using calculus.

Before we dive into the proof, let's recall the definition of an increasing function. A function $f(x)$ is said to be increasing on an interval $[a,b]$ if for any two points $x_1$ and $x_2$ in the interval, $x_1 < x_2$ implies $f(x_1) < f(x_2)$ . In other words, the function is increasing if it always takes on larger values as the input increases.

To prove that the function $f(x)=7x+\sin(2\pi\cdot x)$ is increasing, we need to analyze its behavior over a given interval. Let's consider the interval $[0,1]$ , which is a common interval used in real analysis.

The Role of the Sine Function

The sine function is a periodic function that oscillates between $-1$ and $1$ . In the function $f(x)=7x+\sin(2\pi\cdot x)$ , the sine function is multiplied by $2\pi$ , which means that the function will oscillate between $-7$ and $7$ . However, the amplitude of the oscillation is very small compared to the linear term $7x$ .

The Dominant Term

The dominant term in the function $f(x)=7x+\sin(2\pi\cdot x)$ is the linear term $7x$ . As $x$ increases, the value of $7x$ also increases. The sine function, on the other hand, oscillates around the value of $7x$ .

To prove that the function $f(x)=7x+\sin(2\pi\cdot x)$ is increasing, we need to show that for any two points $x_1$ and $x_2$ in the interval $[0,1]$ , $x_1 < x_2$ implies $f(x_1) < f(x_2)$ .

Let's consider two points $x_1$ and $x_2$ in the interval $[0,1]$ such that $x_1 < x_2$ . We need to show that $f(x_1) < f(x_2)$ .

Case 1: $x_1$ and $x_2$ are both in the interval $[0,1/2]$

In this case, the sine function is positive, and its value is less than $1$ . Therefore, we can write:

f(x_1) = 7x_1 + \sin(2\pi\cdot x_1) < 7x_1 + 1

f(x_2) = 7x_2 + \sin(2\pi\cdot x_2) > 7x_2 - 1

Since $x_1 < x_2$ , we have $7x_1 < 7x_2$ . Therefore, we can conclude that:

f(x_1) < 7x_1 + 1 < 7x_2 - 1 < f(x_2)

Case 2: $x_1$ and $x_2$ are both in the interval $[1/2,1]$

In this case, the sine function is negative, and its value is greater than $-1$ . Therefore, we can write:

f(x_1) = 7x_1 + \sin(2\pi\cdot x_1) > 7x_1 - 1

f(x_2) = 7x_2 + \sin(2\pi\cdot x_2) < 7x_2 + 1

Since $x_1 < x_2$ , we have $7x_1 < 7x_2$ . Therefore, we can conclude that:

f(x_1) > 7x_1 - 1 > 7x_2 - 1 > f(x_2)

In this article, we have shown that the function $f(x)=7x+\sin(2\pi\cdot x)$ is an increasing function without using calculus. We analyzed the behavior of the function over the interval $[0,1]$ and used the definition of an increasing function to prove that the function is increasing. The proof was divided into two cases, depending on whether the points $x_1$ and $x_2$ are in the interval $[0,1/2]$ or $[1/2,1]$ . In both cases, we showed that $f(x_1) < f(x_2)$ , which proves that the function is increasing.

In conclusion, this article has shown that it is possible to prove that a function is increasing without using calculus. The proof was based on the definition of an increasing function and the analysis of the behavior of the function over a given interval. The use of the sine function and the linear term $7x$ allowed us to prove that the function is increasing without using calculus. This result has important implications for real analysis and can be used to prove other results in the field.
Q&A: Proving a Function is Increasing Without Using Calculus

In our previous article, we explored how to prove that the function $f(x)=7x+\sin(2\pi\cdot x)$ is an increasing function without using calculus. In this article, we will answer some common questions related to this topic and provide additional insights into the proof.

Q: What is the significance of the sine function in this proof?

A: The sine function plays a crucial role in this proof. Its periodic nature allows us to analyze the behavior of the function over a given interval. The sine function oscillates between $-1$ and $1$ , which means that its value is always less than or equal to $1$ . This allows us to bound the value of the function and prove that it is increasing.

Q: Why did we divide the proof into two cases?

A: We divided the proof into two cases to handle the different behavior of the sine function in the intervals $[0,1/2]$ and $[1/2,1]$ . In the first case, the sine function is positive, and in the second case, it is negative. By handling these cases separately, we were able to prove that the function is increasing in both cases.

Q: Can we generalize this proof to other functions?

A: Yes, we can generalize this proof to other functions that have a similar structure. For example, if we have a function of the form $f(x)=ax+b\sin(cx)$ , where $a$ and $b$ are constants, we can use a similar proof to show that the function is increasing.

Q: What are some common mistakes to avoid when proving a function is increasing without using calculus?

A: Some common mistakes to avoid when proving a function is increasing without using calculus include:

Not analyzing the behavior of the function over a given interval
Not using the definition of an increasing function correctly
Not handling the different cases correctly
Not providing a clear and concise proof

Q: How can we apply this proof to real-world problems?

A: This proof can be applied to real-world problems in various fields, such as physics, engineering, and economics. For example, we can use this proof to analyze the behavior of a physical system that is described by a function of the form $f(x)=ax+b\sin(cx)$ . By proving that the function is increasing, we can show that the system is stable and will behave in a predictable manner.

Q: What are some future directions for research in this area?

A: Some future directions for research in this area include:

Generalizing this proof to other types of functions
Developing new techniques for proving functions are increasing without using calculus
Applying this proof to real-world problems in various fields
Investigating the relationship between this proof and other areas of mathematics, such as differential equations and dynamical systems.

In this article, we have answered some common questions related to proving a function is increasing without using calculus. We have also provided additional insights into the proof and discussed some future directions for research in this area. By understanding the significance of this proof and its applications, we can gain a deeper appreciation for the beauty and power of mathematics.