How To Prove That The Function $f(x)=7x+\sin(2\pi\cdot X)$ Is An Increasing Function Without Using Calculus (derivative)??

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.

What is an Increasing Function?

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.

Analyzing the Function

The given function is $f(x)=7x+\sin(2\pi\cdot x)$. To prove that this function is increasing, we need to show that for any two points $x_1$ and $x_2$ in the interval, $x_1 < x_2$ implies $f(x_1) < f(x_2)$. Let's start by analyzing the behavior of the function over a small interval.

Using the Definition of an Increasing Function

Let $x_1$ and $x_2$ be any two points in the interval $[a,b]$, where $x_1 < x_2$. We need to show that $f(x_1) < f(x_2)$. Using the definition of the function, we have:

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

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

To show that $f(x_1) < f(x_2)$, we can use the following inequality:

$$7x_1 + \sin(2\pi\cdot x_1) < 7x_2 + \sin(2\pi\cdot x_2)$$

Using the Properties of Sine Function

The sine function has a periodic nature, with a period of $2\pi$. This means that the value of the sine function repeats every $2\pi$ units. We can use this property to simplify the inequality above.

Let $k$ be an integer such that $2\pi\cdot x_1 + 2\pi\cdot k < 2\pi\cdot x_2$. Then, we can write:

$$\sin(2\pi\cdot x_1) = \sin(2\pi\cdot x_2 - 2\pi\cdot k)$$

Using the property of the sine function, we know that:

$$\sin(\theta - 2\pi\cdot k) = \sin(\theta)$$

for any angle $\theta$ and integer $k$. Therefore, we can simplify the inequality above as:

$$7x_1 + \sin(2\pi\cdot x_1) < 7x_2 + \sin(2\pi\cdot x_2)$$

Using the Linearity of the Function

The function $f(x)$ is a linear function, with a slope of $7$. This means that the function is always increasing, and the value of the function increases linearly with the input.

Let $x_1$ and $x_2$ be any two points in the interval $[a,b]$, where $x_1 < x_2$. Then, we can write:

$$f(x_2) - f(x_1) = 7(x_2 - x_1)$$

Since the function is increasing, we know that $f(x_2) - f(x_1) > 0$. Therefore, we can conclude that:

$$7(x_2 - x_1) > 0$$

Conclusion

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 used the definition of an increasing function and analyzed the behavior of the function over a small interval. We also used the properties of the sine function and the linearity of the function to simplify the inequality and conclude that the function is increasing.

Final Answer

The final answer is: $\boxed{Yes}$
Q&A: Proving the Function $f(x)=7x+\sin(2\pi\cdot x)$ is an Increasing Function 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 frequently asked questions related to this topic.

Q: What is the definition of an increasing function?

A: An increasing function is a function that always takes on larger values as the input increases. In other words, if $x_1 < x_2$, then $f(x_1) < f(x_2)$.

Q: Why is it important to prove that a function is increasing?

A: Proving that a function is increasing is important because it helps us understand the behavior of the function. For example, if a function is increasing, we know that the value of the function will always increase as the input increases.

Q: How do we prove that a function is increasing without using calculus?

A: We can prove that a function is increasing without using calculus by analyzing the behavior of the function over a small interval. We can use the definition of an increasing function and the properties of the function to simplify the inequality and conclude that the function is increasing.

Q: What are some common properties of increasing functions?

A: Some common properties of increasing functions include:

• The function is always increasing, and the value of the function increases linearly with the input.
• The function has a positive slope.
• The function is continuous and differentiable.

Q: Can we use calculus to prove that a function is increasing?

A: Yes, we can use calculus to prove that a function is increasing. One way to do this is by finding the derivative of the function and showing that it is always positive.

Q: What are some common mistakes to avoid when proving that a function is increasing?

A: Some common mistakes to avoid when proving that a function is increasing include:

• Assuming that the function is increasing without analyzing the behavior of the function over a small interval.
• Using calculus to prove that a function is increasing without showing that the derivative is always positive.
• Failing to consider the properties of the function, such as continuity and differentiability.

Q: Can we use numerical methods to prove that a function is increasing?

A: Yes, we can use numerical methods to prove that a function is increasing. One way to do this is by using a computer program to evaluate the function at a large number of points and showing that the value of the function always increases.

Q: What are some real-world applications of increasing functions?

A: Some real-world applications of increasing functions include:

• Modeling population growth: Increasing functions can be used to model population growth, where the population increases over time.
• Modeling economic growth: Increasing functions can be used to model economic growth, where the economy grows over time.
• Modeling physical systems: Increasing functions can be used to model physical systems, such as the motion of an object under the influence of gravity.

Conclusion

In this article, we have answered some frequently asked questions related to proving that a function is increasing without using calculus. We have discussed the definition of an increasing function, the importance of proving that a function is increasing, and some common properties of increasing functions. We have also discussed some common mistakes to avoid when proving that a function is increasing and some real-world applications of increasing functions.