Hey guys! Ever wondered about the ups and downs of the function sin(3x)cos(3x)? It's a fun one to explore, and understanding whether it's increasing or decreasing at different points is key to grasping its behavior. In this article, we'll break down this function, looking at how it changes and what that means. We'll use some cool math concepts, but don't worry, I'll explain everything in a way that's easy to follow. Ready to dive in?

Unraveling the Function: A Trigonometric Adventure

Let's start with the basics. The function sin(3x)cos(3x) is a product of two trigonometric functions, sine and cosine. But what makes the '3x' inside the sine and cosine special? Well, it affects the period of the function. Normally, sine and cosine have a period of 2π, meaning their values repeat every 2π units on the x-axis. However, because of the '3x', our function completes three full cycles in the interval of 2π. This means the period of our function is actually 2π/3. This is crucial because the period affects when the function increases and decreases. When we graph this function, it will oscillate more rapidly compared to the standard sin(x)cos(x) function. This oscillation is the continuous change, the ups and downs we will analyze. Moreover, the function is actually the result of trigonometric identity. This identity is the double angle formula for sine, let's explore it! We know that sin(2θ) = 2sin(θ)cos(θ). If we let θ = 3x, then we have sin(6x) = 2sin(3x)cos(3x). This means that sin(3x)cos(3x) = (1/2)sin(6x). This is awesome because it shows us that our function is essentially a scaled-down version of sin(6x). We can use this to understand the function better, including its critical points. Let's delve into the function's domain and range. Since both sine and cosine functions are defined for all real numbers, sin(3x)cos(3x) is also defined for all real numbers, thus its domain is all real numbers. Let's use the identity we discovered before to find its range. We know that the range of sin(6x) is [-1, 1]. This means that the range of (1/2)sin(6x) is [-1/2, 1/2]. This tells us that the values of the function will always lie between -1/2 and 1/2. Pretty cool, huh? The maximum value is 1/2, and the minimum value is -1/2. So, we've got the basics down, now it's time to dig into the increasing and decreasing parts. The key is understanding how sine and cosine behave and how they interact when multiplied.

Unveiling the Increasing and Decreasing Intervals: A Step-by-Step Guide

Alright, let's get into the heart of the matter: finding the increasing and decreasing intervals of sin(3x)cos(3x). To do this, we'll need to use some calculus. Don't sweat it, I'll keep it simple! The core idea is to find the function's derivative. The derivative tells us the rate of change of the function at any given point. If the derivative is positive, the function is increasing; if the derivative is negative, the function is decreasing; and if the derivative is zero, we've found a critical point, which could be a local maximum, a local minimum, or a point of inflection. First, let's use the product rule. The derivative of sin(3x)cos(3x) with respect to x is: d/dx[sin(3x)cos(3x)] = cos(3x) * 3cos(3x) + sin(3x)* * -3sin(3x)*. Simplifying this, we get: 3cos²(3x) - 3sin²(3x). We can then use the trigonometric identity cos²(θ) - sin²(θ) = cos(2θ), to simplify it further. So, the derivative becomes: 3cos(6x). Now, to find the critical points, we set the derivative equal to zero and solve for x. 3cos(6x) = 0 which means cos(6x) = 0. The cosine function is zero at π/2, 3π/2, 5π/2, and so on. So, we need to find the values of x that make 6x equal to these values. Therefore, 6x = π/2 + nπ, where n is an integer. Dividing by 6, we get x = π/12 + nπ/6. These are our critical points! These are the points where the function potentially changes from increasing to decreasing, or vice versa. Next, we test the intervals between these critical points to see if the derivative is positive or negative. The intervals we need to test are: (-∞, π/12), (π/12, 3π/12), (3π/12, 5π/12), and so on. Let's pick a test value in each interval. For the interval (-∞, π/12), we can choose x = 0. The derivative at x = 0 is 3cos(0) = 3, which is positive. This means that the function is increasing in this interval. For the interval (π/12, 3π/12), we can choose x = π/6. The derivative at x = π/6 is 3cos(π) = -3, which is negative. Therefore, the function is decreasing in this interval. Now, you can keep checking these intervals to find where the function increases and decreases. When you perform the test to each interval you will find that the function alternates between increasing and decreasing. Specifically, sin(3x)cos(3x) is increasing when x is in the intervals (π/12 + nπ/6, 3π/12 + nπ/6), and decreasing when x is in the intervals (3π/12 + nπ/6, 5π/12 + nπ/6), where n is an integer. Awesome! It might seem like a lot of steps, but once you break it down, it's pretty straightforward, right?

Visualizing the Function: The Power of Graphs

Graphs are an awesome way to understand a function's behavior. A graph provides a visual representation that can make it easy to see where a function is increasing, decreasing, or at its maximum and minimum values. Imagine plotting sin(3x)cos(3x) on a graph. The x-axis represents the input values of x, and the y-axis shows the output values of the function. The graph will look like a wave that oscillates between -1/2 and 1/2. We already know the function has a period of 2π/3, so you'll see the wave repeat itself over intervals of 2π/3. To find the increasing and decreasing intervals visually, look for the following: Where the graph is going upward, the function is increasing, where the graph is going downward, the function is decreasing, the peaks are the local maxima, and the valleys are the local minima. Plotting the critical points we calculated earlier (x = π/12 + nπ/6) helps you. These points mark the transition between increasing and decreasing intervals. Specifically, at these points, the slope of the curve momentarily becomes zero. They are the turning points on your graph. It is important to know that the graph of sin(3x)cos(3x) is symmetrical. This means that the increasing and decreasing intervals will be mirrored across the graph. Visual aids, like graphing calculators or online graphing tools, can be extremely useful here. You can input the function and zoom in to get a closer look at these intervals and critical points. Being able to see the graph alongside your calculations can really help solidify your understanding of how the function behaves. Using a graphing tool is also a great way to verify your calculations and ensure that you're correctly identifying the intervals. Trust me, it makes things a whole lot easier!

Real-World Applications and Beyond: Where Does This Matter?

Okay, so why should we care about whether sin(3x)cos(3x) is increasing or decreasing? Well, understanding this can be useful in several real-world applications. Trigonometric functions like sine and cosine are used to model periodic phenomena – things that repeat in cycles. In engineering, for example, they are used to analyze circuits and signal processing. In physics, they are used to model waves, like sound and light. Think about a guitar string vibrating; its movement can be described using a sine wave. The function sin(3x)cos(3x), which is a scaled and shifted version of sin(6x), is also used to model the phenomena. Knowing where the function increases and decreases can help you understand the changes happening in the model. This is especially useful if you're trying to identify the points of maximum and minimum oscillation or the rate of change at certain times. Moreover, these functions are fundamental to more complex mathematical models. Understanding them is a gateway to grasping more advanced concepts. They are also essential in fields like image processing and computer graphics, where you can describe the intensity of light or color in pixels using trigonometric functions. So, by understanding sin(3x)cos(3x), you're building a foundation for various applications.

Key Takeaways: Recap and Reflections

Alright guys, let's recap everything we've covered! We've explored the function sin(3x)cos(3x) and investigated its increasing and decreasing intervals. We began by understanding the function's structure, recognizing that it's a product of sine and cosine and that it can also be expressed as (1/2)sin(6x). This tells us its period is 2π/3, and its range is [-1/2, 1/2]. We then used calculus, specifically the derivative, to find the critical points and the intervals where the function increases and decreases. The derivative is 3cos(6x), and we found that the critical points are located at x = π/12 + nπ/6, where n is an integer. With this information, we could identify the increasing intervals as (π/12 + nπ/6, 3π/12 + nπ/6) and the decreasing intervals as (3π/12 + nπ/6, 5π/12 + nπ/6). We also touched on the usefulness of graphing to visualize these behaviors, and finally, we looked at some real-world applications where understanding this function can be valuable. So, next time you see this function, you'll know exactly how it behaves! Thanks for sticking with me. I hope you found this guide helpful and that you now have a better grasp of the oscillations of sin(3x)cos(3x)!