Question

Graph the function \(f(x)=\cos x+\frac{1}{50} \sin 50 x\) using the windows given by the following ranges of \(x\) and \(y\). (a) \(-5 \leq x \leq 5,-1 \leq y \leq 1\) (b) \(-1 \leq x \leq 1,0.5 \leq y \leq 1.5\) (c) \(-0.1 \leq x \leq 0.1,0.9 \leq y \leq 1.1\) Indicate briefly which \((x, y)\) -window shows the true behavior of the function, and discuss reasons why the other \((x, y)\) -windows give results that look different. In this case, is it true that only one window gives the important behavior, or do we need more than one window to graphically communicate the behavior of this function?

Short answer

Both (a) and (c) windows are needed: (a) for overall form and (c) for detailed oscillations.

Step-by-step solution

Understanding the function

The function given is \( f(x) = \cos x + \frac{1}{50} \sin 50x \). It is a combination of a basic cosine function with a slowly varying component and a fast oscillating sine function with a smaller amplitude. This means we have two periodic behaviors superimposed: one slow varying and one rapidly oscillating.

Setting Graph Window for Part (a)

For the interval \(-5 \leq x \leq 5\) and \(-1 \leq y \leq 1\), graph the function to capture a broader view of both slow and rapid oscillations. The cosine term dominates, embedded with rapid oscillations due to the sine term.

Setting Graph Window for Part (b)

For the interval \(-1 \leq x \leq 1\) and \(0.5 \leq y \leq 1.5\), graph to magnify a smaller period of the function, with special interest in capturing both parts of the function potentially more distantly, honing in on a slightly shifted y-range.

Setting Graph Window for Part (c)

For the interval \(-0.1 \leq x \leq 0.1\) and \(0.9 \leq y \leq 1.1\), focus on visualizing the rapid oscillations of the sine component. In this narrow window, the cosine component will appear nearly linear, allowing the sine's rapid oscillations to be observed clearly.

Analyzing Graphs from Different Windows

Part (a) shows the overall behavior of the function but the details of the fast oscillations might appear as 'noise'. Parts (b) and (c) reveal the fine behavior of more localized and rapid features, with part (c) being particularly insightful in observing the sine oscillations.

Assessing the Graph Windows

While part (a) gives a broad view appropriate for initial understanding, part (c) is necessary for capturing the true behavior of the rapid oscillations. Both windows are needed: (a) for general structure and (c) for detail, indicating that multiple windows are necessary for a full understanding of this function's behavior.

Key concepts

Function Behavior

The function we are considering, \( f(x) = \cos x + \frac{1}{50} \sin 50x \), presents a blend of two different types of trigonometric behavior. One part, \( \cos x \), is a simple cosine function, known for its smooth and regular waves that repeat every \(2\pi\). The other part, \( \frac{1}{50} \sin 50x \), introduces a sine function. However, this sine has a much faster oscillation (due to the factor \(50\) within the argument of the sine function) but a smaller amplitude (due to the coefficient \(\frac{1}{50}\)). Together, these components create a function that varies slowly in terms of the cosine curve but possesses rapid fluctuations courtesy of the sine term. Understanding this mixed behavior is crucial in accurately graphing and interpreting the function in different views.

Graph Windows

Selecting the appropriate graph window is pivotal in showcasing different aspects of the function's behavior. Different windows will significantly affect what we observe:

• Large Windows: These offer a broad view, allowing us to see the more significant structure of the function dominated by the slowly varying cosine curve.
• Narrower Windows: These concentrate on a smaller portion of the function, emphasizing the rapid oscillations of the sine component that might be missed in broader views.

Each window can provide unique insights, from understanding the subtle details to viewing the overarching pattern.

Cosine and Sine Functions

Cosine and sine functions are the building blocks of trigonometry.

• Cosine Functions: Cosine functions create smooth, wave-like curves that repeat after every \(2\pi\). They are key for modeling periodic processes.
• Sine Functions: Sine functions, meanwhile, follow a similar repeated pattern but start at a different phase, or point in the cycle, compared to the cosine.

When combined, as in our function, both the cosine and sine functions contribute their periodic properties. They create rich and complex patterns that can model more complicated real-world phenomena, thanks to the interplay of different frequencies and amplitudes.

Rapid Oscillations

Rapid oscillations occur when a trigonometric function has a high frequency, meaning it completes many cycles in a short period. In the function \( f(x) = \cos x + \frac{1}{50} \sin 50x \), the term \( \frac{1}{50} \sin 50x \) represents the rapid oscillations. The sine function here oscillates fifty times faster than the cosine component, due to the factor \(50\) in its argument. Despite their rapid nature, these oscillations have an amplitude of only \(0.02\), making them rapidly fluctuating yet subtle in their vertical impact. Observing these oscillations requires zooming in on small sections, otherwise, they might blend into the overarching cosine wave or appear as noise. Utilizing small windows can effectively reveal these oscillations, giving us a microscopic view of their behavior and how they interact with the slower, larger amplitude variations of the cosine component.