Question

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=x+0.01 \sin 50 x $$

Short answer

The graph of the function \(f(x) = x + 0.01\sin{(50x)}\) can be plotted using a suitable viewing window. We find that a good viewing window for the graph is \([0,\frac{\pi}{25}] \times [-0.01,0.13]\) or, in decimal form: \([0,0.125] \times [-0.01,0.13]\). Use a graphing calculator or software to plot the function within this viewing window to observe a combination of linear growth and small oscillation created by the sinusoidal term.

Step-by-step solution

1. Identify the given function

The given function is \(f(x) = x + 0.01\sin{50x}\).

2. Analyzing the function

The function is a combination of a linear element, \(x\), and an oscillating element, \(0.01\sin{50x}\). The linear element determines the overall growth of the function, while the oscillating element gives the function small perturbations as x varies. We know that \(\sin(x)\) oscillates between -1 and 1, so the term \(0.01\sin{50x}\) will oscillate between -0.01 and 0.01 with a frequency of 50 times the frequency of \(\sin{x}\).

3. Choosing an appropriate viewing window

Since the small oscillating term \(0.01\sin{50x}\) affects the graph's appearance, we want to choose a viewing window that captures it. We also want to make sure the linear growth from the \(x\) term is visible. The period of the \(\sin{50x}\) function is given by: \(\frac{2\pi}{k}\), where \(k\) is the coefficient of x (in this case, \(k = 50\)). The period is then equal to: \(\frac{2\pi}{50} = \frac{\pi}{25}\). To capture one full period of the sinusoidal term, let's choose a window on the x-axis as \([0,\frac{\pi}{25}]\). To find the bounds on the y-axis, let's consider the minimum and maximum the function can take within our chosen x-axis range. The minimum occurs when \(f(x) = x - 0.01\), and the maximum occurs when \(f(x) = x + 0.01\). When \(x = 0\), the minimum value will be -0.01 and the maximum value will be 0.01. When \(x = \frac{\pi}{25}\), the minimum value is \(x - 0.01\) and the maximum value is \(x + 0.01\). So the y-axis bounds will be between -0.01 and \(0.13\), approximately. Thus, a suitable viewing window for the graph is \([0,\frac{\pi}{25}] \times [-0.01,0.13]\) or, in decimal form: \([0,0.125] \times [-0.01,0.13]\).

4. Plotting the graph

Using a graphing calculator or software, plot the function \(f(x) = x + 0.01\sin{(50x)}\) using the viewing window of \([0,0.125] \times [-0.01,0.13]\). The graph will show the linear growth and small oscillations added by the sinusoidal term. If possible, zoom in on the graph to examine the oscillations more closely. This should help visualize the influence of the sinusoidal term on the overall function.

Key concepts

Oscillating Functions

When studying mathematical functions, we come across different behaviors that functions can exhibit. One such behavior is oscillation, observed in oscillating functions. Oscillating functions are characterized by their recurrent up and down movement between two values within a specific interval, much like the motion of a pendulum or a wave.

Understanding oscillating functions is crucial in many areas of science and engineering, particularly in physics where they describe phenomena like sound waves or alternating currents. The example function from the exercise, involving both a linear and an oscillating term, is a wonderful demonstration of how such functions can be combined.

Mathematically, the amplitude of oscillation in our example is determined by the coefficient before the sine function. In this case, the maximum amplitude is 0.01, which means our function won't oscillate more than 0.01 units above or below the line dictated by the linear term.

Sinusoidal Functions

A sinusoidal function is a specific type of oscillating function and is well-represented by the sine and cosine functions, which are fundamental in the study of trigonometry. These functions depict smooth, periodic oscillations and are defined for all real numbers, making them continuous and infinitely differentiable.The function in the exercise includes a sinusoidal part, specifically a sine function multiplied by a constant factor. The key features of this sinusoidal component include its amplitude - the height of the wave peaks, its period - the distance over which the function repeats, and its frequency - how often the oscillation occurs over a fixed interval. A higher frequency, as in our case with the multiplier of 50 in the sine function, results in more oscillations over a given interval. The step-by-step solution carefully analyzes these features to understand fully how the function behaves.

Viewing Window in Graphs

Selecting an appropriate viewing window is critical when graphing functions, particularly for oscillating functions where the details can be subtle. The viewing window is the portion of the coordinate plane that we choose to display our graph. The right window can make all the difference in understanding the graph's behavior and appreciating its nuances.

In the step-by-step solution for graphing our function, the period of the oscillation is used to determine an appropriate width for the x-axis, ensuring that at least one complete cycle of the sinusoidal part is visible. For the y-axis, calculating the maximum and minimum values of the function within the chosen x-axis range ensures that all the significant changes in value are visible.

This precise approach to choosing a viewing window allows for an accurate representation of the function, highlighting both the linear aspect and the intricate oscillations superimposed upon it, as shown by the combination of linear growth and the small amplitude sinusoidal wave.