Question

The Sawtooth Curve An oscilloscope often displays a sawtooth curve. This curve can be approximated by sinusoidal curves of varying periods and amplitudes. (a) Use a graphing utility to graph the following function, which can be used to approximate the sawtooth curve. $$ f(x)=\frac{1}{2} \sin (2 \pi x)+\frac{1}{4} \sin (4 \pi x) \quad 0 \leq x \leq 4 $$ (b) A better approximation to the sawtooth curve is given by $$ f(x)=\frac{1}{2} \sin (2 \pi x)+\frac{1}{4} \sin (4 \pi x)+\frac{1}{8} \sin (8 \pi x) $$ Use a graphing utility to graph this function for \(0 \leq x \leq 4\) and compare the result to the graph obtained in part (a). (c) A third and even better approximation to the sawtooth curve is given by \(f(x)=\frac{1}{2} \sin (2 \pi x)+\frac{1}{4} \sin (4 \pi x)+\frac{1}{8} \sin (8 \pi x)+\frac{1}{16} \sin (16 \pi x)\) Use a graphing utility to graph this function for \(0 \leq x \leq 4\) and compare the result to the graphs obtained in parts (a) and (b). (d) What do you think the next approximation to the sawtooth curve is?

Short answer

The next approximation is likely \[ f(x) = \frac{1}{2} \sin (2 \pi x) + \frac{1}{4} \sin (4 \pi x) + \frac{1}{8} \sin (8 \pi x) + \frac{1}{16} \sin (16 \pi x) + \frac{1}{32} \sin (32 \pi x) \]

Step-by-step solution

Graphing the First Function

Use a graphing utility to plot the function \[ f(x) = \frac{1}{2} \sin (2 \pi x) + \frac{1}{4} \sin (4 \pi x) \] over the interval \(0 \leq x \leq 4\). To do this, open your graphing utility, input the function, and specify the range for \(x\).

Observing the First Graph

Observe the plot of the first function. Notice the general shape of the curve and how closely it resembles a sawtooth wave.

Graphing the Second Function

Next, graph the function \[ f(x) = \frac{1}{2} \sin (2 \pi x) + \frac{1}{4} \sin (4 \pi x) + \frac{1}{8} \sin (8 \pi x) \] over the same interval \(0 \leq x \leq 4\). Input this new function into your graphing utility and plot it.

Comparing the First and Second Graphs

Compare the graph of the second function to the graph of the first function. Notice that the second graph appears to be a better approximation of a sawtooth curve with sharper peaks and valleys.

Graphing the Third Function

Now graph the function \[ f(x) = \frac{1}{2} \sin (2 \pi x) + \frac{1}{4} \sin (4 \pi x) + \frac{1}{8} \sin (8 \pi x) + \frac{1}{16} \sin (16 \pi x) \] over the interval \(0 \leq x \leq 4\). Input this function into your graphing utility and plot it.

Comparing All Graphs

Compare the graph of the third function to the graphs of the first and second functions. Notice how the third function provides an even more accurate approximation of a sawtooth curve with even sharper features.

Predicting the Next Approximation

Based on the pattern observed from the previous approximations, the next approximation to the sawtooth curve could be given by the function \[ f(x) = \frac{1}{2} \sin (2 \pi x) + \frac{1}{4} \sin (4 \pi x) + \frac{1}{8} \sin (8 \pi x) + \frac{1}{16} \sin (16 \pi x) + \frac{1}{32} \sin (32 \pi x) \]

Key concepts

sinusoidal function

A sinusoidal function is a type of smooth, periodic oscillation like sine or cosine. In mathematical terms, it oscillates between two values, following the formula \(y = A \sin(Bx + C) + D\). Here, A represents the amplitude, B affects the period of the waveform, C indicates the phase shift, and D is the vertical shift. Sinusoidal functions resemble smooth curves like waves and are fundamental in various fields such as physics and engineering.
In the context of the sawtooth curve approximation, each term in the given functions \(f(x)\) is a sinusoidal function with different amplitudes and frequencies. Combining these sinusoidal functions helps create a more complex wave pattern that approximates the sawtooth shape. For instance: \[f(x) = \frac{1}{2} \sin(2 \pi x) + \frac{1}{4} \sin(4 \pi x)\] The function here effectively layers multiple sine waves to achieve a closer approximation of the desired shape.

graphing utility

A graphing utility is an essential tool to visualize mathematical functions. It can be a graphing calculator, computer software, or an online platform that accurately plots functions based on user inputs. Using a graphing utility helps to understand the behavior of complex functions more interactively.
To graph the given functions for the sawtooth curve approximation, follow these steps:

• Open your chosen graphing utility.
• Input the function, such as \(f(x) = \frac{1}{2} \sin(2 \pi x) + \frac{1}{4} \sin(4 \pi x)\).
• Specify the interval range, for example \(0 \leq x \leq 4\).
• Observe and analyze the plotted graph.

Comparing different graphs generated using increasingly complex functions allows a better understanding of how additional sinusoidal terms improve the sawtooth curve approximation.

trigonometric series

A trigonometric series is a series of terms involving trigonometric functions like sine and cosine. These series are particularly useful in approximating periodic functions due to their inherent periodicity.
In the case of the sawtooth curve approximation, a trigonometric series of the form: \[ f(x) = \frac{1}{2} \sin(2 \pi x) + \frac{1}{4} \sin(4 \pi x) + \frac{1}{8} \sin(8 \pi x) + \frac{1}{16} \sin(16 \pi x)\] effectively adds together multiple sine functions with smaller and smaller amplitudes and higher and higher frequencies. Each term added makes the approximation closer to the actual sawtooth curve. The pattern continues and, by including more terms, the resulting graph becomes sharper and better defined.
Understanding trigonometric series helps us appreciate how complex waveforms can be constructed by summing simple sinusoidal components.