A Square Wave Function and Fourier Series Project: Figure 5-7j shows the graph of The

Chapter 5, Problem C.2

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A Square Wave Function and Fourier Series Project: Figure 5-7j shows the graph of The smaller sinusoid pulls the larger one closer to the x-axis at some places and pushes it farther away at others. The terms in the given equation form this partial sum of a Fourier series. y = cos x cos 3x + cos 5x cos 7x + cos 9x cos 11x The more terms that are added to the partial sum, the more the result looks like a square wave. For instance, the graph of the partial sum with 11 terms is shown in Figure 5-7k. Figure 5-7k Jean-Baptiste-Joseph Fourier, French mathematician (17681830) a. Plot the graph of the partial sum with 11 terms and show that it is similar to the graph in Figure 5-7k. Find the sum and sequence commands that will allow you to write the equation without typing in all of the terms. b. If you plot the partial sum with 12 terms, some of the high points and low points in Figure 5-7k are reversed. So if you average the 11-term and the 12-term sums, you get a better square-wave pattern. Do this on your grapher. (If you perform some clever calculations first, you can find a relatively easy way to add one term to the 11-term sum that gives the average without having to compute the 12-term sum.) c. Plot the average of the 50-term sum and the 51-term sum. How close does the graph come to a square wave? d. When the square wave is on the high portion of its graph, there is an axis about which it oscillates. Find the y-value of this axis. Explain how you found your answer. How is the answer related to ? e. Plot the 10th partial sum of the Fourier sine series y = sin x + sin 2x + sin 3x + + sin 10x From your graph, figure out why the result is called a sawtooth wave pattern. T