Chapter 5.4, Problem 49PE

In this section, we used the product-to-sum formulas to write sums and differences of trigonometric functions. Another type of expression involving the sum or difference of sine and cosine terms is a Fourier series named after Jean-Baptiste Joseph Fourier (1768-1830). A Fourier series is an expression of the form $A_0+\left(A_1\cos x+B_1\sin x\right)+\left(A_2\cos 2x+B_2\sin 2x\right)+\left(A_3\cos 3x+B_3\sin 3x\right)+\mathrm{...}$ where $A_1,A_2,A_3,\mathrm{...}$ and $B_1,B_2,B_3,\mathrm{...}$ , are constants. Fourier proved that any continuous function can be represented by a Fourier series. In Exercises 49-50, we use Fourier polynomials (a finite number of terms from a Fourier series) to model a "saw-tooth" wave and a "square" wave.

A “saw-tooth" wave is a periodic wave that rises linearly upward and then drops sharply. In music, such waves are generated in digital synthesizers to give high-quality sound without distortion. Given,

$f\left(x\right)\frac{1}{2}-\frac{1}{\pi }\left(\frac{\sin \pi x}{1}+\frac{\sin 2\pi x}{2}+\frac{\sin 3\pi x}{3}+\cdot \cdot \cdot \right)$

a. Graph the first three terms of the function on the window $\left[-4,4,1\right]$ by $\left[-1,1.5,0.5\right]$ .

b. Graph the first five terms of the function on the window $\left[-4,4,1\right]$ by $\left[-1,1.5,0.5\right]$ .