Advanced Calculus fi..

Advanced Calculus, Fifth Editionpointed out at the end of Section 7.1. The phase angle cr effectively determines howmuch the curve y = sin nx has to be shifted to the left or right to match the givenoscillation; the amplitude A, merely adjusts the vertical scale.It should be remarked that the process of decomposing a periodic phenomenoninto its component simple harmonic parts is used in a great variety of commonexperiences. The rattling of an old-fashioned vehicle corresponds to a high-frequency(large n) component with large amplitude; we automatically separate this from a lowfrequencyvibration, or swaying. In a less precise sense, a large day-to-day fluctuationin weather conditions also corresponds to a high-frequency component of largeamplitude; the seasonal changes are of low frequency and are much less disturbing.PROBLEMS1. Find the Fourier series for each of the following functions:I) G(X)=;-5-$, IT -x5~50; G(x)=f +$-g, 0sxsn(It is suggested that the first few partial sums be graphed and compared with the functionin each case.)2. It follows from the fundamental theorem of Section 7.3 that if f (x) is defined between0 and 2n and is piecewise very smooth in that interval, then f (x) can be represented bya series of form (7.1) in that interval.a) Show that the coefficients a, and b, are given by the formulas:2n1 2na,=L/ f(x)cosnxdx. b,,=--1 f (x) sin nx dx.x 0b) Extend this result to a function defined from x = c to x = c + 2n, where c is anyconstant.3. Using the results of Problem 2(a), find Fourier series for the following functions:4. Determine which of the following functions are periodic and find the smallest period ofthose that are periodic:a) sin5x b) cos 5 c) sinxxd) x sinx e) sin 3x + sin 5x . f) sin $ + sin 4g) sin x + sin xx5. The result of Problem l(a) implies that for 0 < x < x,Use x = i n in this equation to show thatsin 3xsinx+ - +...3