Consider a function f(x) defined for x[0,L], not necessarily periodic. To approximate f(x) using Fourier series, we can extend f(x) to the interval x[L,0] so as to get either an odd or even function. The even extension may be more advantageous in some situations, while in other cases the odd one can be better. We will investigate both scenarios in this exercise, using truncated series. a) Consider f(x)=ex,0x1. Sketch (draw or plot) the odd and the even extensions of f(x) in the interval 1x1. b) Based on your sketch, answer: which of the extensions should be easier to approximate with a Fourier partial sum? In other words, if we fix the number N of terms in the series, for which case (even or odd extension) can we expect a better approximation of f(x) ? Hint: use your plots to assess the regularity (smoothness) of each extended function. Consider a function f(x) defined for x[0,L], not necessarily periodic. To approximate f(x) using Fourier series, we can extend f(x) to the interval x[L,0] so as to get either an odd or even function. The even extension may be more advantageous in some situations, while in other cases the odd one can be better. We will investigate both scenarios in this exercise, using truncated series. a) Consider f(x)=ex,0x1. Sketch (draw or plot) the odd and the even extensions of f(x) in the interval 1x1. b) Based on your sketch, answer: which of the extensions should be easier to approximate with a Fourier partial sum? In other words, if we fix the number N of terms in the series, for which case (even or odd extension) can we expect a better approximation of f(x) ? Hint: use your plots to assess the regularity (smoothness) of each extended function