Figure 2: Basis functions (y_{n} = sin(nx)) for (n = 1, 2, 3, 4, 5)
The integrals defined in line (3) are meaningful numbers for any integrable function (f). Treating a given function as if it were a combination of periodic functions opens the door to Fourier analysis, a powerful tool with many applications in science, engineering, and mathematics. Present exact calculations where they are requested below, but use suitable software to produce the corresponding plots.
(a) (1 mark) Make three plots showing (y = f(x)) and (y = S_{N}(x)) on the same axes, where
(S_{N}(x) = sum_{n=1}^{N} B_{n}(f) sin(nx))
Use (N = 3) for the first plot, (N = 5) for the second, and (N = 11) for the third.
(b) (1 mark) Let (widetilde{S}_{N}(x) = sum_{n=1}^{N} B_{n}(g) sin(nx))
Use (N = 3) for the first plot, (N = 5) for the second, and (N = 11) for the third.
The integrals defined in line (3) are meaningful numbers for any integrable function (f). Treating a given function as if it were a combination of periodic functions opens the door to Fourier analysis, a powerful tool with many applications in science, engineering, and mathematics. Present exact calculations where they are requested below, but use suitable software to produce the corresponding plots.
(a) 1 mark Let (f) for (0 < x < pi). Find a formula for (B_{n}f) valid for every integer (n > 1).
(b) 1 mark Make three plots showing (y = f(x)) and (y = S_{N}(x)) on the same axes, where
(S_{N}(x) = B_{n}f) sin(nx)).
Use (N = 3) for the first plot, (N = 5) for the second, and (N = 11) for the third.
(c) 1 mark Let (g = sqrt{x}) for (0 < x < pi). Find a formula for (B_{g}) valid for every integer (n > 1).
(d) 1 mark Make three plots showing (y = g) and (y = S_{N}(x)) on the same axes, where
(S_{N}(x) = B_{n}(g) sin(nx)).
Use (N = 3) for the first plot, (N = 5) for the second, and (N = 11) for the third.
