Question

3. (5) For the following problems, either fill in the blank or answer true/false. a) The functions f(x)= x - 1 and g(x)=x* are orthogonal on -T,] b) The product of an odd function and an even function is c) To expand the function f(x)= -x' on [-1, 1). in an appropriate Fourier Series we could use a series. d) Since f(x) = x2 on (0,2) is not an even function, it cannot be expanded in a Fourier cosine series. 5, -<x<' will converge to e) The Fourier series of f(x at x = 0. -1, 0sx<a

Answer 1

g) = f(n) g(x) dhe Esta - xay due = [-T 70 Hence, fins and gens are not orthogonal on L-A, 1] (6) het fins be an even function and gen) be an odd function. Then, Fr, fl-x) = fun and $(-4)=-g(K), So, (f.g) ta) - fol. 91-4). - Fins. (- gini) =- Fenigin) =-(5.2) (x) Hence, fg in an odd function So, the product of odd function an even function in odd function. 1

fino = -23 So, f(-x) = -(-x}3 = x3 = - fou Heme, f in an odd function To enhand the function f12) = -23 on [-1, "], in an afforafonate Jeannier series we could use a sine series (d) sinn, all terms in 5 0,21 in of name ingine in v HE 20, 21, 220; we ran enfand it in a Fearnier carine veries. [ the given onde afflies in internal dike [-1, 1] or [-D, TJ ete ]. Hence, it in fabe. Here, we make it an add function on 1-22) by: fix) = Gut, dans2 1-22, 72 Enco