Consider the function ft defined on the interval 0 le t lt

Question

Question 1 - Consider the function f(t) defined on the interval 0 ≤ t < 1 by f (t) = t(1 - t).

(a) Sketch the graph of the odd extension f_odd of f for -3 ≤ t ≤ 3, and hence state the fundamental period of the odd extension.

(b) Sketch the graph of the even extension f_even of f for -3 ≤ t ≤ 3, and hence state the fundamental period of the even extension.

Question 2 - Consider the periodic function f(t) with fundamental interval -π ≤ t ≤ π that is defined by

          -t-π     for    -π ≤ t < 0

f(t) =

          t - π    for    0 ≤ t < π,

f(t + 2π) = f(t).

(a) Sketch the graph of the function f for -3π ≤ t ≤ 3π, and hence state whether the function is even, odd, or neither even nor odd.

(b) Calculate the Fourier series for f(t).