Fourier series

Question 1

Consider the function f(t) defined on the interval 0 ≤ t < 1 by

f (t) = t(1 − t).

1. (a)  Sketch the graph of the odd extension fodd of f for −3 ≤ t ≤ 3, and

   hence state the fundamental period of the odd extension. [4]

2. (b)  Sketch the graph of the even extension feven 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).