Please help me with (b) (c) (d) (e) of question 2

2. Consider the following function: f(x) = 1 re (0, ;) 10 re (,a) (2) (a) Find both the sine and cosine series representations of f. (Hint: first make f either odd or even, then make it periodic and find the Fourier series of the new funciton) (b) Let f be the initial data for the standard heat equation on a bar of length a with a tempurature on both end points fixed to be -. Sketch (on the samge graph) what you think u(x, 0), u(x, 1), u(x, 10), and u(x, 100) look like. (c) Let f be the initial data for the standard heat equation on a bar of length a with two insulated ends. Sketch (on the samge graph) what you think u(x, 0), u(x, 1), u(x, 10), and u(x, 100) look like. (d) Solve the heat equation on the bar with two insulated ends. Compare the actual solution with your graph in part (c). (e) Let a = 1. The value of u(.75, 0) is 0. Approximate the value of u(.75, .1). Is this evidence for or against the claim the heat travels instantaneously