This question concerns a cantilevered beam of length 1, that is clamped at x = 0, and free at the other end x = 1

This question concerns a cantilevered beam of length 1, that is clamped at x = 0, and free at the other end x = 1. The equation for such a beam subjected to a vertical load w(x) at each point x along its length is 
d'y = —kw(x), y(0) = y'(0) = 0 = y"(/) = ym(/), 
dx4 where y(x) is the deflection of the beam at x and k is a constant. You may assume that the weight of the beam is negligible, and that the load w(x) decreases linearly from too at x = 0 to zero at x = 2, and remains zero from x = i to 1. (a) Find a function f (x) in terms of wo and the Heaviside function that extends the definition for w(x) to (0, oo). Note that, as the Laplace transform is an integral from 0 to oo, you'll need to extend the definition for w(x) to (0, oo) in order to apply the Laplace transform to w(x). (b) Solve the initial value problem with y"(0) = a, ym(0) = b. (c) Evaluate y" (1) and ym (1) in your solution, obtaining equations for a and b. Hence find a and b. 
(d) Solve the initial value problem with the same initial conditions when the function f (x) is replaced by 
w(x) 0 < x < 1; g(x) = w(21 — x) 1 < x < 21; { 0 for all x > 2/, that also extends w(x) to (0, oo). Is your solution the same as that obtained from the previous extension f (x) for x E [0, 1]? (The point of this question is less of a physical problem than a mathematical one, which is to determine the effects, if any, that the extension function and its domain have on the Laplace transform. Hence this part of the problem may not have a physical interpretation.) 
 

Consider the periodic functions that are defined by 
and 
0 < t < 27r, f (t) t f(t + 27r) for all t, 
{ t2 g (t) = g (t + 27r) 
0 < t < 27r; for all t, 
(a) Calculate the Fourier series for f and g. (b) Hence find the two polynomials that the two series 
0. si n nt ) n n=1 
I cos nt n2 n=1 
respectively converge to on the interval (0, 27r). (c) Produce good quality graphs of the limit functions in (b), showing at least three periods. (d) Use Maple or MATLAB (or another alternative) to plot <94, 86, sg on the same graph, where sN(t) is the truncation of the each series in (b) at the Nth term. Is there any overshoot?