. (i) Show that the half-range Fourier Cosine Series on the interval (0, L] of the function f(x)=1-\frac{x}{L} is given by i) A metal bar of length L with insulated

ends is initially heated. The temperature of the bar 0(x, t) is governed by the one-dimensional heat equation \frac{\partial \theta}{\partial t}=D \frac{\partial^{2} \theta}{\partial x^{2}} where D is a constant. It is subject to the two boundary conditions \frac{\partial \theta}{\partial x}(x=0, t)=0 \quad \text { and } \quad \frac{\partial \theta}{\partial x}(x=L, t)=0 . (a) Use the method of separation of variables to find the general solution in the form \theta(x, t)=A_{0}+\sum_{n=1}^{\infty} A_{n} e^{-\beta_{n} t} \cos \left(\frac{n \pi x}{L}\right) where the coefficients An, n = 0, 1, 2,... are unknown constants,and the form of the constants B, should be found. (b) Find the solution where the temperature also satisfies the initial con-dition \theta(x, t=0)=1-\frac{x}{L} O Explain how this solution behaves as t 0o and sketch the solution of 0 against r for increasing t. f(x)=\frac{1}{2}+\sum_{k=1}^{\infty} \frac{4}{(2 k-1)^{2} \pi^{2}} \cos \left(\frac{(2 k-1) \pi}{L} x\right) \text {. }