A model for the temperature u = u(x, t) u(r, 0) = f(z) in a thin metal rod is given by the (initial temp. distribution) One-Dimensional Heat Equation : w(0,!) =0 a(1,t) =0 au at 2x2 In this formula, x represents the distance from one end of the rod (measured in cm), t represents time (measured in sec), and k is a constant ; the output temperature u = u(x, t) is given in K (Kelvins). Use the heat function : u(x, t) = e Ft sin(wx) for parts (a) -(c) , where B, w are positive constants. (a) Compute both first-order partial derivatives of u = u(x, t) (b) Verify Clairaut's Theorem : Uxt = Utx (c) Verify that u(x, t) satisfies the 1-D Heat Equation ut = Kuxx ; what is the formula for k