Let f(x)=1-4(-x) for x = [0, 1]. (i) Calculate the coefficients Cn 2 ff(y) sin(nry)dy for...

 

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Let f(x)=1-4(-x) for x = [0, 1]. (i) Calculate the coefficients Cn 2 ff(y) sin(nry)dy for all n = 1,2,... (show your work). (ii) Determine a formula for u = u(x, t) so that for all a [0, 1] and all t 0: au(x, t) = u(x, t) u(0, t) = u(1, t) = 0 u(x,0) = f(x) Writing down the formula suffices. You do not need to justify your answer. (iii) What is the physical meaning of u(x, t)? Explain in one brief sen- tence. (iv) Sketch the graphs of the three functions u(x, 0), u(x, 1), and u(x, 100) by hand, in the same coordinate system (no computer allowed). Make sure you label the coordinate axes. You do not need to justify your answer. Let f(x)=1-4(-x) for x = [0, 1]. (i) Calculate the coefficients Cn 2 ff(y) sin(nry)dy for all n = 1,2,... (show your work). (ii) Determine a formula for u = u(x, t) so that for all a [0, 1] and all t 0: au(x, t) = u(x, t) u(0, t) = u(1, t) = 0 u(x,0) = f(x) Writing down the formula suffices. You do not need to justify your answer. (iii) What is the physical meaning of u(x, t)? Explain in one brief sen- tence. (iv) Sketch the graphs of the three functions u(x, 0), u(x, 1), and u(x, 100) by hand, in the same coordinate system (no computer allowed). Make sure you label the coordinate axes. You do not need to justify your answer.