Homework 8, due November 12, 2014.

[goodwine]

Reading: Chapter 11, sections 1 and 2.

Exercises: 11.1 (numbers 2 and 5 only) and 11.3 (numbers 1 and 3 only).

[goodwine]

Someone asked me:

For 11.1 question 2, when I solved for a_n I ended up with (-1/(n*pi))*(-cos(4n*pi/3)-cos(2*n*pi/3)+2cos(2*n*pi)). I was wondering for this equation if it is possible to add -cos(4*n*pi/3) with -cos(2*n*pi/3), since for every n=1,2,3,4.... they will equal each other. I was just confused on this since by a_n seemed a lot more complicated with more terms then what we were solving for in class and the book and perhaps I was solving for a_n wrong.

My other question was pertaining to problem 11.3 question 3, since instead of having the partial derivative of u with respect to t we have u(x,0)=0 and in the equations below instead of u(x,0) we have the partial of u with respect to t. I was wondering for this problem can I take the antiderivative of the partial of with respect to t where the resulting values would be {0, 0<x<3 ; x, 3<x<4 ; 0, 4<x<10}. If you can't do that what approach to this problem should I take to solve it.

For the first question, if they are always equal, then you may combine them. It's not absolutely necessary, though, because what you basically do is plot them and Matlab will just evaluate them to be the same number.

For the second question, the answer is no, for two reasons. First, it's MUCH easier just to use the velocity term (g(x)) in the integral for a(n) because they are constants and it's easier to integrate. Secondly, if you do integrate it, you don't know the constant of integration, so then when you integrate the result (f(x)) you can't evaluate that integral.

[goodwine]

What is alpha as given in the homework 11.3? I remember you said it contained T_hat and rho but in what way?

Just compare the first boxed equation in section 11.1.4 with Equation 11.4.