1. During the quiz you may use your textbook, my notes, and your own notes. 2. No calculators or other electronic devices are permitted. 3. Please make sure your cell phones are quiet during class and off during quizzes. 4. I'll be in my office from 8:15 to 9 on Thursday if you have last minute questions about the quiz. 5. Some students in the class are entitled to extra time, so please do not discuss the solution to the quiz until 11:30, after all of the papers are turned in. For Quiz 2, be able to: -- Apply the Maximum Principle, Minimum Principle, and the stronger variant of the Maximum Principle to draw conclusions about the behavior of the solution to an ODE-BVP. -- Given a function, compute its sup-norm, L_2 norm, or H^1 norm. -- Determine whether a function is in L_2, H^1, or C (the space of continuous functions). -- Use a Green's function to compute a solution to an ODE-BVP. -- Derive the weak form of the problem from the strong. -- Show that a given bilinear form is coercive. -- Determine the regularity of a solution to an ODE-BVP. -- Solve the Unquizzes. -- Show that the finite difference matrix is tridiagonal. -- Bound a finite difference solution using the discrete maximum principle. -- Show that if a(u,phi_j) = (f, phi_j) for j = 1,...,M-1, then a(u,v) = (f,v) for all v in S_h. -- Use Theorem 5.2. -- Suppose we change the boundary conditions from u(0)=u(1)=0 to u'(0)=u'(1)=0. Assume that b(x) = 0 and c(x) > 2 for x in [0,1]. The solution exists and is unique. Write the weak formulation. (Use test functions v in H^1.) What can you conclude about existence and uniqueness of the solution to the weak problem? Write the linear systems corresponding to finite difference and finite element approximations to this problem. What can you say about existence and uniquenss of the discrete solutions? How close are they to the solution to the ODE?