MATH 21A Final: Math 21a Final Exam 4 Fall 2009
Document Summary
Answers which are illegible for the grader cannot be given credit: show your work. Except for problems 1-3, we need to see details of your computation: no notes, books, calculators, computers, or other electronic aids can be allowed, you have 180 minutes time to complete your work. For any two nonzero vectors ~v, ~w the vector ((~v ~w) ~v) ~v) is parallel to. The cross product satis es the law (~u ~v) ~w = ~u (~v ~w). If the curvature of a smooth curve ~r(t) in space is de ned and zero for all t, then the curve is part of a line. The curve ~r(t) = (1 t)a + tb, t [0, 1] connects the point a with the point b. For every c, the function u(x, t) = (2 cos(ct) + 3 sin(ct)) sin(x) is a solution to the wave equation utt = c2uxx. The length of the curve ~r(t) = (t, sin(t)), where t [0, 2 ]
