MATH 21B Final: MATH 21B Harvard 21b Fall 10Final
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Tth 11:30 sebastian vasey: start by printing your name in the above box and check your section in the box to the left, do not detach pages from this exam packet or unstaple the packet, please write neatly. Answers which are illeg- ible 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. The distance from (1, 2, 1) to (3, 2, 1) is ( 2, 4, 2). The plane y = 3 is perpendicular to the xz plane. All functions u(x, y) that obey ux = u at all points obey uy = 0 at all points. ~f dr 2 . y f (x, y) dx(cid:17) dy. x f (x, y) dy(cid:17) dx =
