MATH 21A Midterm: MATH 21A Harvard 21a Fall 16Practice1
Document Summary
Answers which are illeg- ible for the grader cannot be given credit: show your work. Problem 1) (20 points) no justi cations are needed. The vector h2, 3, 6i has a length which is an integer. The surface x2 + y2 + z2 2x = 3 is a sphere. If ~v ~w is negative, then the angle between ~v and ~w is acute ( = smaller than /2). The level curves f (x, y) = 1 and f (x, y) = 0 do not intersect for f (x, y) = (xy + cos(x))6. For any nonzero ~a, the equation ~a ~x = ~b always has a solution ~x. For two non-parallel ~a,~b, the equation (hx, y, zi ~a) ~b = 1 de nes a plane. The curvature of ~r(t) = ht3, 1 t3, t3i is 0 if t = 1.
