APPM 2350 Midterm: appm2350spring2016exam1_sol
Document Summary
Thus (u + v) and (u v) are orthogonal if and only if u and v are the same length. (d) not always true: consider the curve parameterized by r(t) = (cid:10)cos(t2), sin(t2), 0(cid:11). This is a unit circle in the xy-plane and thus lies on the surface of the unit sphere centered at the origin. The unit tangent and normal vectors of r(t) are given by. T(t) = (cid:10) sin(t2), cos(t2), 0(cid:11) , n(t) = (cid:10) cos(t2), sin(t2), 0(cid:11) The acceleration is given by a(t) = (cid:10) 4t2 cos(t2) 2 sin(t2), 4t2 sin(t2) + 2 cos(t2)(cid:11) = 2t(t) + 4t2n(t) (e) not always true: see the counterexample in (d). 2. (a) when = 0 the surface becomes x2 + y2 = 1, which is a circular cylinder. When 6= 0 we have z = and downward if > 0).
