MATH 214 Sample Final Spring 2016 PART I. MULTIPLE-CHOICE QUESTIONS MC1. The points on the curve ~r (t) = ht3 3t, t + 1, t3 3t2 i, t R, where the tangent lines are parallel to the yz-plane are the following: (A) (2, 2, 2) & (0, 1, 0) (D) (2, 0, 4) & (0, 1, 0) (B) (0, 1, 0) & (2, 3, 4) (E) (0, 1, 0) & (2, 3, 4) (C) (2, 2, 2) & (2, 0, 4) MC2. If x + ay = b is an equation for the osculating plane of the curve ~r (t) = h2 sin 3t, t, 2 cos 3ti, t R at the point (0, , 2) then (A) a = 6, b = 6 (B) a = 6, b = 6 (C) a = 6, b = 6 (D) a = 6, b = (E) a = 6, b = MC3. The limit x2 cos(y) equals: (x,y)(0,0) x2 + y 2 (A) 1 (B) 0 lim (C) 21 (D) 1 (E) does not exist MC4. If z is a function of (x, y) and x3 + y 3 + z 3 + 6xyz 2 = 2, Then (A) 0 (B) 1 (C) 1 (D) 2 1 (E) 2 z x evaluated at(1, 0, 1) equals: PART II. LONG ANSWER QUESTIONS LA1. Show that the curve ~r (t) = hsin t, 2 sin t, 2 cos ti, t R, lies at the intersection between two cylinders and a plane. Find the point(s) (in Cartesian coordinates) on the curve at which the curvature 5 is k = . 4 ~ = (5 sin t)~i + (5 cos t)~j + 12t~k. Find the point on the curve at a distance LA2. Consider the helix r(t) 26 units along the curve from the point (0, 5, 0) in the direction of increasing arc length. LA3. Find an equation for the tangent plane to the surface x3 + y 3 + z 3 = 4xyz at the point (1, 1, 1). LA4. For a given function f (x, y), it is noted that at the point P = (1, 1) the directional derivative f f in the direction towards (0, 0) is 1, while the directional derivative towards (1, 2) is -1. Find and x y at P. LA5. Find the local minimum and maximum values and saddle point(s) of f (x, y) = 3x2 y + y 3 3x 3y 2 + 2 2 LA6. Find the absolute maximum and absolute minimum values of f (x, y) = 4x + 6y x2 y 2 on the set D = {(x, y)|0 x 4, 0 y 5} 2