MATH-3030, Spring-2017 Quiz #6 Your name: MATH-3030 - Math Models for CS - CRN 15775 Quiz 6 - Sections C.14.1-C.14.2, C.15.4, C.16.2 Directions: Do the quiz on your own paper and attach to this quiz assignment sheet. Clearly show all your work. For full credit, all your answers must be justified by your work (except for clearly short-answer questions or where directed otherwise). 1. Sketch the curve traced out by the tip of the vector function r(t) = (3 cos t)i(4 sin t)j, where 0 t 2. (Assume r(t) is always in the standard position, i.e. with its initial point at the origin.) Indicate the direction in which the curve is traversed as t increases. [Hint: review the topic of parametric curves from Calculus II.] 2. Find the first and second derivatives of f (t) = et cos ti sin 2tj + 3k. 3. Evaluate: Z1 \u0012 0 1 i + sec2 tj + tet k dt. 1 + t2 \u0013 4. (8 points) Find all (first order) partial derivatives of the function f (x, y, z, w) = xy sin z + yz cos(xw) xy 2 z 3 . w4 Make sure to use correct notation for each derivative. 5. (10 points) Find all first order and all second order partial derivatives of the function h(p, q) = 3p2 arctan(2q) 4q 3 ln p. Make sure to use correct notation for each derivative. 6. Find the directional derivative of f (x, y) = x2 ey yex at the point P (0, 2) in the direction of 3i 4j. 7. Find the unit vector in the direction in which f (x, y, z) = x2 + 2xyz yz 2 increases most rapidly at P (1, 1, 2) and give the rate of change of f in that direction. - 1