ALABAMA A & M UNIVERSITY, COLLEGE OF ENGINEERING, TECHNOLOGY, & PHYSICAL SCIENCES Calculus III Quiz #2 Fall 2016 Instructor: Dr. Ncube Due Date: October 26, 2016 (in class) Full Name: Student Number: 1. Find the equation of the tangent plane to the surface z = xy at (1, 2, 2). 3x 2y Show that this plane passes through the origin. 2. Calculate the directional derivative of g(x) = `n (x + eyz ) at (0, 1, 0) in the direction from (0, 1, 0) to (2, 3, 1). 1 3. An engineer wishes to build a railroad up a mountain that has the shape of an elliptic paraboloid z = c ax2 by2 , where a, b, c are positive constants. At the point (1, 1), in what directions may the track be laid so that it will be climbing with a slope of 0.03 (i.e. a vertical rise of 0.03m for each horizontal metre)? Make a sketch showing a few level curves, the gradient vector z at (1, 1), and the two possible directions for the track. Work out the details using a = 3b, b = 0.015. Grade: 2 10