MA 265 Midterm: 4-12f-test3

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31 Jan 2019
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Egr 265, math tools for engineering problem solving. Problem 1 (9+9 points) (a) let g(x, y, z) = x2y2 cos(2z). Find its third order partial derivative gxyz. (b) for the function f (x, y) = xex2+y2. Problem 2 (9+9 points) (a) for the function h(x, y) = x + 2y nd a unit vector in the direction of steepest descent at the point p (1, 4). Also, nd the rate of descent in this direction. (b) find the directional derivative of h(x, y) at p (1, 4) in the direction of the vector from p (1, 4) to q(3, 6). Evaluate rc(y 1) ds, where c is the quarter circle of radius 1, centered at the origin, starting at (1, 0) and ending at (0, 1). Find the work done by the force eld. F (x, y) = 2xyi + yeyj along the curve c parameterized by x = t3, y = t, 0 t 1.

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