MATH 210 Study Guide - Final Guide: Tangent Space, Scilab, Unit Vector
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Solution: (a) by de nition, the gradient of f (x, y, z) is: F = hfx, fy, fzi fx = y cos(xy 8) + fy = x cos(xy 8) . 2x (y z)2 fz = z + 1 (y z)2. The partial derivatives evaluated at (4, 2, 0) are: fx(4, 2, 0) = 2 cos((4)(2) 8) + fy(4, 2, 0) = 4 cos((4)(2) 8) . 2(4) (2 0)2 = 2 fz(4, 2, 0) = . 3(x 4) + 2(y 2) + (z 0) = 0 (c) by de nition, the directional derivative of f (x, y, z) at (4, 2, 0) in the direction of u is: From part (b), we have f (4, 2, 0) = h3, 2, 1i. Recalling that u must be a unit vector, we multiply h 2, 1, 0i by the reciprocal of its magnitude. | h 2, 1, 0i| h 2, 1, 0i =
