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Gradients and Directional Derivatives Reading assignment: 2.6 Recommended exercises: 2.6: 1, 3, 5, 8, 9, 11, 12, 13, 16, 18, 29, 33, 35 For each problem 1 – 3, in part a) find the gradient of the given function, and in part b) find the directional derivative in the direction of the given v at the given point P. 1) f ( x, y) e x sin y P (0, ) a) b) v = i + j, 2) g ( x, y) x 2 y 3 a) b) v = 3i + 4 j, P(1, 1) 3) h( x, y, z ) a) x y z b) v = 2i j + 2k, P (1,1, 1) 4) Find an equation for the tangent plane to the surface z = x3 – 3xy2 at an arbitrary point (a, b, c) on the surface. This surface is called a monkey saddle. Perhaps if you graph it you'll see why! 5) Find a tangent plane to the surface which is defined implicitly by the equation xy2 + yz2 + zx2 = 3 at the point (1, -2, 1). 6) Consider the function f ( x, y, z ) ze x sin y . At the point ( 1, ,1) it has value 2 the change in this value if you move 0.1 units in the direction of u (2,3, 6) . 1 . Estimate e 7) Let f ( x, y) 4 x 2 y . (a) Sketch and label the level curves z = 2, 1, 0,1, 2 . [Neatness and accuracy count] (b) Find f (1,1) (c) Find a unit vector u such that Du f (1,1) 0 . (d) Sketch and label f (1,1) and u with initial point (1,1) (Sketch these vectors in the coordinate plane from part (a)). 3 4 4 3 8) Let u , and v , . Suppose that at some point p ( x, y ) , Du f 6 and 5 5 5 5 Dv f 17 . a. Find f at p . b. Show, in general, that f 2 Du f Dv f whenever u and v are perpendicular. 2 2 9) Consider the sphere ( x a)2 y 2 z 2 2 and the circular cone z 2 2 x 2 2 y 2 . Find a such that the two surfaces are tangent. [Two surfaces are tangent at a common point if and only if they share a common tangent plane at that point.]