ps3 - a . (c) Is there a direction in which the directional...

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University of Hong Kong Fall 2011 Math 1813 Mathematical Methods for Actuarial Science Instructor: Siye Wu Problem Set 3 (due at 17:00, Friday, 28 October) I. (a) For f ( x, y ) = e xy 2 , compute f x , f xx , f xy and f xyx . Compute f y , f yx , f yxx and f xxy indepen- dently, and verify the identities f xy = f yx and f xyx = f yxx = f xxy in this case. (b) Find ∂y ∂x at the point ( x, y, z ) T = ( - 3 , 1 , - 1) T if y is a function of x and z de±ned by the equation xy + z ln y - y 2 + 4 = 0. II. Let f : R 3 R be a continuously di²erentiable function. Let v 1 = (2 , 4 , 5) T , v 2 = (3 , 4 , 4) T , v 3 = ( - 2 , - 2 , 3) T and a = (1 , 2 , 3) T . The directional derivatives of f at the point a along v 1 , v 2 and v 3 are respectively 1, - 2 and 3. (a) Find the directional derivative of f at the point a along (3 , 3 , 1) T . (b) Find the direction in which the function f increases most rapidly at
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Unformatted text preview: a . (c) Is there a direction in which the directional derivative of f at a is less than-7? III. Let f ( x, y ) = x 4 + 4 xy + y 4 . (a) Find the equation of the plane tangent to the surface z = f ( x, y ) at the point (1 , 1 , 6) T . (b) Find and classify the critical points of f ( x, y ). IV. Let g : R → R be a twice di²erentiable function, and let c be a nonzero constant. Show that the function v = v ( t, x, y, z ) given by v ( t, x, y, z ) = 1 r g ( t-r c ) , where r = r x 2 + y 2 + z 2 satis±es the equation ∂ 2 v ∂x 2 + ∂ 2 v ∂y 2 + ∂ 2 v ∂z 2 = 1 c 2 ∂ 2 v ∂t 2 ....
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