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Unformatted text preview: 1 . Given f ( x,y,z ) = yze xy , a) find the maximal increase rate at (0,1,1) and the direction in which it occurs. b) Find the rate of change of f ( x,y,z ) at (0,1,1) toward (2 , 1 , 2) . c) Write an equation of the tangent plane to the level surface f ( x,y,z ) = yze xy = 1 at (0 , 1 , 1). Answer . a) It occurs in the direction of ∇ f (0 , 1 , 1) and the maximal rate is ∇ f (0 , 1 , 1)  . We find f x = y 2 ze xy ,f y = ze xy + xyze xy ,f z = ye xy , and ∇ f (0 , 1 , 1) = h 1 , 1 , 1 i , ∇ f (0 , 1 , 1)  = √ 1 2 + 1 2 + 1 2 = √ 3 . b) The vector toward (2 , 1 , 2) is ~v = h 2 , 2 , 1 i and the unit vector toward (2 , 1 , 2) is ~u = h 2 , 2 , 1 i /  ~v  = D 2 3 , 2 3 , 1 3 E . The rate of change is the directional derivative D ~u f (0 , 1 , 1) = ∇ f (0 , 1 , 1) · ~u = 2 3 + 2 3 + 1 3 = 1 3 . c) The normal vector to the level surface is ∇ f (0 , 1 , 1) = h 1 , 1 , 1 i and the equation is x + y 1 + z 1 = 0 or x + y + z 2 = 0 2 . The height above the sea level at a point (....
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 '07
 KAMIENNY
 Math, Calculus, Derivative, Rate Of Change, Cos, Spherical coordinate system, Multiple integral, yex dxdy

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