math226spring08mt2s

math226spring08mt2s - 1. Given f ( x,y,z ) = 3 x + 2 yz, a)...

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Unformatted text preview: 1. Given f ( x,y,z ) = 3 x + 2 yz, a) Find the maximal increase rate of f at (1 ,- 1 ,- 1) and the direction in which it occurs b) Find the rate of change of f at (1 ,- 1 ,- 1) in the direction of the vector ~ b = h 1 , 2 , 2 i . c) Write the equation of he tangent plane to the surface S : 3 x +2 yz = 5 at (1 ,- 1 ,- 1) . Answer. a) f x = 3 ,f y = z yz ,f z = y zy and f (1 ,- 1 ,- 1) = h 3 ,- 1 ,- 1 i is the direction of the maximal increase rate and the rate itsel is | f (1 ,- 1 ,- 1) | = q 3 2 + (- 1) 2 + (- 1) 2 = 11 . b) The unit vector in ~ b direction is ~u = ~ b | ~ b | = D 1 3 , 2 3 , 2 3 E , the rate of change is f (1 ,- 1 ,- 1) ~u = 1- 2 3- 2 3 =- 1 3 . c) The equation is 3( x- 1)- ( y + 1)- ( z + 1) = 0 . 2 . a) Find the work done by the force ~ F ( x,y ) = h 3 e y x 2 ,y i to move an object along the curve C : y = x 3 , x 1 . Answer. We parametrize C : x = t,y = t 3 , t 1 . We have ~ r ( t ) = h t,t 3 i ,~ r ( t ) = h 1 , 3 t...
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math226spring08mt2s - 1. Given f ( x,y,z ) = 3 x + 2 yz, a)...

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