Solutions6

# Solutions6 - ± Critical Point = 1 2 2 D = 2 x 2 y 2-1 = 1...

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MATH 214 A1 - Assignment 6 Not for marks (1) Find the value of the partial derivatives ∂z ∂x and ∂z ∂y at the point (1,1,1) for the following relations: (a) z 3 - xy + yz + y 3 - 2 = 0 ± ∂z ∂x = 1 4 , ∂z ∂y = - 3 4 ² (b) 1 x + 1 y + 1 z - 1 = 0 ± ∂z ∂x = - 1 = ∂z ∂y ² (2) Let w = x 2 e 2 y cos(3 z ). Find the value of dw dt at the point (1 , ln 2 , 0) on the curve x = cos t , y = ln( t + 2), z = t (4) (3) Find the maximum of the directional derivative of f ( x,y ) = y 2 x at (2 , 4) and the direction in which it occurs. ± max = 32 , direction = h - 1 2 , 1 2 i ² (4) Find all points at which the direction of fastest change of the function f ( x,y ) = x 2 + y 2 - 2 x + 4 y is parallel to h 1 , 1 i . (All points on the line y = x + 1) (5) Find the critical point of f ( x,y ) = xy +2 x - ln( x 2 y ) in the ﬁrst quadrant and show that it takes a minimum there.
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Unformatted text preview: ± Critical Point = ( 1 2 , 2 ) , D = 2 x 2 y 2-1 = 1 , f xx = 2 x 2 > ² (6) Among all closed rectangular boxes of volume 27 m 3 , ﬁnd the dimensions of the box with the smallest surface area: (a) First solving for z as a function of x and y , then looking for critical points and using the second derivative test. (b) Second, using Lagrange multipliers. Answer for both is a cube with sides 3 m long (7) Find the extreme values of the function f ( x,y ) = 2 x 2 + 3 y 2-4 x-5 on the region x 2 + y 2 ≤ 16. Max = 47 at (2 , ± √ 12) and Min =-7 at (1 , 0). 1...
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