math20c Meesue Yoo fall 09 quiz5

# math20c Meesue Yoo fall 09 quiz5 - 2,x 2-2 xy 9 z 2> So...

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Quiz 5 Solution for MATH 20C, 2009 1 . Let f ( x,y ) = xe x 2 - y and P = (1 , 1). (a) Find the rate of change of f in the direction of f P . Ans. f = < e x 2 - y + 2 x 2 e x 2 - y , - xe x 2 - y > and at P = (1 , 1), f ( P ) = < 3 , - 1 > . D f ( P ) f ( P ) = ||∇ f ( P ) || = p 3 2 + 1 = 10 . (b) Find the rate of change of f in the direction of a vector making an angle of 45 with f P . Ans. Let u be the unit vector making an angle of 45 with f P , then D u f ( P ) = ||∇ f ( P ) || cos θ where θ = π/ 4 = 10 · 2 2 = 5 . 2 . Find a vector normal to the surface 3 z 3 + x 2 y - y 2 x = 1 at P = (1 , - 1 , 1). Ans. Let F ( x,y,z ) = 3 z 3 + x 2 y - y 2 x. Then F ( P ) is normal to the level surface F ( x,y,z ) = 1 at P . F = < 2 xy - y
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Unformatted text preview: 2 ,x 2-2 xy, 9 z 2 > So, at P = (1 ,-1 , 1), the normal vector to the given level surface is ∇ F ( P ) = <-3 , 3 , 9 > . 3 . Calculate ∂w ∂z where w is implicitly deﬁned by x 2 w + w 3 + wz 2 + 3 yz = 0. Ans. Let F ( x,y,z,w ) = x 2 w + w 3 + wz 2 + 3 yz . Then for the given F ( x,y,z,w ) = 0, ∂w ∂z =-F z F w . F z = 2 wz + 3 y, F w = x 2 + 3 w 2 + z 2 . Hence, ∂w ∂z =-2 wz + 3 y x 2 + 3 w 2 + z 2 ....
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