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pm2aSolns - Math 265(Butler Practice Midterm II —...

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Unformatted text preview: Math 265 (Butler) Practice Midterm II — A (Solutions) 1. Find the directional derivative of f ( x,y,z ) = xz 2- 3 xy + 2 xyz- 3 x + 5 y- 17 from the point (2 ,- 6 , 3) in the direction of the origin. To find a directional derivative we note that D u f = ∇ f · u . So we need to find u and ∇ f . Since we are going from the point (2 ,- 6 , 3) towards (0 , , 0) then a vector pointing in the appropriate direction is h- 2 , 6 ,- 3 i . This is not a unit vector since kh- 2 , 6 ,- 3 ik = √ 4 + 36 + 9 = √ 49 = 7, but by scaling we can make it a unit vector so that u = 1 7 h- 2 , 6 ,- 3 i . The gradient is ∇ f ( x,y,z ) = h z 2- 3 y + 2 yz- 3 ,- 3 x + 2 xz + 5 , 2 xz + 2 xy i . Evaluating at the point (2 ,- 6 , 3) we have ∇ f (2 ,- 6 , 3) = h 3 2- 3 · (- 6) + 2 · (- 6) · 3- 3 ,- 3 · 2 + 2 · 2 · 3 + 5 , 2 · 2 · 3 + 2 · 2 · (- 6) i = h- 12 , 11 ,- 12 i . Therefore the desired directional derivative will be given by ∇ f · u = h- 12 , 11 ,- 12 i · 1 7 h- 2 , 6 ,- 3 i = 24 + 66 + 36 7 = 126 7 = 18 . 2.2....
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pm2aSolns - Math 265(Butler Practice Midterm II —...

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