prac2asol

prac2asol - MIT OpenCourseWare http:/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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± ± ± ± ± ± ± 18.02 Practice Exam 2 A Solutions Problem 1. a) f = ( y 4 x 3 ı + x ˆ ; at P , f = a− 3 , 1 A . b) Δ w ≃ − 3 Δ x + Δ y . Problem 2. dh Δ h a) By measuring, Δ h = 100 for Δ s 500, so . 2. ds u ˆ Δ s b) Q is the northernmost point on the curve h = 2200; the vertical distance between consecutive ∂h Δ h 100 level curves is about 1/3 of the given length unit, so ≃ − . 3. ∂y Δ y 1000 / 3 Problem 3. f ( x, y, z ) = x 3 y + z 2 = 3 : the normal vector is f = a 3 x 2 y, x 3 , 2 z A = a 3 , 1 , 4 A . The tangent plane is 3 x y + 4 z = 4. Problem 4. a) The volume is xyz = xy (1 x 2 y 2 ) = xy x 3 y xy 3 . Critical points: f x = y 3 x 2 y y 3 = 0, f y = x x 3 3 xy 2 = 0. b) Assuming
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prac2asol - MIT OpenCourseWare http:/ocw.mit.edu 18.02...

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