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prac2Asol

prac2Asol - 18.02 Practice Exam 2 A Solutions Problem 1 a f...

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18.02 Practice Exam 2A – Solutions Problem 1. a) f = ( y 4 x 3 ı + x ˆ ; at P , f = (− 3 , 1 ) . b) Δ w ≃ − 3 Δ x + Δ y . Problem 2. a) By measuring, Δ h = 100 for Δ s 500, so parenleftbigg dh ds parenrightbigg ˆ u Δ h Δ s . 2. b) Q is the northernmost point on the curve h = 2200; the vertical distance between consecutive level curves is about 1/3 of the given length unit, so ∂h ∂y Δ h Δ y 100 1000 / 3 ≃ − . 3. Problem 3. f ( x, y, z ) = x 3 y + z 2 = 3 : the normal vector is f = ( 3 x 2 y, x 3 , 2 z ) = ( 3 , 1 , 4 ) . 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 x > 0 and y > 0, the equations can be rewritten as 1 3 x 2 y 2 = 0, 1 x 2 3 y 2 = 0. Solution: x 2 = y 2 = 1 / 4, i.e. ( x, y ) = (1 / 2 , 1 / 2). c) f xx = 6 xy = 3 / 2, f yy = 6 xy = 3 / 2, f xy = 1 3 x 2 3 y 2 = 1 / 2. So f xx f yy f 2 xy > 0, and f xx < 0, it is a local maximum.
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