prac2Asol

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

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Unformatted text preview: 18.02 Practice Exam 2 A Solutions Problem 1. a) f = ( y 4 x 3 ) + x ; at P , f = ( 3 , 1 ) . b) w 3 x + y . Problem 2. a) By measuring, h = 100 for s 500, so parenleftbigg dh ds parenrightbigg u h 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 h 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....
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