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prac2A - 18.02 Practice Exam 2 A Problem 1(10 points 5 5...

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18.02 Practice Exam 2A Problem 1. (10 points: 5, 5) Let f ( x,y ) = xy - x 4 . a) Find the gradient of f at P : (1 , 1). b) Give an approximate formula telling how small changes Δ x and Δ y produce a small change Δ w in the value of w = f ( x,y ) at the point ( x,y ) = (1 , 1). Problem 2. (20 points) On the topographical map below, the level curves for the height function h ( x,y ) are marked (in feet); adjacent level curves represent a difference of 100 feet in height. A scale is given. a) Estimate to the nearest .1 the value at the point P of the directional derivative parenleftbigg dh ds parenrightbigg ˆ u , where ˆ u is the unit vector in the direction of ˆ ı +ˆ . b) Mark on the map a point Q at which h = 2200, ∂h ∂x = 0 and ∂h ∂y < 0. Estimate to the nearest .1 the value of ∂h ∂y at Q . P 1900 2000 1000 2200 2100 Problem 3. (10 points) Find the equation of the tangent plane to the surface x 3 y + z 2 = 3 at the point ( - 1 , 1 , 2).
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Problem 4. (20 points: 5,5,5,5) A rectangular box is placed in the first octant as shown, with one corner at the origin and the three adjacent faces in the coordinate planes. The opposite point
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