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Unformatted text preview: 18.02 Practice Exam 2 A 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 x and y produce a small change 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 u , where u is the unit vector in the direction of + . 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). Problem 4. (20 points: 5,5,5,5)...
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This note was uploaded on 04/18/2008 for the course 18 18.02 taught by Professor Auroux during the Fall '08 term at MIT.
- Fall '08