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 18 02 Practice Exam 2 B Problem 1 Let f x y x2 y 2 x a 5 Find f at 2 1 b 5 Write the equation for the tangent plane to the graph of f at 2 1 2 c 5 Use a linear approximation to nd the approximate value of f 1 9 1 1 d 5 Find the directional derivative of f at 2 1 in the direction of Problem 2 10 On the contour plot below mark the portion of the level curve f 2000 on f which 0 y Problem 3 a 10 Find the critical points of w 3x2 4xy y 2 12y 16x and say what type each critical point is b 10 Find the point of the rst quadrant x 0 y 0 at which w is largest Justify your answer Problem 4 Let u y x v x2 y 2 w w u v a 10 Express the partial derivatives wx and wy in terms of wu and wv and x and y b 7 Express xwx ywy in terms of wu and wv Write the coe cients as functions of u and v c 3 Find xwx ywy in case w v 5 Problem 5 a 10 Find the Lagrange multiplier equations for the point of the surface x4 y 4 z 4 xy yz zx 6 at which x is largest Do not solve b 5 Given that x is largest at the point x0 y0 z0 nd the equation for the tangent plane to the surface at that point Problem 6 Suppose that x2 y 3 z 4 1 and z 3 zx xy 3 a 8 Take the total di erential of each of these equations b 7 The two surfaces in part a intersect in a curve along which y is a function of x Find dy dx at x y z 1 1 1
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