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Math 251 Questions for Ch.12 from past exams 1. G ( x, y , z ) 3x y z 2 J. Lewis 2 a) Find an equation of the tangent plane to G ( x, y, z ) 22 at the point P ( 2, 1, 3) . b) Find the rate of change of G in the direction of w 2, 6, 4 from the point, P( 2, 1, 3) . c) Find the maximum rate of change of G from the point P ( 2, 1, 3) . 2. Use Clairaut's theorem to show there is no function, f(x, y) with continuous partial derivatives and for which f x ye x 2y and f y xe x 2y . 3. Find and classify, by the 2nd derivative test, the four critical points of f ( x, y ) x 3 xy 2 12 xy 27 x . 4. Give parametric equations for the tangent line to the curve of intersection of the surfaces 4 x 9 y 36 z 0 and 4 x 2 y z 48 at the point P(3, 2, 2). You do not need to find the curve. 2 2 2 2 2 2 5. f ( x , y ) xe xy a) Show that the graph of f has no local max or min. b) Verify Clairaut's theorem in this case, that is show f xy f yx . 6. f ( x, y ) x 3 y 2 a) Find the directional derivative of f(x,y) in the direction of the vector i 4 j from the point P ( 1, 2, 4) . b) What is the maximum rate of change of f(x,y) from the point P ( 1, 2, 4) ? c) Find the equation of the tangent plane to the surface at the point P ( 1, 2, 4) . 7. f ( x, y ) x 3 xy 2 6 xy . Find all critical points. For each critical point, give the conclusion of the 2nd derivative test or show the test has no conclusion. 8. Use differentials to approximate ( 26 3 9 ) . 2 9. z x e 3 y x x( s, t ) 10. z f ( x, y ) xs t 3 2 y y ( s, t ) y s t 4 Find 4 2z s 2 . 2z Find . t s