Problems covered in recitation on Monday Oct 6 2003 October 8 2003 On Monday we talked about Lagrange multipliers We also did the following problems Problem 1 Let w x y x3 y 2 Find a parametric equation for the line tangent to the curve w 1 at the point 1 1 Use the gradiant Solution We compute w h3x2 y 2 2x3 yi so we have w 1 1 h3 2i Now the curve w 1 is a level curve for w and we know that w is perpendicular to level curves So if we rotate w by 90 we get a vector tangent to the curve This rotated vector is w 1 1 h 2 3i A parametric equation for a line passing through 1 1 in the direction of h 2 3i is given by x t 2t 1 y t 3t 1 Problem 2 Find the point of the plane x 2y 3z 1 closest to the origin Solution We use Lagrange multipliers Our constraint equation is g x y z 1 where g x y z x 2y 3z We want to minimize distance to the origin which is really the same as minimizing the square of the distance get rid of that nasty square root So the function we want to minimize is f x y z x2 y 2 z 2 Now we do Lagrange multipliers We look for solutions to f g 1 We compute f g h2x 2y 2zi h1 2 3i Comparing the components of these vectors we see that we have a system of equations 2x 2y 2 2z 3 Of course we also have the constraint equation x 2y 3z 1 Solving for x y and z in terms of we get x 2 y and z 3 2 Plugging into the constraint equation we see that 2 2 9 2 7 1 so 1 7 Therefore x 1 14 y 1 7 z 3 14 2
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