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Take Home stuff - MAC 2313 Homework Set I on Optimization...

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MAC 2313 -- Homework Set I on Optimization, due Friday, March 18th Name _____________________________________________ Last Four Digits of UF Id. No. ______________________ The following problem set is worth 25 points in the total course grade. Please show your work. A deduction of 5 points will be made for late papers. (1). Find the points on the surface x 2 y 2 z = 1 which are closest to the origin by the method of substitution (e.g., by the techniques of Section 14.7). [You should find 4 answers here]. (2). Find the points of the surface x 2 y 2 z = 1 which are closest to the origin by the method of Lagrange multipliers (e.g., the method of Section 14.8). (3). Consider the plane x + y + 2z = 6 and point P = (1,1,1). a). Using the method of Lagrange multipliers (Section 14.8), find the point Q in the plane which minimizes the distance from P to the plane. Using the distance formula, calculate the distance from the point to the plane by calculating dist(Q,P). b). Now use the vector analysis methods of Section 12.5 to calculate the distance from the point to the plane to verify that max-min techniques yield a consistent result with our early techniques.

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