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265L13 - LESSON 13 Lagrange Multipliers READ Section 15.8...

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LESSON 13 Lagrange Multipliers READ: Section 15.8. NOTES: At the end of the last lesson, we looked at the problem of finding the maximum value of f ( x, y ) = x 2 + y for points ( x, y ) on the circle x 2 + y 2 = 1. The idea used there was to parameterize the circle as x = cos t , y = sin t , 0 t 2 π . Then those component functions were plugged into f for x and y . The result was a function of just one variable ( t ). Now we can determine the max and min of f along the boundary in the old-fashioned calculus I way. In this lesson, we will look at another way to determine the max and min of f along the boundary. To help visualize the problem, imagine we are walking about on the surface given by equation f ( x, y ) = x 2 + y , but we can’t walk just anywhere on the surface: we are constrained to the part of the surface directly above the circle x 2 + y 2 = 1 in the x , y -plane. The question we are trying to answer is: under that constraint, what is the highest point we will reach and what is the lowest point we will reach as we go all the way around above the circle. The question is usually posed as: what are the maximum and minimum values of x 2 + y subject to the constraint x 2 + y 2 = 1? Lagrange devised a method for solving such problems. The method can be explained geometrically. Suppose the maximum value on the surface z = f ( x, y ) directly above the constraint curve g ( x, y ) = 0 is M . (In our example above, the constraint curve would be given by equation g ( x, y ) = x 2 + y 2 - 1 = 0.) Imagine the level curve for the surface produced by slicing through the surface at z = M . Now down in the x , y -plane, the constraint curve and the level curve will intersect, since there is a spot on the surface with z = M somewhere above the constraint curve. Call the point of intersection ( x 0 , y 0 ). If there is any justice in life, the level curve and the constraint curve ought to be tangent to each other at the point where they intersect, since if the constraint curve crossed over the level curve, then that would mean there would be points above the constraint curve that were higher than M , which we assumed was the maximum value above the constraint curve. So it seems reasonable that the level curve and the constraint curve are tangent to each other at the
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