HH_15_3 - (11/25/08) Math 10C. Lecture Examples. Section...

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Unformatted text preview: (11/25/08) Math 10C. Lecture Examples. Section 15.3. Lagrange multipliers Imagine that the curve in Figure 1 is a mirror and that a viewer at point F 2 is looking at the image in the mirror of an object at point F 1 . According to Fermats principle from physics, the image will be the point P on the mirror such that the total distance f ( P ) = PF 1 + PF 2 (1) that the light travels from the object to the viewer is a minimum a that point. For each number c that is greater than the distance F 1 F 2 between the object and the viewer, the level curve PF 1 + PF 2 = c of the distance f ( P ) is an ellipse. Figure 2 shows eight of these ellipses. Example 1 Why can you expect the ellipse to be tangent to the mirror at the point P ? FIGURE 1 FIGURE 2 Answer: The solution is the answer To express the result of Example 1 with formulas, we introduce xy-axes, as in Figure 3, and let f ( x, y ) be the sum (1) of the distances from P = ( x, y ) to F 1 and F 2 . We also assume that the mirror....
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HH_15_3 - (11/25/08) Math 10C. Lecture Examples. Section...

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