math1a - quiz 9 - solns - Math 1A Quiz 9 Solutions April...

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Math 1A Quiz 9 Solutions April 21st, 2008 1. Find the area of the largest rectangle with sides parallel to the coor- dinate axes which can be inscribed in the ellipse x 2 + y 2 4 = 1. Solution: Let ( x, y ) be the coordinates of the corner of the rectangle which lies in the first quadrant. Then the area of the rectangle is A = 4 xy . We want to maximize A with respect to the constraint x 2 + y 2 4 = 1. Solving this constraint for y gives y = 4 - 4 x 2 = 2 1 - x 2 (Note we don’t have to worry about signs here since we only care about positive x and y ). So we get the formula A = 4 xy = 8 x 1 - x 2 . Now, taking derivatives gives: A 0 ( x ) = 8 p 1 - x 2 + 8 x 1 2 · 1 1 - x 2 · ( - 2 x ) = 8 p 1 - x 2 - 8 x 2 1 - x 2 = 8 - 16 x 2 1 - x 2 . The critical points of A ( x ) occur either when A 0 ( x ) = 0 or A 0 ( x ) does not exist. These correspond to the roots of the numerator and denominator of A 0 , respectively. The roots of the numerator are at x = ± 2 2 (we only care about the positive solution) and
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This note was uploaded on 09/17/2011 for the course MATH 1A taught by Professor Wilkening during the Spring '08 term at University of California, Berkeley.

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math1a - quiz 9 - solns - Math 1A Quiz 9 Solutions April...

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