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0216 - I run √ 60 2 x 2 on grass and then run on sidewalk...

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Math 31A 2010.02.16 MATH 31A DISCUSSION JED YANG 1. Optimisation 1.1. Exercise 4.6.15. Find the dimensions of the rectangle of maximum area that can be inscribed in a circle of radius r . Solution. Let x and y be the side lengths. By Pythagorean theorem, they satisfy x 2 + y 2 = 4 r 2 . We want to maximize A = xy . Solve for y = 4 r 2 - x 2 . So A = x 4 r 2 - x 2 . We can maximize A 2 = x 2 (4 r 2 - x 2 ) = 4 r 2 x 2 - x 4 . Differentiate to get ( A 2 ) = 8 r 2 x - 4 x 3 . Critical points are 8 r 2 x - 4 x 3 = 0 so x = 0 or x 2 = 2 r 2 , that is, x = r 2. End points x = 0 , 2 r . We get A (0) = A (2 r ) = 0 and A ( r 2) = 2 r 2 . So area is maximized when x = y = r 2, that is, we have a square. square 1.2. Catching a bus. I am at one corner of a rectangular park with sides 60 and 300 metres. The bus stop is at the opposite corner. I can run on the grass at 5 m/s, and I can skateboard on the sidewalk (along the long side) at 13 m/s. I want to get to the bus stop by running across the park and then skateboarding on the sidewalk. Where should I run to in order to get there the fastest? Solution. Let x be the distance that I skip on the long sidewalk. In otherwords,
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Unformatted text preview: I run √ 60 2 + x 2 on grass, and then run on sidewalk for 300-x . Dividing by the rates, we get the total travel time is y = 300 − x 13 + √ 60 2 + x 2 5 . Di±erentiating, we get y ′ =-1 13 + x 5 √ 60 2 + x 2 . Setting equal to 0, we solve and get x = 25. Note that if there were a sidewalk on the short side too, then it would be faster to skate along the sidewalk for the entire time instead. Moral: Sometimes it is not good to cut corners. s 2. Basic Integration 2.1. Exercise 4.8.58. Let f ′′ ( t ) = t-cos t , f ′ (0) = 2, f (0) =-2. Find f ′ and f . Solution. By integrating, we see f ′ ( t ) = t 2 / 2-sin t + c . Since f ′ (0) = c = 2, we get f ′ ( t ) = t 2 / 2-sin t + 2. Integrating again to get f ( t ) = t 3 / 6 + cos t + 2 t + d . Since f (0) = 1 + d =-2, we get d =-3, yielding f ( t ) = t 3 /y + cos t + 2 t-3. s...
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