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Unformatted text preview: MA1C ANALYTIC RECITATION 4/30/09 MIDTERM REVIEW 1. Interesting Problem I remember this problem from a freshman math prize exam that I took. Suppose there is a rubber band with one end tied to a post in the ground, and the other end tied to the tail of a kangaroo. The two ends start 1 meter apart. There is a flea sitting on the post, which jumps 1 cm onto the stretched rubber band. The kangaroo then jumps 1 m, stretching the rubber band accordingly. The flea then jumps 1 cm toward the kangaroo, and so on. Does the flea catch the kangaroo? The right way to think about the problem (well, the way I thought about it, which certainly might not be right!) is to consider the ratio between the distance from the post to the flea and the post to the kangaroo. The kangaroo jumping does not change this ratio, so the only thing we have to figure out is how much the flea jump changes the ratio. The ratio starts at 1cm 100cm = 1 / 100. After one jump by the kangaroo and one by the flea, we have the ratio 2cm+1cm 200cm , and so on. In general, if we have the ratio x y , then the kangaroo jump gives us x y +100 y y + 100 (its the same, but written to keep the correct number of centimeters on the top and bottom) and the flea jump gives x y +100 y + 1 y + 100 = x y + 1 y + 100 That is, if the kangaroo is at y , then after the two jumps we add 1 y +100 to the ratio. Therefore after n jumps weve added n i =1 1 100 i +100 = 1 100 n i =1 1 i +1 to the ratio. However, a comparison with the series 1 2 i shows that this sum diverges. That is, there is an n such that n i =1 1 i +1 > 99, which makes the ratio larger than 1 / 100 + (99 / 100) = 1, i.e. the flea has caught the kangaroo. 2. Note About Measure I sent out a proof that if a function is bounded in a rectangle and has a set of discontinuities which is of measure zero, then the function is integrable. This was really meant as kind of info for life, rather than a direction to think about measure too much in this class. I think that most of the problems in this course are more easily solved using the idea of content, especially since they are designed with content in mind. On the last homework, many people apparently got the idea from that theorem I sent out that measure was the only way to come at it. This was probably my fault. Anyway that was a fine way to do it, but there are many facts that are not true about measure and Riemann integration. The one that came up most often is: if two functions differ on a set of measure zero, their Riemann integrals are not necessarily equal. For example, the characteristic function of the rationals differs from zero on a set of measure zero, but does not have integral zero (it has no Riemann integral). The fact about integrability if the discontinuities have measure zero is really kind of a special thing. It kind of comes from the fact that in a compact set, often things can be made finite....
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