|Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:||(263)||
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Problem 1. Is there a non-zero polynomial defined on the interval and with values in the interval so that it and all of its derivatives are integers at both the point 0 and the point ? In either case, prove your answer in detail. (Hint: How did we prove the irrationality of ?)
Solution. There isn't. Had there been one, we could reach a contradiction as in the proof of the irrationality of . Indeed we would have that , hence the integral is not an integer. But repeated integration by parts gives
Problem 2. Compute the volume of the ``Black Pawn'' on the right -- the volume of the solid obtained by revolving the solutions of the inequalities and about the axis (its vertical axis of symmetry). (Check that and hence the height of the pawn is ).
Solution. This is the area of the rotation solid with radius bounded by and . Thus
Problem 5. Do the following series converge? Explain briefly why or why not:
Solution. hence by the vanishing test the series cannot converge.
Solution. . The latter is a multiple of the harmonic series which doesn't converge, hence the original series doesn't converge either.
Solution. Ignoring the first two terms of the series, which don't change convergence anyway,
Solution. The function is positive at and simple differentiation shows that for , hence it is increasing, and hence it is positive for all . Thus which is summable as was shown in class.
Solution. That's a tough one. Here's a solution inspired by the solution to Problem 20 of Spivak's Chapter 23, which by itself is inspired by the proof of the divergence of the harmonic series:
The results. 75 students took the exam; the average grade is 47.4, the median is 46 and the standard deviation is 23.55.