## Calculus wrap-up

• Twice as many students showed $\sum_n(-1/2)^n$ converges by showing that $|(-1/2)^n|=(1/2)^n<1/n^2$ for n sufficiently large (and so the series converges absolutely by comparison with a p-series for $p>1$) than by observing that the series in question is a geometric series.
• Many of my students do multiplication via lattice multiplication instead of the “usual” long multiplication that students of my generation learned in school.  (I am familiar with this other method because I’ve discussed it with D’s mother, who had to learn it to teach it to middle school students.)  I suspect that this would be incomprehensible to most of my peers, but I also am pretty convinced that it’s a better algorithm.
• The best test question is here.
• Best student comment: “I don’t think I can assume mass to be evenly distributed.  I mean, Central Park undoubtedly has less than the part covered by all those skyscrapers.”
• Unintentionally challenging test question: True or false?  If $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0$ then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)$, provided the latter limit exists.
• Of the 37 male students in my class, there were 8 first names that appeared twice: Anthony, Austin, Jacob, Joseph, Joshua, Nick, Ryan, and John.  The 13 female students all had distinct names.
• The final chart of correlations between the various assignments is as follows:
 Psets Ex1 Ex2 Ex3 Final Psets 1 .159 .261 .287 .599 Ex1 1 .157 .495 .513 Ex2 1 .208 .455 Ex3 1 .455 Final 1

Following my earlier post, what’s remarkable about this to me is that the correlations between all the assignments (excluding the final exam) are so low — only the correlation between hour exams 1 and 3 is even remotely in the neighborhood I would expect.  Also, the fact that problem sets correlate better with the final exam than any of the hour exams do is bizarre, given the relative laxity of our p-set grading.