Clark College's spring quarter starts Monday April 3rd. My Trig and Calc II Syllabi and Class Calendars are now updated and available at https://www.integreat.ca/OL/

Happy Pi Day! 3.14, March 14th. Here's one of my favourite, especially since it involves integration not the more often given summations: \[ \pi = 16\int_{0}^{1} \frac{x-1}{x^4-2x^3+4x-4}dx \]

Fun Q for my Calculus I.

Problem: Evaluate \(lim_{n\rightarrow\infty} \frac{1}{n^3}⋅\frac{𝑛(𝑛+1)(2𝑛+1)}{6}\)

Which attempt is wrong and WHY?

Attempt 1:

\[lim_{𝑛\rightarrow\infty} \frac{1}{n^3}⋅\frac{𝑛(𝑛+1)(2𝑛+1)}{6}\]

\[=lim_{𝑛\rightarrow\infty} \frac{(𝑛+1)(2𝑛+1)}{6𝑛^2}\]

\[=lim_{𝑛\rightarrow\infty} \frac{𝑛+1}{6𝑛}⋅\frac{2𝑛+1}{𝑛}\]

\[=lim_{𝑛\rightarrow\infty} \left(\frac{1}{6}+\frac{1}{6n}\right)⋅\left(2+\frac{1}{n}\right)\]

\[=\left(\frac{1}{6}+0\right)⋅\left(2+0\right)\]

\[=\frac{1}{3}\]

Attempt 2:

\[lim_{𝑛\rightarrow\infty} \frac{1}{n^3}⋅\frac{𝑛(𝑛+1)(2𝑛+1)}{6}\]

\[=lim_{𝑛\rightarrow\infty} \frac{1}{n^3}⋅lim_{n\rightarrow\infty} \frac{𝑛(𝑛+1)(2𝑛+1)}{6}\]

\[=0⋅lim_{𝑛\rightarrow\infty} \frac{𝑛(𝑛+1)(2𝑛+1)}{6}\]

\[=0\]

Personally I would not have evaluated using even the "right" attempt here, there is a shorter correct approach (pre L'Hopital's Rule). Can you find it?

Fun read. "Lane formation doesn't require conscious thought—the participants of the experiment were not aware that they had arranged themselves into well-defined mathematical curves." and "At a glance, a crowd of pedestrians attempting to pass through two gates might seem disorderly but when you look more closely, you see the hidden structure. Depending on the layout of the space, you may observe either the classic straight lanes or more complex curved patterns such as ellipses, parabolas, and hyperbolas." #curves #conics https://phys.org/news/2023-03-lane-hidden-chaotic-crowds.html

If only my serviettes were 8"x11" I would fold them like this every dinner time. https://www.youtube.com/watch?v=Q2LhERIVALI

Long read investigating what is going on behind the scenes with #ChatGPT #ArtificialIntelligence #NeuralNets #Modeling https://writings.stephenwolfram.com/2023/02/what-is-chatgpt-doing-and-why-does-it-work/

"Probability and random processes might seem not to have much to do with the the analysis of higher dimensions in space, but mathematics is as much a creative art as a science." https://phys.org/news/2023-02-mathematicians-spheres-4d-spaces-scope.html

Evaluating int(f(x)*g(x))dx via integration by parts when f and g are each proportional to their second derivative? Here's a cool shortcut to avoid having to integrate twice presented by @JohnDCook@twitter.com https://www.johndcook.com/blog/2023/01/30/avoid-ibp-twice/

Fun read. 3-dimensional surface in 5-dimensional space. #topology "Black holes are some of the most perplexing predictions of Einstein’s equations — 10 linked nonlinear differential equations that are incredibly challenging to deal with. In general, they can only be explicitly solved under highly symmetrical, and hence simplified, circumstances." https://www.quantamagazine.org/mathematicians-find-an-infinity-of-possible-black-hole-shapes-20230124/#

Fun short read from Nature mag. "To the mathematical-theory builder, abstraction is not a destination, but a journey. As Steingart puts it, ‘abstract’ is not an adjective but a verb: ‘to abstract’. In the 1930s, owing largely to the influence of German mathematician Emmy Noether, mathematicians began to construct axiomatic systems that were increasingly abstract and general." https://www.nature.com/articles/d41586-023-00087-0

Good 2022 overview video (12 mins) well worth watching. One quote (by Dr. Huh) stood out to me, that #mathematics provides “the ability to search for beauty outside oneself, to try to grasp something external, objective and true.” #topology https://www.quantamagazine.org/the-biggest-math-breakthroughs-in-2022-20221222/

"Zenzizenzizenzic is an obsolete form of mathematical notation representing the eighth power of a number (that is, the zenzizenzizenzic of x is x^8), dating from a time when powers were written out in words rather than as superscript numbers. This term was suggested by Robert Recorde, a 16th-century Welsh writer of popular mathematics textbooks, in his 1557 work The Whetstone of Witte; he wrote that it “doeth represent the square of squares squaredly”." Lovely! https://www.thesouthafrican.com/news/weird-word-of-the-day-zenzizenzizenzic-20-december-2022/

Fun fact: 69 is the only natural number whose square (4761) and cube (328509) use every decimal digit from 0–9 exactly once. Think about how you might prove it.

Fun read full of math dad puns via @physorg_com@twitter.com https://phys.org/news/2022-12-statistics-star-crunches-christmas-math.html

"Often mathematics inspires music but here we see music inspiring new mathematical ideas." Iannis Xenakis' 'Nomos Alpha' was performed by cellist Arne Deforce for Birmingham Contemporary Music Group's Music & Maths Festival, celebrating the composer's centenary. Be sure to expand and read the comments. https://www.youtube.com/watch?v=fWlceolQMGQ

“If stopping depended only on how much of a stop signal MLR received then it could be thought of as a form of integration; quantity of the signal would be what mattered. But it doesn’t because integration by itself isn’t enough for rapid control. Instead, MLR accumulates difference between 2 well-timed signals, which mirrors the way a derivative is calculated: by taking difference between 2 infinitesimally close values to calculate slope of a curve at a point.” #calculus https://www.quantamagazine.org/the-brain-uses-calculus-to-control-fast-movements-20221128/

Four enriching videos on some classic problems of #infinity including Hilbert’s Hotel, Zeno’s Paradox, and Cantor’s Theorem. Via @NOVAeducation @NOVApbs https://www.pbs.org/wgbh/nova/article/math-thought-experiment-zero-infinity/