# Category Archives: Calculus/Analysis

## baby rudin

Define a real valued function on the reals by $f(x):=\begin{cases} 0, & x \not\in \mathbb{Q} \\ \frac{1}{q}, & x = \frac{p}{q}, \gcd(p,q)=1, q >0 \end{cases}$. Where is this function continuous? The problem appeared in Hyman Bass’s article in the … Continue reading

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## Speed Index

A clearly written page explaining how one can measure the speed in which a website loads by integrating the area under the curve. link

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## 3D printing

seems to be the rage these days, and the natural question is that if there is anything that you cannot print. The short answer is NO. Thanks to Fubini’s Theorem.

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## Fake curves

Interesting post on how a supposed trig function is actually not trig. I have to admit I used to do things like this and use the arc tool (on powerpoint) to create curves that only look vaguely like the actual … Continue reading

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## Hardy on Number Theory

The elementary theory of numbers should be one of the very best subjects for early mathematical instruction. It demands very little previous knowledge; its subject matter is tangible and familiar; the processes of reasoning which it employs are simple, general … Continue reading

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## gnuplot with PGF/tikz

One downside about writing calculus lecture notes has always been drawing graphs. They just take up too much time. What I used to do was to use whatever software at my disposal (usually powerpoint) and created jpeg to embed into … Continue reading

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## Manga guide to calculus

by Kojima and Togami (Illustrator). This is a follow up to the well received Manga guide to Statistics, which aims to use japanese manga to teach academic subjects. Having read the book I must say, I enjoyed the back story … Continue reading

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## Mnemonic Trig (or Trick)

$\cos (3x) = 4\cos^3(x) -3 \cos(x)$ The way to remember it in Hokkien (a local dialect commonly spoken in Singapore, Taiwan and southern parts of China): Cor Sah ($1.30) = Si Cor Sah ($4.30) – Sah Cor (\$3). You can … Continue reading

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## The Calculus of Friendship

by Steven Strogatz. The subtitle is “What a teacher and a student learned about life while corresponding about math” and it very aptly sums up the book. Except there is a twist. It turned out the teacher was a high … Continue reading

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## 12 Days of Christmas

Nice problem and solution at squarecirclez about counting the total number of presents. Triangular numbers appears! It seems that this year’s Simpsons Christmas special was a live action parody/tribute to Sesame Street and the Muppets. It had them singing the … Continue reading

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## Application of Harmonic function

Cool article about Pixar and some applications of harmonic function. Would have been useful last year when I taught the maximum principle but didn’t have much else to say about it. Moving Remy in Harmony: Pixar’s Use of Harmonic Functions … Continue reading

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## Breaking up is infinitely hard

I was reading this light-hearted article by jeremy au yong in the local papers. It’s mainly about how the economy is bad and he was retrenching his girlfriend. As compensation he offered a gift. The value of this gift will … Continue reading

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We should all be familiar with the method of separation of variables for first order ordinary differential equations. Here’s a neat example from John Starrett’s article in Amer. Math. Monthly. $\displaystyle \frac{dy}{dx} = \frac{ y^3 + x^2y -x -y}{ … Continue reading Posted in Calculus/Analysis, Problems | Leave a comment ## The Arithmetic vs Geometric Mean Trick One classical trick is the following: Since [tex] (x -y)^2 \ge 0,$ we have $x^2 – 2 x y + y^2 \ge 0 \implies x^2 + 2 x y + y^2 \ge 4xy$ Taking root we obtain … Continue reading

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## Silly Riddle

What’s the difference between jumping down from the 2nd floor versus the 20th floor? Ans: Splat! Aaaaaaahhhhh! (2nd floor) Aaaaaaaahhhhh! Splat! (20th floor) What if you jump from the 10th floor? It all depends on how long you remain in … Continue reading

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My latexrender (installed by my wife with much pain) has gone to disuse. So I’m putting up this problem posted by pAt84. Version 1 $\mbox{\Large \mu = \int_0^1 \frac{ \big( \sum_{j=0}^n a_j t^j \big) \big( \sum_{i=1}^n i b_i t^i \big) … Continue reading Posted in Calculus/Analysis, Problems | 1 Comment ## Stoke’s Theorem and Vector Analysis When I first learnt this 9 years ago, I hated it. It was a “here’s the formula, use it” kind of course. Second time around, I learn the subject in the context of differential forms and boy what a world … Continue reading Posted in Calculus/Analysis | 6 Comments ## Elementary Proofs No one shall expel us from the paradise that Calculus has created. – tpc The fact that an analytic solution to number theoretical problems exist is bewildering. And often, the analytic proof is much simpler than the elementary proof. The … Continue reading Posted in Calculus/Analysis, Number Theory, Problems | 2 Comments ## Irresistible Integrals A book by George Boros and Victor Moll. The authors bill the book as a guide to the evaluation of integrals, but it strucked me as a nice guide to old-fashioned (nineteen century) analysis, in the spirit of Gauss and … Continue reading Posted in Books, Calculus/Analysis | 1 Comment ## Irritating Integral It took me almost a week to prove [tex]\int_0^{\pi} \log (1+4\cos^2(u)) du = 2\pi \log{\phi}$ where $\phi = (\sqrt{5}+1)/2$. A time consuming but fruitful exercise. I had tried all the integration tricks I know, resorted to computer packages Maple and … Continue reading

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