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Category Archives: Calculus/Analysis
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
Posted in Applications, Calculus/Analysis, Web
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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.
Posted in Applications, Calculus/Analysis
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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
Posted in Calculus/Analysis, Teaching, Technology, Trigonometry
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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
Posted in Calculus/Analysis, Number Theory, Quotes/People
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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
Posted in Calculus/Analysis, General, Technology
1 Comment
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
Posted in Books, Calculus/Analysis
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Mnemonic Trig (or Trick)
[tex]\cos (3x) = 4\cos^3(x) -3 \cos(x)[/tex] 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
Posted in Calculus/Analysis, Fun Stuff, Trigonometry
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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
Posted in Books, Calculus/Analysis
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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
Posted in Calculus/Analysis, Fun Stuff, Number Theory, Problems, Teaching, Technology
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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
Posted in Applications, Calculus/Analysis, Complex Numbers, Teaching, Technology
1 Comment
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
Posted in Calculus/Analysis, Fun Stuff
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Separating the variables
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. [tex] \displaystyle \frac{dy}{dx} = \frac{ y^3 + x^2y -x -y}{ … Continue reading
Posted in Calculus/Analysis, Problems
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The Arithmetic vs Geometric Mean Trick
One classical trick is the following: Since [tex] (x -y)^2 \ge 0, [/tex] we have [tex] x^2 – 2 x y + y^2 \ge 0 \implies x^2 + 2 x y + y^2 \ge 4xy [/tex] Taking root we obtain … Continue reading
Posted in Calculus/Analysis
1 Comment
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
Posted in Calculus/Analysis
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Integration problem from pAt84
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 [tex]\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}[/tex] where [tex]\phi = (\sqrt{5}+1)/2[/tex]. A time consuming but fruitful exercise. I had tried all the integration tricks I know, resorted to computer packages Maple and … Continue reading
Posted in Calculus/Analysis, Problems
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