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IntMath Newsletter: Applets, Digital Einstein

By Murray Bourne, 15 Jan 2015

15 Jan 2015

In this Newsletter:

1. New interactive applets
2. Resource: Digital Einstein
3. Math puzzles
4. Final thought - Passionately curious

Happy New Year, readers!

I received a delightful mail from Den shortly after the last Newsletter went out. This is what he said:

Again Mr. Bourne let me compliment you on producing such a high quality newsletter. Yours is the only news letter that I subscribe to.

Thank you very very much for producing such valuable content for math tutors!

Den Ducoff

Thank you for your mail, Den, and I'm glad you find the Newsletters valuable!

1. New interactive applets

(a) Polar to rectangular calculator

Polar to rectangular forms of complex number calculator

Here's a new complex numbers online calculator that converts polar form to rectangular form and vice-cersa. It also displays a graph so the concepts make more sense. See

Polar to Rectangular Online Calculator

(b) Differentiation graphs

Interactive graphs of derivatives involving discontinuities

This applet allows you to explore the graph of a derivative involving discontinuities. You can animate the path of a point on both the original graph and the derivative.

Differentiation interactive applet

This next applet is related, but involves continuous functions only:

Interactive Graph showing Differentiation of a Polynomial Function

2. Digital Einstein

Portrait of Einstein

Princeton University has released The Collected Papers of Albert Einstein, which give a fascinating insight into the original thinking of this oft-quoted genius.

It's not all about the Theory of Relativity, of course. The following quote caught my eye, because he's talking about a subject that trips up most beginner calculus students — what happens to those "infinitely small quantities" as they "tend to zero", and which are vital to how calculus works? Eseentially, he says not to worry too much about it:

Whether one, along with Leibnitz, Poisson, Herbert, et al., seriously wants to take the infinitesimally small for a truly indivisible element, or one wants, along with others, to take it only for a useful fiction, so as thereby supposedly to eliminate all metaphysical difficulties, and conveniently and quickly introduce the calculus, is irrelevant for the calculus, for the one as much as the other leads to the goal.

[From Volume 1: The Early Years, 1879-1902 (English translation supplement) Page 4]

3. Math puzzles

The puzzle in the last IntMath Newsletter asked about the cost of a chocolate bar. The correct answer with explanation was given by Chris, Thomas, Tomas, and Francisco.

New math puzzle: The sequence xn is defined as:

xn = n3 − 9n2 + 631 (where n is an integer)

What is the highest value of n such that xn > xn+1?

Leave your responses here.

Correction: Last mail (10 Dec 2014 edition), I missed giving proper recognition to all those who responded to the exponential and log functions puzzle (in the 28 Nov 2014 Newsletter). I corrected it here.

4. Final thought - Passionately curious

On the home page of Digital Einstein (mentioned above), they've included a great Einstein quote:

"I have no special talents. I am only passionately curious." [Einstein]

A truly intelligent person is one who is keen to learn as much as they can about a wide variety of topics, and who asks a lot of questions. I've also observed that the best teachers provide good role models for their students' learning. They question things, are willing to admit they don't know something, and have good ideas on how they might find the answers.

Until next time, enjoy whatever you learn.

See the 14 Comments below.

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