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IntMath Newsletter: First principles applet, resources

By Murray Bourne, 28 Nov 2017

28 Nov 2017

In this Newsletter:

1. New applet: Calculus first principles
2. Resource: ImmersiveMath
3. Math in the news: Proof School, GPU problem solvers
4. Math movie: Inspiration
5. Math puzzle: Intersecting parabolas
6. Final thought: Tenants

1. New applet: Calculus first principles

Sometimes the early concepts (usually called "first principles") of calculus can be a bit confusing.

Calculus first principles applet

This new applet allows you to explore the concepts of differentiation and integration from first principles.

Calculus first principles interactive applet

2. Resource: ImmersiveMath

This is "the world's first linear algebra book with fully interactive figures".

immersive math example

Immersive Linear Algebra contains some excellent animations and manipulatives that will help you to understand vectors, dot products, matrices, and determinants.

The pages take a while to load, but it's worth it!

3. Math in the news:

(a) Proof School

A few years back I did some mathematics curriculum consultancy work for the newly-established School of Science and Technology, Singapore. Their approach is to incorporate various technologies into an interdisciplinary mix of art, design, media and environmental studies, on top of the national curriculum. I was reminded of that experience when I came across this article on Proof School, in California.

Proof School

Proof School is designed for "students who love math".

Here is some background about the school, which has roots in the "math circle" concept, where kids can feel comfortable in a supportive and challenging math environment:

A School Built Entirely Around the Love of Math

(b) Innovative approach to solving massive problems on an ordinary PC

Russian scientists realized that if they used the GPU (graphics processor unit) of an ordinary PC, they could solve "260 million complex double integrals on a desktop computer within three seconds only". The best a supercomputer can achieve when solving the same problems is 2 to 3 days - and at a much greater expense.

Nvidia GEForce GPU

The GPU used a Nvidia GPU designed for use in game consoles.

See: Russian scientists use a PC to solve complex problems tens of times faster than a supercomputers

4. Math Movie: Exceptional teachers

(This is a bit late for last month's Teachers' Day, but still...)

Here's Stephen Hawking talking about his inspirational mathematics teacher:

"I have to admit – I wasn’t the best student – but with Dikran Tahta's support I became a professor of mathematics at Cambridge, in a position once held by Isaac Newton."

Mr Tahta - Stephen Hawking's math teacher

See the short video:

Stephen Hawking On The Teacher That Changed His Life

5. Math puzzles

The puzzle in the last IntMath Newsletter asked about different formulas for producing a given sequence of numbers.

Correct answers with explanation were provided by Phil, Karam and Chris.

New math puzzle: Intersecting parabolas

A quadratic graph with vertical axes has the general form:

y = ax2 + bx + c

Two parabolas with vertical axes can intersect in at most 2 points.

Now, consider the following graph. Here, I've rotated one of the parabolas:

Intersecting parabolas

The equation of that rotated parabola has the form:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

We can see the two parabolas intersect in 4 points, which is the maximum number of intersection points.

Or is it?

How is it possible for the coordinates of the five points (−1, 6), (3, −2), (1, 1), (0, 4) and (2, 0) to satisfy both of these quadratic equations?

2x2xyy2 − 4x + 4y = 0

6x2 + xyy2 − 16x + 2y + 8 = 0

You can leave your responses here.

6. Final thoughts: Tenants

Plastic garbage

Here's an apt quote from Rose Bird, Chief Justice of the California Supreme Court:

"We have probed the earth, excavated it, burned it, ripped things from it, buried things in it, chopped down its forests, leveled its hills, muddied its waters, and dirtied its air. That does not fit my definition of a good tenant. If we were here on a month-to-month basis, we would have been evicted long ago."

[Hat-tip to PiPo for alerting me to some of the items in this Newsletter.]

Until next time, enjoy whatever you learn.

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