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IntMath Newsletter: Inverse animations, finding quintic functions

By Murray Bourne, 29 Mar 2016

28 Mar 2016

In this Newsletter:

1. Inverse trigonometric graph animations
2. How do you find the function for a given quintic curve?
3. Math in the news
4. Math puzzles
5. Math movies: 3Blue1Brown
6. Final thought: Keep trying

1. Inverse trigonometric graph animations

Inverse trigonometric graph animations

These animations demonstrate how the inverse of a function is a reflection in the line y = x.

Inverse trigonometric function graph animations

2. How do you find the function for a given quintic curve?

find the function for a given quintic curve

A quintic curve is a polynomial of degree 5. Given such a curve, how do you work backwards to find the original function expression?

How to find the equation of a quintic polynomial from its graph

3. Math in the news

a. Bitcoin transaction delays

Earlier this month, the cryptocurrency Bitcoin network "reached its capacity, causing transactions around the world to be massively delayed, and in some cases to fail completely", according to Bitcoin transaction delays: Is block size increase the final answer?

A bitcoin is a 256 bit binary number that has met certain "conditions". The "conditions" only allow for some 21 million coins, and some 13 million coins have so far been discovered i.e. "mined". The value of bitcoins has varied between less than $1 up to $1000 between 2011 (when it was first released) to now, when it is around $415.

Bitcoin historical value chart
Value of Bitcoins since launch

Bitcoins are organized via the Block Chain, a distributed database on thousands of computers, that records the birth and every transaction of every coin.

The problems with transaction time result from the system only being able to process about 7 transactions per second. With a backlog of some 20,000 transactions on March 3rd, people were waiting in stores for over 30 minutes. There has been a lot of discussion about how to fix the technical problems, including increasing the memory allocation.

Bitcoins, or something like them, are part of our future, since they offer a way to reduce the stranglehold of the big banks. See other potential uses for bitcoins in "15 More Amazing Ways Bitcoin Changes the Future of Money".

b. With Math I Can

With Math I Can

Too many people see mathematics as just a set of formulas, algorithms and calculations. Sadly, they miss what it's really all for, end up with negative views and feel they will never "get" it.

Amazon was encouraging a math "growth mindset" through its With Math I Can strategy, but it has since disappeared.

Growth mindset is based on Stanford University-based Jo Boaler's research on inquiry based learning, gender equity and math anxiety.

On the site, we read that Boaler's colleague Carol Dweck found that...

... students who believed that their ability and intelligence could grow and change (otherwise known as growth mindset) outperformed those who thought that their ability and intelligence were fixed.

I believe addressing student emotions and attitudes toward math is one of the keys to success. However, Boaler's work has not been without controversy (see Your Brain on Maths: Educational Neurononsense Revisited).

c. Shape derivatives to achieve 27% cost efficiencies

Electric motor

Ever heard of shape derivatives? In their work to improve electric motor efficiency, researchers at Johannes Kepler University used shape derivatives to optimize the cost functional. See:

Improving electric motor efficiency via shape optimization

The published paper:

Shape optimization of an electric motor subject to nonlinear magnetostatics [PDF]

[Hat tip to Pi Po for alerting me to these stories.]

4. Math puzzles

The puzzle in the last IntMath Newsletter asked about the maximum value for a particular fraction involving 3 distinct integers.

Correct answers with the most convincing (and elegant) explanations (those which adequately explained why) were given by: Don, Tomas and Ben. There were plenty of correct answer submissions, but not all adequately investigated all the possibilities.

A lot of the time we can "see" the answer to a mathematical problem, but we need to explain it to others (who don't have such insights) in such a way that we cover all the possibilities.

New math puzzle

Two girls need to go to the next town, 40 km away, as quickly as possible, and they have one bicycle between them. Peta walks at 4 km/h and Noni walks at 8 km/h, while both cycle at 16 km/h. Only one girl can ride the bicycle at one time.

What is the shortest time they can take to reach the next town?

You can leave your responses here.

5. Math movies: 3Blue1Brown

3Blue1Brown logo

Grant Sanderson has produced some great math videos at 3Blue1Brown. You'll find videos on Fractals, Music, Binary Counting and Calculus. They are well-expressed and thought-provoking.

See: 3Blue1Brown

6. Final thought: Keep trying

On the WithMathICan site, there's a great quote from Thomas Edison:

"Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time."

Until next time, enjoy whatever you learn.

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