December 12, 2017 Tuesday
Bedtime Story
Two Polynomial Examples
Column 2 in the difference engine is set to
a value derived from the first and higher derivatives of the polynomial.
Then the following columns from 3 to N are
set to values derived from the (n-1) first and the higher derivatives of the
polynomial.
So you see that the machine is not very
user friendly as I am sure that the above statements will not make any average
age much wiser on the use of the machine.
It might help perhaps if at this moment I
pop in a couple of examples of polynomials along with the accompanying columns
of finite differences.
Consider a polynomial function
f(x) = x2 + 1
So let us make the four columns for this
one
x f(x) diff 1(x) diff 2(x)
0 1 1 2
1 2 3 2
2 5 5 2
3 10 7 2
In order for the machine to work and work
accurately, at least first 3 values of x have to be calculated mentally along
with the accompanying the first and second differences on the columns.
Please note that in the second order
polynomial, we are getting a constant in column of difference 2.
Then some values have to be fed in manually
into the machine my mechanically rotating the shafts till the rights numbers
are in place in the right columns.
From then on the machine is programmed so
to speak.
We only need to rotate the crank to get
sequential values of the polynomial equations.
Have a look at yet another polynomial
equation of third order and see how the columns work out for that one.
f(x) = x3 – 2x2 + 1
Let us write down the columns of finite
difference for this polynomial as well.
x f(x) diff 1(x) diff 2(x) diff 3(x)
1 0 1 8
6
2 1 9 14 6
3 10 23 20 6
And so on.
Note that in the third degree polynomial
equation the constants are appearing in the diff 3 column.
In a five degree polynomial equation the
constants will appear in the difference 5 column and that will be the recurrent
pattern for any polynomial of ay degree.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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