**David devised a scheme to find a polynomial
rule for any number of input and output pairs of numbers you come up with, from
your rule. [He has written this up and it is in the hands of the NCTM for
publication].**

Don spent over a month trying to understand what David did. After 55 years of teaching math, Don had never seen anything like this. He worked hard to try examples using David's method, then encouraged David to write this up.

Don made up the rule y=**ã**** ^{x}**
+ x, to test David's method and figured out the approximations for each
y-value to 8 digit accuracy, using

N[**ã**** ^{0}**
+ 0, 8]

**1**

N[**ã**** ^{1}**
+ 1, 8]

**3.71828**

N[**ã**** ^{2}**
+ 2, 8]

**9.38906**

N[**ã**** ^{3}**
+ 3, 8]

**23.085537**

Don's rule y=**ã**** ^{x}**
+ x, gives the following pairs of numbers: (x,y),
(0,1), (1,

He used the command **Expand** on David's work (**not shown here**) and
got the following function:

1 + 2.93311x - 1.06036x^{2 }+ 0.845536 x^{3}

Don used *Mathematica* below to show that when the ** x-value** is substituted into the
function (using .. /.x-> ), the **
y-value** comes out as it should.

1 + 2.9331069780437167`x - 1.0603608348801465`x** ^{2}**
+ 0.8455356852954754`x

**1**

1 + 2.9331069780437167`x - 1.0603608348801465`x^{2}^{
}+ 0.8455356852954754`x** ^{3}**/.x®

**3.71828**

1 + 2.9331069780437167`x - 1.0603608348801465`x^{2}^{
}
+ 0.8455356852954754`x** ^{3}**/.x®

**9.38906**

1 + 2.9331069780437167`x - 1.0603608348801465`x^{2}^{
}+ 0.8455356852954754`x** ^{3}**/.x®

**23.0855**

Don graphed David's found function (a cubic polynomial) and
Don's
exponential function in *Mathematica*, from 0 to 8.
The graphs show that from 0-3 they
are exactly the same, then Don's exponential
function jumps up faster when x>3. So David's method of finding a rule for
the 4 pairs of numbers -or any number of pairs- really works, and the arithmetic
in David's scheme was much, much, easier using *Mathematica*.

Plot[{1+2.933106978043716`x-1.0603608348801465`x^{2}^{
}+ 0.845535685295475`x** ^{3}**,