Gods Gift to Calculators: The Taylor Series

It is incredible how far calculators have come since my parents were in
college, which was when the square root key came out. Calculators since then
have evolved into machines that can take natural logarithms, sines, cosines,
arcsines, and so on. The funny thing is that calculators have not gotten any
"smarter" since then. In fact, calculators are still basically limited to the
four basic operations: addition, subtraction, multiplication, and division! So
what is it that allows calculators to evaluate logs, trigonometric functions,
and exponents? This ability is due in large part to the Taylor series, which
has allowed mathematicians (and calculators) to approximate functions,such as
those given above, with polynomials. These polynomials, called Taylor
Polynomials, are easy for a calculator manipulate because the calculator uses
only the four basic arithmetic operators.

So how do mathematicians take a function and turn it into a polynomial
function? Lets find out. First, lets assume that we have a function in the form
y= f(x) that looks like the graph below.

We\'ll start out trying to approximate function values near x=0. To do
this we start out using the lowest order polynomial, f0(x)=a0, that passes
through the y-intercept of the graph (0,f(0)). So f(0)=ao.

Next, we see that the graph of f1(x)= a0 + a1x will also pass through x=
0, and will have the same slope as f(x) if we let a0=f1(0).

Now, if we want to get a better polynomial approximation for this
function, which we do of course, we must make a few generalizations. First, we
let the polynomial fn(x)= a0 + a1x + a2x2 + ... + anxn approximate f(x) near x=0,
and let this functions first n derivatives match the the derivatives of f(x) at
x=0. So if we want to make the derivatives of fn(x) equal to f(x) at x=0, we
have to chose the coefficients a0 through an properly. How do we do this?
We\'ll write down the polynomial and its derivatives as follows.

fn(x)= a0 + a1x + a2x2 + a3x3 + ... + anxn

f1n(x)= a1 + 2a2x + 3a3x2 +... + nanxn-1

f2n(x)= 2a2 + 6a3x +... +n(n-1)anxn-2

.
.
f(n)n(x)= (n!)an

Next we will substitute 0 in for x above so that
a0=f(0) a2=f2(0)/2! an=f(n)(0)/n!

Now we have an equation whose first n derivatives match those of f(x) at
x=0.

fn(x)= f(0) + f1(0)x + f2(0)x2/2! + ... + f(n)(0)xn/ n!

This equation is called the nth degree Taylor polynomial at x=0.
Furthermore, we can generalize this equation for x=a instead of just