How A Calculator Computes Functions 
Polynomials Powering Our World (Watch
The Video!)
Did you know that every time you pick up a calculator and use
function key like a square root key, an exponent function, or other
scientific function, you are calculating a polynomial approximation?
A calculator is really a computer. And computers can only
add, subtract, multiply, and divide. So how would the calculator
evaluate the square root of 1.2? Or find 3.1^{2.4}? It
can't store up all the answers since there are an infinite number of
potential answers. Instead, the calculator uses what is called a
Taylor Polynomial Approximation.
So What is a Taylor Polynomial Approximation?
Given a differentiable function f(x), the Taylor Polynomial
Approximation at a value x = c is:
where f '(c) is the first derivative evaluated at x=c, f ''(c) is the
second derivative evaluated at x=c, and f^{(n)}(c) is the nth
derivative evaluated at x=c. The fewer terms used, the less
accurate this approximation is. Only an infinite number of terms
provides the exact value of f(x).
The 4th Degree Taylor Polynomial Approximation of
at x = 1
Watch
the video! As we include more and more terms, the
approximation gets better and better. In fact, when all 5 terms
of the polynomial above are included, this approximation evaluates the
square root of 1.5 correctly out to two decimal places! The only
operations needed to evaluate a polynomial are addition, subtraction,
multiplication, and division  all possible by a computer.
Do Calculators Really Use The Taylor Polynomial?
I inquired recently with Texas Instruments and asked. Here is
the reply I got.
Polynomials Power Our World!
Without the Taylor Polynomial or other polynomial approximation, there
would be no way for scientific calculators and computers to perform
the calculations needed to guide our spaceships and aircraft. Banks
would have no quick way to calculate compound interest.
Technologywise, we would be functioning at the precomputer level
using slide rules and square root tables, an era we left behind in
about the year 1965.
