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.  Technology-wise, we would be functioning at the pre-computer level using slide rules and square root tables, an era we left behind in about the year 1965.