Introduction

This is a short introduction to the notion of using polynomials to approximate more complicated functions. It is entirely informal, with the intent of motivating a careful study of infinite series prior to learning about Taylor polynomials and Taylor series.

1 A Very Basic Approximating Polynomial

Consider the following algebra centering on polynomial multiplication,

The approximation in the last step is valid if {x}^{n+1} is small, which will be the case if − 1 < x < 1 and n is large. Keep those conditions in mind as we continue.

If we assume x\mathrel{≠}1 and divide both sides of the above by 1 − x we obtain

2 Approximating a Rational Function

We can use the approximation above to create an approximation of a rational function (a fraction of two polynomials). With a systematic use of partial fractions, this method can be extended to more complicated examples. Consider the following:

Employ equation (1) where we replace x by {x}^{2},

Notice that our approximation should again be best when n is large and now we would require − 1 < {x}^{2} < 1 which simply translates back to − 1 < x < 1. In the demonstration below we only plot the function for − 1 < x < 1. The full graph would have vertical asymptotes at x = −1 and x = 1 and has two branches below the x-axis — one for x < −1 and another for x > 1.

3 Approximating a Transcendental Function

We begin with equation (1), replacing x by − {t}^{2}, then form a definite integral that equals the inverse tangent. Again, we would expect larger values of n, with − 1 < x < 1, to yield better approximations. First,

We will now use a definite integral and the derivative of the inverse tangent in a novel way,

In the demonstration below we have drawn attention to the value of the function and the approximating polynomial at x = 1. It is of course debatable if the approximation is even valid at x = 1, but we will examine this question carefully later. We know that in a 45-45-90 right triangle, the two non-hypotenuse sides are equal. Expressing angles in radians we formulate this fact as \tan \left ( {\pi \over 4}\right ) = 1, or equivalently, \arctan (1) = {\pi \over 4}. So if we evaluate our approximating polynomial at x = 1 we should get a reasonable approximation of {\pi \over 4}, and by extension, multiplying by 4 we could obtain an estimate of \pi . Consider that \pi is defined as the ratio of a circle’s circumfrence to its diameter. Could we derive the necessary trigonometric facts, limits, derivatives and integrals used above without ever computing an actual value for \pi ? I think so. Hmmmmmmmmm. The “real” value of {\pi \over 4}, and the approximation, are given below, in addition to being pinpointed by specific points on the plot.

4 An Approximation Valid Everywhere

Our previous approximating polynomials were each valid, at best, on the interval − 1 < x < 1. We will change our approach for this final example by simply producing a very interesting polynomial and examining its properties. Recall that “n-factorial” is defined by n! = n(n − 1)(n − 2)\cdots 3\cdot 2\cdot 1, and by convention 0! = 1. Consider the polynomials, indexed by their degree n,

Each of these polynomials has a derivative, which we will compute. (Notice how fractions with factorials simplify nicely.)

So {P}_{n}(x) and its derivative, {P}_{n}'(x), are very similar, differing only in the term {{x}^{n}\over n!} .

Even for fixed values of x greater than 1, if we let n get large enough, the denominator of this fraction will overwhelm the numerator, and this difference will be small and tend to zero. So {P}_{n}(x) is very nearly equal to its derivative. Do we know any functions like this? Ah, yes, {e}^{x}! The demonstration below examines the possibility that these polynomials might be good approximations to the exponential function.