The general method for producing a polynomial approximation of a function is as follows. First, choose the point that you want your approximation to be centered around.

*(When you’re solving a problem, you should expand your approximation around a point within the region that you’re most interested in, because Taylor Polynomial approximations are most accurate near their center, but for simplicity’s sake I’ll just pick x = 0 as the center for now, and give a more general formula later.)*Once you've chosen the point that you want to expand about (in our case,*x*= 0), you can start writing down polynomial approximations. The simplest (and least accurate) polynomial we could use to approximate a function would just be a constant, equal to the value of the function at*x*= 0 (i.e.*P*(_{0}*x*) =*f*(0), where "*P*" is a polynomial function and the subscript indicates the order of the polynomial).Zero-Order Taylor Polynomial (magenta) for f(x) (blue) |

As you can probably tell, a zero-order polynomial usually won't be sufficient to provide a useful approximation of a function. It is, of course, exactly equal to the function at

*x*= 0, but it quickly diverges from the function as we move away from*x*= 0, since the function is changing its value while our approximation is not. We can do better by using a polynomial that not only has the same*value*as the function at*x*= 0, but also changes at the same rate (i.e. has the same value of its first derivative). We do this by setting*P'*(*x*) =*f'*(0) and then integrating:*Remember that the derivative of f evaluated at a singlepoint ( x = 0) is a constant, not a function of x. |

And our approximation gets a whole lot better:

First-Order Taylor Polynomial (magenta) for f(x) (blue) |

Of course,

*f*(*x*) isn't linear, so our first-order approximation still diverges from the function eventually. If we only need to consider small values of*x*, this approximation may be sufficient, but if the situation calls for accuracy at larger values of*x*, we need to go deeper...To the second dream lev- ...err, derivative. |

The linear approximation eventually diverges from

*f*because, while*P*_{1 }has a constant slope, the slope of*f*is changing. To get a better approximation, we need a polynomial that not only has the same value and slope as*f*at*x*= 0, but also is changing its slope at the same rate as*f*(*x*) (i.e.*P''*(*x*) =*f''*(0)).As before, we find the constants of integration by setting the nth derivative of P equal to the nth derivative of f at x = 0. |

Second-Order Taylor Polynomial (magenta) for f(x) (blue) |

For increasingly better approximations, we repeat this process of setting the

*n*th derivative of*P*equal to the*n*th derivative of*f*at*x*= 0 and then integrating, for ever-increasing values of*n*, so that:
In fact, if we let

*n*go to infinity, and if*f*(*x*) is a continuous, analytic function, then the Taylor Series is exactly equivalent to the function:
Earlier I promised I'd give a generalized formula to allow for the Taylor Polynomial to be centered around

*any*point. To do this, just replace "0" with "*c*," and "*x*"
(=

*x*- 0) with "*x*-*c*":
Obviously, you're never actually going to calculate an infinite number of terms, but luckily in many cases you can get a pretty good approximation of a function with just a few terms.

Now, you might be wondering when you'd ever want to approximate a function as a polynomial. It turns out that in certain problems, either the function that you're working with isn't explicitly known, or it's just more difficult to work with than a polynomial. For example, if you're only dealing with small angles, it'll often make it easier on you to approximate sin

Now, you might be wondering when you'd ever want to approximate a function as a polynomial. It turns out that in certain problems, either the function that you're working with isn't explicitly known, or it's just more difficult to work with than a polynomial. For example, if you're only dealing with small angles, it'll often make it easier on you to approximate sin

*x*as just*x*, and cos*x*as 1 -*x*^{2 }(It's this approximation that allows us to describe the motion of a pendulum bob as simple-harmonic, as long as the maximum displacement angle is small).^{}