About the Applet

The Taylor series, centered at a number \(a\), of an infinitely differential function is the sum \[ \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!}(x-a)^k = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \frac{f^{(4)}(a)}{4!}(x-a)^4 + \cdots \] The first \(n\) terms of this series form a polynomial, that can be taken as an approximation of the function \(f\) near \(a\). The Maclaurin series is the Taylor series centered at 0. Hence, the \(n\)th degree Maclaurin polynomial approximating a function \(f\) near 0 has the form \[ \sum_{k=0}^n \frac{f^{(k)}(0)}{k!}x^k = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots + \frac{f^{(n)}(0)}{n!}x^n \] Note that if the function being approximated is a polynomial, its Taylor series will have only a finite number of nonzero terms, and will be the function itself.

For some functions, the Maclaurin polynomial approximation will be quite poor outside of a small neighborhood of zero, regardless of degree. An example of this can be seen with the function \(\ln(1+x)\), where the approximation has relatively error for \(|x|\) < 1, even for low degree, while there is substantial error for values of \(x\) greater than 1. Other functions, such as sine and cosine, have approximations which are fairly accurate on larger intervals as the degree of the polynomial increases.

Using the Applet:
After choosing a function to approximate from the pull down menu, use the slider to change the order of the approximating Maclaurin polynomial.