When using a Taylor polynomial of degree n centered at c to approximate the value of a function f at x, there is an error because the polynomial does not exactly mimic the function (unless, of course, f is a polynomial of degree less than or equal to n). We can bound this error using the Lagrange remainder (or Lagrange error bound). The remainder is: where M is the maximum of the absolute value of the (n + 1)th derivative of f on the interval from x to c. The error is bounded by this remainder (i.e., the absolute value of the error is less than or equal to R). Note that R depends on how far x is away from c, how big n is, and on the characteristics of f.
See About the calculus applets for operating instructions. |
The applet shows the Taylor polynomial with n = 3, c = 0 and x = 1 for f (x) = e^{x}. To compute the Lagrange remainder we need to know the maximum of the absolute value of the 4th derivative of f on the interval from 0 to 1. Since the 4th derivative of e^{x} is just e^{x}, and this is a monotonically increasing function, the maximum value occurs at x = 1 and is just e. So: Note in the applet that the actual error is about 0.052. The Lagrange remainder is a bound on the error, not the actual error itself. It just says that the error, whatever it is, will be less than the Lagrange remainder.