next up previous
Next: About this document ... Up: Taylor Polynomials Previous: Maple Commands

Exercises

  1. For the following functions and base points, determine what minimum order is required so that the Taylor polynomial approximates the function to within a tolerance of $0.1$ over the given interval.
    1. $f(x) = \exp(x)$, base point $a=0$, interval $[-2,2]$.
    2. $\displaystyle f(x) = \ln(1+x)$, base point $a=0$, interval $[-0.9,0.9]$.
    3. $\displaystyle f(x) = \frac{\sin(x)}{x^2+1}$, base point $a=0$, interval $[-0.9,0.9]$.
    4. $\displaystyle f(x) = \frac{x}{(x+1)^2}$, base point $a=1$, interval $[-0.5,2.5]$.
    5. $\displaystyle f(x) = \frac{1}{x^2}$,base point $a=2$, interval $[0.2,3.8]$. (Hint: You may want to use axes=none at the end of this plot command to see what is happening in the plot.)
  2. For the function, $f(x) = 1/(x-1)$, use the TayPlot command to plot the function and multiple Taylor polynomial approximations of various orders with base point $a=0$ on the same graph over the interval $-3 \leq x \leq 3$; use a y-range from $-5$ to $5$.
    1. If you increase the order of the Taylor polynomial, can you get a good approximation at $x=-1.5$?
    2. Can you make a good guess at the radius of convergence of the Taylor series for $f$?
  3. Repeat all of exercise 2 using $f(x)=\displaystyle \frac{x}{\sqrt{x^2+4}}$ and the same ranges.

    A theorem from complex variables says that the radius of convergence of the Taylor series of a function like $f$ is the distance between the base point ($a=0$ in this case) and the nearest singularity of the function. By singularity, what is meant is a value of $x$ where the function is undefined. Where is $f$ undefined? Is the distance between this point and the base point consistent with your guess of the radius of convergence from the plot?

next up previous
Next: About this document ... Up: Taylor Polynomials Previous: Maple Commands
William W. Farr
2006-04-03