Mathematics 150BL, Fall 2003

Assignment 9 -- Taylor and Maclaurin Series

Preliminary Remarks and examples:

If a function $f$ is $\ n+1$ times continuously differentiable at a point $a$ we can compute its Taylor polynomial of degree $n$ at $a$ as follows:

MATH where $R_{n}(x,a)$ is a remainder term which satisfies

MATH with $\ M$ the maximum value of MATH for all $t$ such that MATH.

The Taylor polynomial is an approximation of the function; the remainder term measures the error in this approximation. For example, the Taylor polynomial of degree $4$ at $0$ for the function $f(x)=\frac{1}{1-x}$ is computed as follows:

MATH

MATH

MATH

MATH

Putting these terms together we get MATH

The class software can do the entire computation for us, by using Compute+Power Series and choosing 5 for the number of terms and $(x-0)$ for the power.

MATH

The term MATH represents the remainder, and signifies that the remainder is a fifth order term.

In the figure below both $f(x)$ and its 4th degree Taylor polynomial about zero are plotted. (When you do this you should make sure that you do not include the MATH term in the plot.)

$f$
graphics/f3b9__27.png

We see that close to 0 the two curves are identical, further away they no longer match. If we want a better measure of the actual error we plot

MATH
graphics/f3b9__29.png

We see that in the interval MATH this error is less than 0.05.

The Taylor series of $f$ at $a$ is MATH. The special case of $a=0$ is called the Maclaurin series of $f$.

Problems

      1. Use the class software to find the Taylor polynomials up to degree 6 for $g(x)=\cos x$ centered at $a=0$. Graph $g$ and these polynomials on a common screen.

      2. Evaluate $g$ and these polynomials at $x=\frac{\pi }{4},$ $\frac{\pi }{2},$ and $\pi $.

      3. Comment on how the Taylor polynomials converge to $g(x)$.

      1. Use the class software to find the Taylor polynomials up to degree 3 for $r(x)=\frac{1}{x}$ centered at $a=1$. Graph $r$ and these polynomials on a common screen.

      2. Evaluate $r$ and these polynomials at $x=0.9$ and $1.3$.

      3. Comment on how the Taylor polynomials converge to $r(x)$.

    More Problems

    Use the class software to find the Taylor polynomial of degree 7 for the given functions around the indicated points. Plot the function together with the Taylor polynomial in a single graph. Then plot the actual error: MATH. Choose your units carefully and estimate the size of the error.

    1. $f(x)=\sqrt{1+x},$ $a=0$

    2. $h(x)=e^{x},$ $a=0$

    3. $k(x)=e^{-x},$ $a=0$

    4. Would the Taylor polynomials of degree 5 be better or worse approximations to the functions? Explain. What about the degree 9 Taylor polynomials? Explain.

    5. Find the Maclaurin series for $f(x)=e^{x}$. Then use it to find a series which is equal to $e$.