M150B Lab power series2_27_08.htm

M150B Lab

Assignment #2

Feb 28, 2008

Part 2:

Recall: If a function is times continuously differentiable at a point we can compute its Taylor polynomial of degree at as follows:

$P_{n}(x)$ MATH

and

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

with the maximal value of for all such that $\leq$ .

The Taylor polynomial is an approximation of the function, and the remainder term measures the error in this approximation.

For example, we can find the Taylor polynomial of degree 5 at 0 for the function $f(x)=\sin x$ .

Find the first five derivatives of $\sin x$ and evaluate each at :

$f(0)=\sin 0=0$

$f^{^{\prime }}(0)$ = $\cos 0=1$

Scientific Notebook can do the entire computation for us, by using Compute+Power Series and choosing 6 for the number of terms and for the power:

The remainder

The notation (called the big Oh notation) signifies that the remainder is

of sixth order term.

Example: Recall the Taylor series of $\sin x:$

MATH

The 5th degree Taylor polynomial is

$P_{5}(x)=$ Graph $\sin x$ and $P_{5}(x)$

M150B Lab power series2_27_08__39.png

Now we want to study Look at the graph:

M150B Lab power series2_27_08__42.png

$P_{5}(x)$ appears to be a "pretty good" approximation to near

How can we quantify "pretty good?" Perhaps we can ask a specific question, such as for which is The "pretty good" is translated as

and $P_{5}(x)$ $\$ are within a tolerance of

M150B Lab power series2_27_08__52.png

Solve

Solution

M150B Lab, Assignment #2, Part 2, Due Feb 28

Consider each of the following functions, complete exercises A-D below:

1. $\ a=0$

2. $f_{2}(x)=\ln x,$

3.

A. Find $P_{2}(x)$ and $P_{5}(x)$ for each of the $\ f_{i},$

B. Graph $f_{i}$ and $P_{2}(x)$ and $P_{5}(x)$ on the same graph,

C. Graph and

D. Find for which ,

and find for which , for

You can do this from the graph or analytically.

This document created by Scientific Notebook 4.0.