Introduction
Theory
HOWTO
Examples
Questions
Matlab
Maple

# Introduction

One of the five tools which we will use in this course are
Taylor series. These have been covered in your Calculus courses,
so this topic is a general overview. For further help, see your
calculus text.

# Theory

In your calculus course, you have probably seen Taylor series written as

where *x*_{0} is a point around which we are
expanding, and *x* is some point in the neighbourhood. In this
course, we will rewrite this as:

where *x* is some point around which we are expanding the Taylor
series and *h* is a *small* value. To recall some
terminology, the approximation f(*x* + *h*) = f(*x*) is called a zeroth-order
Taylor-series approximation, while f(*x* + *h*) = f(*x*) + f^{(1)}(*x*) *h* is
a first-order Taylor-series approximation.

To view the usefulness of Taylor series, Figures 1, 2, and 3 show the 0th-, 1st-, and 2nd-order
Taylor series approxiations of the exponential function f(*x*) = *e*^{x} at *x* = 0. While the
approximation in Figure 1 becomes poor very quickly, it is quite apparent that the linear, or 1st-order,
approximation in Figure 2 is already quite reasonable in a small interval around *x* = 0. The quadratic,
or 2nd-order, approximation in Figure 3 is even better.

Figure 1. The zeroth-order Taylor series approximation of *e*^{x} around *x* = 0.

Figure 2. The first-order Taylor series approximation of *e*^{x} around *x* = 0.

Figure 3. The second-order Taylor series approximation of *e*^{x} around *x* = 0.

We will use Taylor series for two purposes:

- To linearize a system, using the 1st-order Taylor-series approximation, and
- To perform error analysis on numerical method.

The most useful feature is that in many cases we can use a 1st-order Taylor
series, that is, a linear polynomial, to approximate a function locally. This
is a result of the observation that any differentiable function, locally, looks
like a straight line. To observe this, take a look at the
HOWTO on differentiable functions.
Thus, if we are in the neighbourhood of a point and we have some knowledge
as to its Taylor series (from our model) then we may successfully approximate
the value of points on the curve in that given neighbourhood.

# Examples

To show how good Taylor series are at approximating a funciton, Figures 4 and 5
show successively higher and higher Taylor series approximations, starting with the zeroth order
Taylor series approximation, of the function f(*x*) = sin(*x*) around the point *x* = 1.

Figure 4. The 0th- through 20th-order Taylor series approximations of f(*x*) = sin(*x*) around *x* = 1.

Figure 5. A cummulative plot of the 0th- through 20th-order Taylor series approximations of f(*x*) = sin(*x*) around *x* = 1.

# HOWTO

Given a function f(*x*) and point *x*, we calculate the
*n*th-order Taylor series around *x* by evaluating the function
and its first *n* derivatives at *x* and writing down:

If you now wish to approximate the value of f(*x* + *h*), simply evaluate
the above series with the given value of *h*.

The error associated with the above Taylor approximation is:

where ξ ∈ [*x*, *x* + *h*] (or [*x* + *h*, *x*] if *h* is negative).

In most cases, engineers use linear approximations:

If *h* is small (e.g., 0.001), then *h*^{2} will be very small (0.000001), and therefore,
unless the second derivative is very large, the error will be resonably small.

# Examples

# Example 1

Rewrite the Taylor series

f(*x*) = f(*x*_{0}) + f^{(1)}(*x*_{0})(*x* - *x*_{0}) + 1/2 f^{(2)}(*x*_{0})(*x* - *x*_{0})^{2} + 1/6 f^{(3)}(*x*_{0})(*x* - *x*_{0})^{3}
for f(*x*_{0} + *h*) where *h* = *x* - *x*_{0}.

Answer: f(*x*_{0} + *h*) = f(*x*_{0}) + f^{(1)}(*x*_{0})*h* + 1/2 f^{(2)}(*x*_{0})*h*^{2} + 1/6 f^{(3)}(*x*_{0})*h*^{3}
# Example 2

Find the 1st-order Taylor series approximation to sin(1.1) given the Taylor
series of sin(*x*) around *x* = 1.

Answer: sin(1.1) ≈ sin(1) + cos(1) ⋅ 0.1 = 0.89550.

# Example 3

Find the 2nd-order Taylor series approximation to sin(1.1) given the Taylor
series of sin(*x*) around *x* = 1.

Answer: sin(1.1) ≈ sin(1) + cos(1) ⋅ 0.1 - 0.5 sin(1) ⋅ 0.1^{2} = 0.89129.

# Questions

# Question 1

Approximate sin(1.1) using a first-order Taylor series expanded around *x* = 1.
What is the relative error of this answer?

Answer: 0.8955012154 and 0.0048

# Question 2

What is the bound on the error of using a first-order Taylor series expanded around
*x* = 0.5 for the function f(*x*) = *e*^{x} when estimating *e*^{x} for
*x* = 0 and *x* = 1?

Answer: [0.125, 0.125*e*^{0.5}] ≈ [0.125, 0.20609] and
[0.125*e*^{0.5}, 0.125*e*] ≈ [0.20609, 0.33979].

(The actual errors are 0.17564 and 0.24520, both of which fall within the bounds found above.)

# Matlab

Taylor series are not applicable to Matlab. They must be computed
manually and programmed.

# Maple

Finding Taylor series in Maple is easier than opening your Calculus text.
The first argument is the function, the second is the variable and the point
around which the Taylor exapnsion is to take place, and the third is the
desired order.

series( sin(x), x = 0, 10 );
series( cos(x), x = 1, 5 );