Taylor (Maclaurin) Approximations
Our goal in this section is to describe one way that calculators
(and computers) can use to compute complicated functions such as
sin, cosine and exponentials. Before doing so, however, we need to
Consider the problem of measuring the side length of a square
and then using the measured date to compute the area of the square.
For example, if one side is measured to be 2.74 inches, then the area
would be computed as (2.74)2
Limits of Functions
In Chapter 4 we described the limit of a sequence in terms of
approximations. We may describe limits of functions similarly.
Example 1. The Acme Geometry Company, manufacturers of
precision geometric gures, has received an o
The reader should note that all of the techniques demonstrated
so far are based on the Bounded Increasing Theorem, which requires
that the an be non-negative.
Any series with positive and negative terms can be written as
Positive Term Series
One very important goal in mathematics is computation. The
number , for example, has been computed to thousands of decimals. How is this done? It turns out that there are some remarkable
formulas that can be used to appro
Max, Min, Sup, Inf
We would like to begin by asking for the maximum of the function
f (x) = (sin x)/x. An approximate graph is indicated below. Looking
at the graph, it is clear that f (x) 1 for all x in the domain of f .
Furthermore, 1 is the s
When doing calculations using approximations, the error can increase. For example, 1.99 approximates 2 with error .01 and 2.98
approximates 3 with error .02. The sum 1.99 + 2.98 = 4.97 approximates 2 + 3 = 5 with error .03. Thi
Limits of Sequences
In this section we study limits of sequences. As a preliminary
denition, we might dene a sequence to be a function whose domain
consists of the set N natural numbers. Thus, when we refer to the
an = n/(n2 3)
Rates of Growth
Inequalities are very useful in comparing rates of growth of functions.
Definition 1. If f and g are two functions, we say that f dominates g if there is a value N such that
f (x) > g (x)
for all x > N .
(See Figure 1.)
In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1.
A thorough mastery of this section is essential as analysis is based on
Numbers, proof and all that jazz.
There is a fundamental dierence between mathematics and other
sciences. In most sciences, one does experiments to determine laws.
A law will remain a law, only so long as it is not contradicted
by experimental e