The Precise Limit Denition
The Limit Denition
We are ready to make precise what we mean the term limit. So far we have
worked with an intuitive understanding of the term:
DEFINITION 5.1 (Informal Denition of Limit). We write lim g( x ) = L and say that th
Still More on Continuity
Continuity on Intervals
As you might expect, a function is continuous on an interval if it is continuous at
every point in the interval. Sounds simple, right? But what about at the endpoints
when the interval is closed? Take a loo
Introduction to Derivatives
5-Minute Review: Instantaneous Rates and Tangent Slope
Recall the analogy that we developed earlier. First we saw that the secant slope of
the line through the two points ( a, f ( a) and ( x, f ( x ) on a curve f was given by
t
More Related Rates
We have seen that many related rate problems involve implicit derivatives of familiar geometric relations. Before reading the examples below, try nding the
derivatives with respect to time t of these relations. The answers are at the en
The Mean Value Theorem
Introduction
Today we discuss one of the most important theorems in calculusthe MVT. It
says something about the slope of a function on a closed interval based on the
values of the function at the two endpoints of the interval. It r
Optimization and Calculus
The IVT: An Important Property of Continuous Functions
While we know continuous functions are nice because they make evaluating limits
easy, we saw earlier in the term that they posses another nice property. On an
intuitive level
More on Continuity
Quick Review
Continuity Checklist. A function f is continuous at a if the following three conditions hold:
1. f ( a) is dened (i.e., a is in the domain of f ).
2. lim f ( x ) exists.
x!a
3. lim f ( x ) = f ( a).
x!a
Remember:
A polynom
Continuity in More Detail
5-Minute Review: Continuity
We have worked off and on with continuous functions. Recall
DEFINITION 8.1 (Continuity at a Point). A function f ( x ) is continuous at a point a if lim f ( x ) =
f ( a). If f is not continuous at a, t
Introduction
Why Calculus?
Why are you studying calculus? For most of you the answer is because it is required for your major. Several disciplines require or suggest calculus, among them
biology, physics, chemistry, architecture, economics, geology, envir
m ath 130
2 .4
day 3 : intro to limits
9
Limits
What we need is some way to make this notion of approaching that we used in
Example 2.1 more precise. In a few days we will give a technical denition. But for
the moment, we will use the following idea.
DEFI
day 4 : working with limits
m ath 130
4 .0
5-Minute Review: Rational Functions
DEFINITION. A rational function1 is a function of the form
y = r(x) =
Here the term rational means ratio
as in the ratio of two polynomials.
1
p( x )
,
q( x )
where p( x ) and
Innite Limits
1-Minute Review: Large Numbers
It will be helpful to remember a couple of simple things about fractions or ratios. Lets use 0+ to indicate a small positive number and 0 to indicate a smallmagnitude negative number. Then
positive number
0+
=
Concavity
Weve seen how knowing where a function is increasing and decreasing gives a
us a good sense of the shape of its graph. We can rene that sense of shape by
determining which way the function bends. This bending is called the concavity
of the funct