In the previous chapter we focused almost exclusively on the
computation of derivatives. In this
chapter will focus on applications of derivatives. It is important to always remember that we
didn’t spend a whole chapter talking about computing derivatives just to be
talking about them. There are many very
important applications to derivatives.
The two main applications that we’ll be looking at in this
chapter are using derivatives to determine information about graphs of
functions and optimization problems.
These will not be the only applications however. We will be revisiting limits and taking a
look at an application of derivatives that will allow us to compute limits that
we haven’t been able to compute previously.
We will also see how derivatives can be used to estimate solutions to
equations.
Here is a listing of the topics in this section.
Rates of Change The point of this section is to remind us of
the application/interpretation of derivatives that we were dealing with in the
previous chapter. Namely, rates of
change.
Critical Points In this section we will define critical
points. Critical points will show up in
many of the sections in this chapter so it will be important to understand
them.
Minimum and Maximum Values In this section we will take a look at some of
the basic definitions and facts involving minimum and maximum values of
functions.
Finding Absolute Extrema Here is the first application of derivatives
that we’ll look at in this chapter. We
will be determining the largest and smallest value of a function on an
interval.
The Shape of a Graph, Part I We will start looking at the information that
the first derivatives can tell us about the graph of a function. We will be looking at increasing/decreasing
functions as well as the First Derivative Test.
The Shape of a Graph, Part II In this section we will look at the
information about the graph of a function that the second derivatives can tell
us. We will look at inflection points,
concavity, and the Second Derivative Test.
The Mean Value Theorem Here we will take a look at the Mean Value
Theorem.
Optimization Problems This is the second major application of
derivatives in this chapter. In this
section we will look at optimizing a function, possibly subject to some
constraint.
More Optimization Problems Here are even more optimization problems.
L’Hospital’s Rule and Indeterminate Forms This isn’t the first time that we’ve looked at
indeterminate forms. In this section we
will take a look at L’Hospital’s Rule.
This rule will allow us to compute some limits that we couldn’t do until
this section.
Linear Approximations Here we will use derivatives to compute a
linear approximation to a function. As
we will see however, we’ve actually already done this.
Differentials We will look at differentials in this section
as well as an application for them.
Newton’s Method With this application of derivatives we’ll see
how to approximate solutions to an equation.
Business Applications Here we will take a quick look at some applications
of derivatives to the business field.