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Online Notes / Calculus I (Notes) / Applications of Derivatives

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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.

Online Notes / Calculus I (Notes) / Applications of Derivatives

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