The Shape of a Graph

The Shape of a Graph - The Shape of a Graph, Part I In the...

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The Shape of a Graph, Part I In the previous section we saw how to use the derivative to determine the absolute minimum and maximum values of a function. However, there is a lot more information about a graph that can be determined from the first derivative of a function. We will start looking at that information in this section. The main idea we’ll be looking at in this section we will be identifying all the relative extrema of a function. Let’s start this section off by revisiting a familiar topic from the previous chapter. Let’s suppose that we have a function, . We know from our work in the previous chapter that the first derivative, , is the rate of change of the function. We used this idea to identify where a function was increasing, decreasing or not changing. Before reviewing this idea let’s first write down the mathematical definition of increasing and decreasing. We all know what the graph of an increasing/decreasing function looks like but sometimes it is nice to have a mathematical definition as well. Here it is. Definition 1. Given any and from an interval I with if then is increasing on I . 2. Given any and from an interval I with if then is decreasing on I . This definition will actually be used in the proof of the next fact in this section. Now, recall that in the previous chapter we constantly used the idea that if the derivative of a function was positive at a point then the function was increasing at that point and if the derivative was negative at a point then the function was decreasing at that point. We also used the fact that if the derivative of a function was zero at a point then the function was not changing at that point. We used these ideas to identify the intervals in which a function is increasing and decreasing. The following fact summarizes up what we were doing in the previous chapter.
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Fact 1. If for every x on some interval I , then is increasing on the interval. 2. If for every x on some interval I , then is decreasing on the interval. 3. If for every x on some interval I , then is constant on the interval. The proof of this fact is in the Proofs From Derivative Applications section of the Extras chapter. Let’s take a look at an example. This example has two purposes. First, it will remind us of the increasing/decreasing type of problems that we were doing in the previous chapter. Secondly, and maybe more importantly, it will now incorporate critical points into the solution. We didn’t know about critical points in the previous chapter, but if you go back and look at those examples, the first step in almost every increasing/decreasing problem is to find the critical points of the function. Example 1
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This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.

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The Shape of a Graph - The Shape of a Graph, Part I In the...

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