5.3 Analyzing Graphs

(This topic is also in Section 5.3 in Applied Calculus or Section 12.3 of Finite Mathematics and Applied Calculus)

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It is easy enough to use graphing technology to draw a graph, but we need to use calculus to understand some of the features we are seeing and also to help us decide where to look. The most interesting features of a graph are the following:

Features of a Graph

    1. The x- and y-intercepts: If y = f(x), find the x-intercept(s) by setting y = 0 and solving for x; find the y-intercept by setting x = 0 and solving for y.

    2. Relative and absolute extrema: Use the techniques of Section 5.1 to locate the relative and absolute extrema.

    3. Points of inflection: Candidates for points of inflection are given by setting the second derivative equal to zero and solving for x.

    4. Behavior near points where the function is not defined: If f(x) is not defined at x = a, consider \displaystyle \lim_{x\to a^-}f(x) and \displaystyle \lim_{x\to a^+}f(x) to see how the graph of f behaves as x approaches a.

    5. Behavior at infinity: Consider \displaystyle \lim_{x\to -\infty}f(x) and \displaystyle \lim_{x\to +\infty}f(x) if appropriate, to see how the graph of f behaves far to the left and right.

Here is an illustration of a graph showing these features:

Note  It is sometimes difficult or impossible to solve all of the equations that come up in Steps 1, 2, and 3 of the above analysis. As a consequence, we might not be able to say exactly where the x-intercept, extrema, or points of inflection are. When this happens, we can use graphing technology to assist us in determining accurate numerical approximations.

Here is an exercise on recognizing these features on a graph:

In the next exercise we are given the equation of the graph only:

Now try some of the exercises in Section 5.3 in Applied Calculus or Section 12.3 in Finite Mathematics and Applied Calculus.

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Last Updated: April, 2008
Copyright © 2007 Stefan Waner