Calculus For Dummies by Mark Ryan

Author:Mark Ryan
Language: eng
Format: epub
ISBN: 9781118791332
Publisher: Wiley
Published: 2014-06-09T09:56:59+00:00

Now let’s take a second trip along f to consider its intervals of concavity and its inflection points. First, consider intervals A and B in Figure 11-11. The graph of f is concave down — which means the same thing as a decreasing slope — until it gets to the inflection point at about .

So, the graph of decreases until it bottoms out at about . These coordinates tell you that the inflection point at on f has a slope of Note that the inflection point on f at is the steepest point on that stretch of the function, but it has the smallest slope because its slope is a larger negative than the slope at any other nearby point.

Between and the next inflection point at , f is concave up, which means the same thing as an increasing slope. So the graph of increases from about to where it hits a local max at . See interval C in Figure 11-11. Let’s take a break from our trip for some more rules.

More rules:

A concave down interval on the graph of a function corresponds to a decreasing interval on the graph of its derivative (intervals A, B, and D in Figure 11-11). And a concave up interval on the function corresponds to an increasing interval on the derivative (intervals C, E, and F).

An inflection point on a function (except for a vertical inflection point where the derivative is undefined) corresponds to a local extremum on the graph of its derivative. An inflection point of minimum slope (in its neighborhood) corresponds to a local min on the derivative graph; an inflection point of maximum slope (in its neighborhood) corresponds to a local max on the derivative graph.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.