Calculus For Dummies (For Dummies (Math & Science)) by Mark Ryan

Author:Mark Ryan [Ryan, Mark]
Language: eng
Format: azw3
ISBN: 9781119297437
Publisher: Wiley
Published: 2016-05-17T16:00:00+00:00

Resuming our trip, after , f is concave down till the inflection point at about — this corresponds to the decreasing section of from to its min at (interval D in Figure 11-11). Finally, f is concave up the rest of the way, which corresponds to the increasing section of beginning at (intervals E and F in the figure).

Well, that pretty much brings you to the end of the road. Going back and forth between the graphs of a function and its derivative can be very trying at first. If your head starts to spin, take a break and come back to this stuff later.

If I haven’t already succeeded in deriving you crazy — aren’t these calculus puns fantastic? — perhaps this final point will do the trick. Look again at the graph of the derivative, , in Figure 11-11 and also at the sign graph for in Figure 11-9. That sign graph, because it’s a second derivative sign graph, bears exactly (well, almost exactly) the same relationship to the graph of as a first derivative sign graph bears to the graph of a regular function. In other words, negative intervals on the sign graph in Figure 11-9 (to the left of and between 0 and ) show you where the graph of is decreasing; positive intervals on the sign graph (between and 0 and to the right of ) show you where is increasing. And points where the signs switch from positive to negative or vice versa (at , 0, and ) show you where has local extrema. Clear as mud, right?

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