Calculus All-In-One for Dummies (+ Chapter Quizzes Online) by Ryan Mark;

Author:Ryan, Mark; [Ryan, Mark]
Language: eng
Format: epub
Publisher: John Wiley & Sons, Incorporated
Published: 2023-03-21T00:00:00+00:00

Thus the concavity goes down, up, down, up. Because the second derivative is zero at –0.085 and 0.085, and because the concavity switches there, you can conclude that there are inflection points at those two x-values. But because both the first and second derivatives are undefined at , you have to check whether there’s a vertical tangent there. You can see that there is by just looking at the graph, but if you want to be rigorous about it, you figure the limit of the first derivative as x approaches zero. Since that equals infinity, you have a vertical tangent at , and thus there’s an inflection point there.

Now plug –0.085, 0, and 0.085 into q to get the y-values, and you’re done.

19 Absolute precision is not the point of this exercise, but your graph should look close to the graph of f shown here.

Note the following features of the graph of f. Your graph should show local mins at and and a local max at . And your graph should show inflection points at roughly and . Finally, your graph should show that f is concave up everywhere to the left of and everywhere to the right of . In between those points, f should be concave down.

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