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Figure 5 illustrates the relationship between the
concavity of the graph of y = cos x and the graph of its second derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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The table on the next slide shows how to check if the points
where f ″ = 0 are points of inflection. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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The information from the table above would normally be displayed
in a sign chart, which is constructed as follows: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Just as a critical point occurs where f ′ does not exist, so a point
of inflection may exist at a point where f ″ does not exist, as illustrated in Figure 7. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Example, Page 243 Determine the intervals on which the function is concave up or down and find the points of inflection. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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There is a distinct relationship between
the graphs of the parent function, its first derivative, and its second derivative. This relationship may be seen in Figure 8. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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As illustrated in Figure 9, the sign of the second derivative at a
critical point tells us whether the critical point is a local minimum or a local maximum. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Example, Page 243 20. Through her website, Leticia has been selling bean bag chairs with monthly sales as recorded below. In a report to her investors, she states, “Sales reached a point of inflection when I started using pay-per-click advertising.” In which month did that occur? Explain. Month 1 2 3 4 5 6 7 8 Sales 20 30 35 38 44 60 90 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Our observation in Figure 9 is formalized in Theorem 3.
Theorem 3 gives us a second means of justifying that a minimum or maximum occurs at a critical point. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Figure 10 illustrates using the Second Derivative Test to find a
local maximum. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Example, Page 243 Find the critical points of f (x) and use the Second Derivative Test to determine whether each corresponds to a local minimum or maximum. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Example, Page 243 Find the critical points of f (x) and use the Second Derivative Test to determine whether each corresponds to a local minimum or maximum. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Use the second derivative test to find the local extrema of
f (x) = x5 – 5x4. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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The signs of the first and second derivative provide us specific
information about the behavior of the parent function that is summarized in the following table: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Example, Page 243 52. Water is pumped into a sphere at a constant rate. Let h (t) be the water level at time t. Sketch the graph of h (t). Where does the point of inflection occur? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Example, Page 243 Sketch the graph of a function satisfying the given condition. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Homework Homework Assignment #25 Read Section 4.5
Page 243, Exercises: 1 – 57 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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