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# lecture24 - Lecture 24(Chapter 4 Section 24 Derivatives and...

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Lecture 24 (Chapter 4, Section 24) Derivatives and the Shape of a Graph, Part I 6 - ? ± Increasing/Decreasing Test If f 0 ( x ) > 0 on an interval, then If f 0 ( x ) < 0 on an interval, then

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2 ex. Find the intervals on which f ( x ) = x 4 ¡ 4 3 x 3 is increasing and decreasing.
3 ex. For which intervals is g ( x ) = x 2 ¡ 1 x 4 increasing and decreasing?

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4 Local Extrema and the ﬂrst derivative First Derivative Test: Suppose that c is a critical number of a continuous function f . 1. If f 0 changes from positive to negative at c , then 2. If f 0 changes from negative to positive at c , then 3. If f 0 does not change signs at c , then
5 ex. Find all local extrema of f ( x ) = x 2 3 ( x + 5).

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6 ex. Find the local maximum and minimum values of f ( x ) = sin x + cos x , 0 x 2
Concavity 6 - ? ± f ( x ) = x 3 Def. If the graph of f lies its tangent lines on an interval I , then it is concave on I . If the graph of

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lecture24 - Lecture 24(Chapter 4 Section 24 Derivatives and...

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