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lecture21

lecture21 - Lecture 21 1 Lecture 21 Part I(Sec 4.2 part II...

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Lecture 21 1 Lecture 21, Part I: (Sec. 4.2, part II) ex. Find the intervals on which f ( x ) = 3 x 5 - 5 x 3 is concave up and down, and each x -value at which the graph of f has an inflection point.

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Lecture 21 2 The Second Derivative Test Extrema and the Second Derivative Suppose continuous function f is differentiable and has a relative extreme value at x = c . Relative Minimum Relative Maximum
Lecture 21 3 The Second Derivative Test for Relative Extrema: For a function f , 1) Find f 0 ( x ) to locate the critical points of f . 2) For each critical point c so that f 0 ( c ) = 0, find f 00 ( c ). We can conclude: 1) If f 00 ( c ) > 0, 2) If f 00 ( c ) < 0, 3) If f 00 ( c ) = 0,

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Lecture 21 4 ex. Use the Second Derivative Test to find the relative extreme values of f ( x ) = 3 x 5 - 5 x 3 .
Lecture 21 5 Note the following about the Second Derivative Test: 1) 2) 3)

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Lecture 21 6 Lecture 21, part II(Sec. 4.3, part I) Asymptotes Vertical Asymptotes ex. Let f ( x ) = 1 x - 2 Def. The line x = a is a vertical asymptote(VA) of the graph of a function f if
Lecture 21 7 ex. Given f ( x ) = 2 - x x + 3 .

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