lecture25a - f (1) = 2, make a possible sketch of f ( x )....

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Lecture 25 (Chapter 4, Section 24) Derivatives and the Shape of a Graph, Part II Recall the following: Increasing/Decreasing Test If f 0 ( x ) > 0 on an interval, then If f 0 ( x ) < 0 on an interval, then Test for Concavity If f 00 ( x ) > 0 for all x on an interval, then the graph of f If f 00 ( x ) < 0 for all x on an interval, then the graph of f
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2 Derivatives and the Shape of a Graph Signs of f 0 Properties of the Shape of the and f 00 graph of f graph of f f 0 ( x ) > 0 f 00 ( x ) > 0 f 0 ( x ) > 0 f 00 ( x ) < 0 f 0 ( x ) < 0 f 00 ( x ) > 0 f 0 ( x ) < 0 f 00 ( x ) < 0
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3 ex. Determine when the graph of f ( x ) = x 4 ¡ 4 3 x 3 is concave up and concave down, and flnd any in±ection points. Make a rough sketch of f ( x ). 6 - ? ± f ¡ 2 3 ¢ = ¡ 16 81
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4 ex. Find all extrema and in±ection points of f ( x ) = e ¡ x 2 . Sketch the graph of f ( x ). 6 - ? ±
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5 ex. Find all relative extrema and in±ection points of the graph of f ( x ) = 2 5 x 5 = 3 ¡ x 2 = 3 . f 00 ( x ) = 4 x + 2 9 x 4 3 .
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6 Sketch the graph of f ( x ) = 2 5 x 5 = 3 ¡ x 2 = 3 . Note that f ( ¡ 1 2 ) … ¡ 0 : 76. 6 - ? ±
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7 ex. Suppose that f is continuous on ( ¡1 ; 1 ). Given the graph of f 0 ( x ), flnd the following: 1. Intervals on which f is increasing/decreasing 2. Local extrema 3. Intervals on which f is concave up/down 4. Points of in±ection
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8 If f ( ¡ 3) = 0, f ( ¡ 2) = 2, f ( ¡ 1) = 3, f (0) = 1 and
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Unformatted text preview: f (1) = 2, make a possible sketch of f ( x ). 6-? 9 We have a useful application of derivatives and the shape of a graph: Second Derivative Test Suppose that f 00 ( x ) is continuous near c . I. If f ( c ) = 0 and f 00 ( c ) then f has a local at c . II. If f ( c ) = 0 and f 00 ( c ) then f has a local at c . ex. Use the Second Derivative Test if possible to nd the local (relative) extrema of f ( x ) = 5 x 3 3 x 5 . 10 Try These! 1) Find the relative extrema and inection points of each of the following functions. Sketch a graph of the function. a) f ( x ) = x 4 2 x 3 + 2 x 1 = ( x 1) 3 ( x + 1) b) f ( x ) = 3 x 2 = 3 + 2 x 2) Use the Second Derivative Test to nd the local maximum and minimum values of f ( x ) = x 4 4 x 3 + 4 x 2 . Conrm by using the First Derivative Test....
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lecture25a - f (1) = 2, make a possible sketch of f ( x )....

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