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Unformatted text preview: c circlecopyrt Kendra Kilmer March 8, 2009 Section 5.2 - Second Derivative and Graphs Definition: For y = f ( x ) , the second derivative of f , provided that it exists, is Other notations for the second derivative are: Example 1: Find the first and second derivative of f ( x ) = 2 x 3 − 14 x 2 + 3 x − 16. Example 2: Discuss the difference in shapes of the graphs of f ( x ) = x 2 and g ( x ) = √ x on [ , ∞ ) Definition: The graph of a function f is concave upward on the interval ( a , b ) if on ( a , b ) and is concave downward on the interval ( a , b ) if on ( a , b ) . 7 c circlecopyrt Kendra Kilmer March 8, 2009 Example 3: Given f ( x ) = − x 3 + 2 x 2 − 3 x + 9 determine the intervals where f ′ ( x ) is increasing and decreasing. Test for Concavity: For a function whose second derivative exists on an open interval ( a , b ) : 1. If f ′′ ( x ) > 0 ( ) for all x on ( a , b ) , then f ( x ) is on ( a , b ) ....
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This note was uploaded on 03/27/2012 for the course MATH 142 taught by Professor Drost during the Fall '08 term at Texas A&M.
- Fall '08