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Unformatted text preview: Chapter 5 Graphing and Optimization Section 2 Second Derivative and Graphs 2 Barnett/Ziegler/Byleen Business Calculus 12e Objectives for Section 5.2 Second Derivatives and Graphs ■ The student will be able to use concavity as a graphing tool. ■ The student will be able to find inflection points. ■ The student will be able to analyze graphs and do curve sketching. ■ The student will be able to find the point of diminishing returns. 3 Barnett/Ziegler/Byleen Business Calculus 12e Concavity The term concave upward (or simply concave up ) is used to describe a portion of a graph that opens upward. Concave down(ward) is used to describe a portion of a graph that opens downward. Concave down Concave up 4 Barnett/Ziegler/Byleen Business Calculus 12e Definition of Concavity A graph is concave up on the interval ( a , b ) if any secant connecting two points on the graph in that interval lies above the graph. It is concave down on ( a , b ) if all secants lie below the graph. up down 5 Barnett/Ziegler/Byleen Business Calculus 12e Concavity Tests Theorem. The graph of a function f is concave upward on the interval ( a , b ) if f ′ ( x ) is increasing on ( a , b ), and is concave downward on the interval ( a , b ) if f ′ ( x ) is decreasing on ( a , b ). For y = f ( x ), the second derivative of f , provided it exists, is the derivative of the first derivative: Theorem. The graph of a function f is concave upward on the interval ( a , b ) if f ′′ ( x ) is positive on ( a , b ), and is concave downward on the interval ( a , b ) if f ′′ ( x ) is negative on ( a , b ). ′′ y = ′′ f ( x ) = d 2 f dx 2 ( x ) 6 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 Find the intervals where the graph of f ( x ) = x 3 + 24 x 2 + 15 x 12. is concave up or concave down. 7 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 Find the intervals where the graph of f ( x ) = x 3 + 24 x 2 + 15 x – 12. is concave up or concave down. f ′ ( x ) = 3 x 2 + 48 x + 15 f ′′ ( x ) = 6 x + 48 f ′′ ( x ) is positive when 6 x + 48 > 0 or x > –8, so it is concave up on the region (–8, ∞ ). f ′′ ( x ) is negative when 6 x + 48 < 0 or x < –8, so it is concave down on the region (– ∞ , –8). 8 3 Functions and Concavity Barnett/Ziegler/Byleen Business Calculus 12e 1. Y = e x 2. Y = ln x 3. Y = x 3 9 Barnett/Ziegler/Byleen Business Calculus 12e Inflection Points An inflection point is a point on the graph where the concavity changes from upward to downward or downward to upward....
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This note was uploaded on 03/31/2011 for the course NREM 203 taught by Professor Bowen during the Fall '10 term at University of Hawaii, Manoa.
 Fall '10
 Bowen

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