Lecture 5.1 - Chapter 5 Graphing and Optimization Section 1...

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Chapter 5 Graphing and Optimization Section 1 First Derivative and Graphs
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2 Barnett/Ziegler/Byleen Business Calculus 12e Objectives for Section 5.1 First Derivative and Graphs The student will be able to identify increasing and decreasing functions, and local extrema. The student will be able to apply the first derivative test. The student will be able to apply the theory to applications in economics.
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3 Barnett/Ziegler/Byleen Business Calculus 12e Increasing and Decreasing Functions Theorem 1. (Increasing and decreasing functions) On the interval ( a , b ) f ’( x ) f ( x ) Graph of f + increasing rising decreasing falling
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4 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 Find the intervals where f ( x ) = x 2 + 6 x + 7 is rising and falling.
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5 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 Find the intervals where f ( x ) = x 2 + 6 x + 7 is rising and falling. Solution: From the previous table, the function will be rising when the derivative is positive. f ( x ) = 2 x + 6. 2 x + 6 > 0 when 2 x > -6, or x > –3. The graph is rising when x > –3. 2 x + 6 < 6 when x < –3, so the graph is falling when x < –3.
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6 Barnett/Ziegler/Byleen Business Calculus 12e f ( x ) - - - - - - 0 + + + + + + Example 1 (continued ) f ( x ) = x 2 + 6 x + 7, f ( x ) = 2 x +6 A sign chart is helpful: f ( x ) Decreasing –3 Increasing (– , –3) (–3, )
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7 Partition Numbers and Critical Values A partition number for the sign chart is a place where the derivative could change sign. This can only happen: - where f itself is not defined , not consistent with textbook - where f is not defined (critical value) - where f is zero (critical value) Definition. The values of
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Lecture 5.1 - Chapter 5 Graphing and Optimization Section 1...

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