Lecture 6 - CC2053: Calculus 6 Introduction to Calculus and...

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1 CC2053: Calculus 6 Introduction to Calculus and Linear Algebra 2011/2012 Semester 1 CALCULUS: Applications of Differentiation
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2 CC2053: Calculus 6 Increasing and Decreasing Functions, Concavity
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3 CC2053: Calculus 6 Increasing and Decreasing Functions Sign properties on an interval ( a, b ) If f is continuous on ( a , b ) and f ( x ) 0 for all x in ( a , b ) , then either f ( x ) > 0 for all x in ( a , b ) or f ( x ) < 0 for all x in ( a , b ).
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4 CC2053: Calculus 6 Increasing and Decreasing Functions A function f ( x ) is increasing on an interval a < x < b if f ( x 2 ) > f ( x 1 ) whenever x 2 > x 1 for x 1 , x 2 in the interval. A function is decreasing on a < x < b if f ( x 2 ) < f ( x 1 ) whenever x 2 > x 1 for, x 1 , x 2 in the interval.
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5 CC2053: Calculus 6 Increasing and Decreasing Functions - Derivative Criteria for Increasing and Decreasing Functions f ( x ) is increasing on an interval where f ( x ) > 0 f ( x ) is decreasing on an interval where f ( x ) < 0
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6 CC2053: Calculus 6 Increasing and Decreasing Functions Example 1 Find the intervals of increase and decrease for the function 7 12 3 2 ) ( 2 3 - - + = x x x x f
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7 CC2053: Calculus 6 Increasing and Decreasing Functions Example 1 (solution) 7 12 3 2 ) ( 2 3 - - + = x x x x f or 0 ) ( = x f ) 1 )( 2 ( 6 ) 2 ( 6 12 6 6 ) ( 2 2 - + = - + = - + = x x x x x x x f 2 - = x 1 = x interval Test number conclusion x < –2 - 3 + increasing –2 < x < 1 0 decreasing x > 1 2 + increasing sign of f ( x ) at
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8 CC2053: Calculus 6 Increasing and Decreasing Functions Example 1 (solution) Arrow diagram 7 12 3 2 ) ( 2 3 - - + = x x x x f The graph of -2 1 x The function is increasing for x < - 2 or x > 1 The function is decreasing for - 2 < x < 1
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9 CC2053: Calculus 6 Increasing and Decreasing Functions Example 2 Find the intervals of increase and decrease for the function 2 ) ( 2 - = x x x f
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10 CC2053: Calculus 6 Increasing and Decreasing Functions Example 2 (solution) 2 ) ( 2 - = x x x f 2 2 2 ) 2 ( ) 4 ( ) 2 ( 2 ) 2 ( ) ( - - = - - - = x x x x x x x x f 0 ) ( = x f is discontinuous at x = 2 at x = 0 or x = 4 interval Test number conclusion 0 < x < 2 1 decreasing 2 < x < 4 3 decreasing x > 4 5 + increasing sign of f ( x ) x < 0 - 1 + increasing ) ( x f
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11 CC2053: Calculus 6 Increasing and Decreasing Functions Example 2 (solution) Arrow diagram The graph of 2 ) ( 2 - = x x x f x 0 4 2 The function is increasing for x < 0 or x > 4 The function is decreasing for 0 < x < 2 or 2 < x < 4
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12 CC2053: Calculus 6 Increasing and Decreasing Functions Relative Extremum • The graph of the function f ( x ) is said to have a relative maximum at x = c if f ( c ) f ( x ) for all x in the interval a < x < b containing c • The graph of the function f ( x ) is said to have a relative minimum at x = c if f ( c ) f ( x ) for all x in the interval a < x < b containing c • The relative maxima and minima of f are called its relative extremum • A point on a graph where a relative extremum occurs is also called a turning point
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Lecture 6 - CC2053: Calculus 6 Introduction to Calculus and...

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