Math1011_Prelim3B - Math1011 Spring 1999 III-B1 The...

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Math1011 Spring 1999 III-B1 The function f is defined by the first of the following three equations. For your convenience, we also give f’(x) and f’’(x). 2 2 2 2 2 2 4 2 ( 3) 1 1 (1 ) (1 ) ( ) '( ) ''( ) x x x x x x x f x f x f x + + + = = = Find the intervals on which ( ) is increasing or decreasing. f x Find the intervals on which f is concave up and concave down Find the local maximum and minimum values of ƒ. Find the inflection points of ƒ.
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Math1011 Spring 1999 B1 The function f is defined by the first of the following three equations. For your convenience, we also give f’(x) and f’’(x). 2 2 2 2 2 2 4 2 ( 3) 1 1 (1 ) (1 ) ( ) '( ) ''( ) x x x x x x x f x f x f x + + + = = = Find the intervals on which ( ) is increasing or decreasing. f x 2 2 2 2 1 (1 ) '( ) 0 when 1 - 0 or roots at x = -1 and x = 1 x x f x x + = = = -1 1 f’(x) - + - Increasing on (-1, 1) when f’(x) > 0 Decreasing on (- , -1) U (1, ) when f’(x) < 0 Find the intervals on which f is concave up and concave down 2 2 4 2 ( 3) 2 (1 ) ''( ) 0 when 0 and x 3 0 or 3 x x x f x x x + = = = = = ± - 3 0 3 f’’(x) - + - + Concave Up on (- 3, 0) U ( 3, ) when f’’(x) > 0 Concave Down on (- ,- 3) U (0, 3) when f’(x) < 0 Find the local maximum and minimum values of ƒ. Check the critical points (x = 1 and x = 1) Second Derivative Test f’’(-1) = +, so x=-1, y=-½ is a local minimum f’’(1) = -, so x=1, y = ½ is a local maximum Find the inflection points of ƒ. Check x = 3 and x = 0 and x = 3 f changes concavity at these points, so all are inflection points. Inflection Points at (- 3, - 3/4), (0, 0), and ( 3, 3/4) (Doc #011p.44.01t)
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Math1011 Fall 2007 Prelim B2 2 1 2 Let g be the function defined by the formula ( ) . Find the local extrema of g and its interval of concavity. x x x g x =
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Math1011 Fall 2007 Prelim B2 2 1 2 Let g be the function defined by the formula ( ) . Find the local extrema of g and its interval of concavity. x x x g x = 2 2 d d dx dx 2 2 2 2 2 2 2 2 2 2 2 (x-2) ( 1) ( 1) (x-2) (x-2) (x-2)(2 1) ( 1)(1) (2 5 2) ( 1) ( 3)( 1) 4 3 (x-2) (x-2) (x-2) (x-2) ( -2) ( 4 3) ( 4 3 Find g'(x): g'(x)= g'(x)= Find g''(x): ''( ) d dx x x x x x x x x x x x x x x x x x x x x g x + + + + = = = = 2 2 4 2 2 2 2 4 4 2 2 2 2 3 3 3 ) ( -2) ( -2) ( -2) (2 4) ( 4 3)2( -2) 2( -2)[( -2) ( 4 3)] ( -2) ( -2) 2[( -2) ( 4 3)] 2[( -4 4) ( 4 3)] 2 ( -2) ( -2) (x-2) ''( ) ''( ) Local Extrema: Critical Points: 1 and d dx x x x x x x x x x x x x x x x x x x x x x x g x g x x x + + + + + = = = = = = 3 3 2 (1-2) 2 (3-2) 3 Singular Points: None (x =2 isn't in the domain of g(x)) End Points: None Use 2nd derivative test to determine local min or max g''(1)= 2 1 is a local maximum g''(3)= 2 3 is a l x x = = − = = =
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