Math1011_Prelim3B

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 22 2 24 2( 3 ) 1 1( 1 ) (1 ) () ' ' ' xx x fx f x f ++ + == = Find the intervals on which ( ) is increasing or decreasing. 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 22 2 24 2( 3 ) 1 1( 1 ) (1 ) () ' ' ' xx x fx f x f ++ + == = Find the intervals on which ( ) is increasing or decreasing. 2 2 1 (1 ) '( ) 0 when 1 - 0 or roots at x = -1 and x = 1 + = -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 3 ) 2 (1 ) ''( ) when 0 and x 3 0 or 3 + = = = ± - 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. xx x gx =
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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. xx x gx = 22 dd dx dx 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) 43 (x-2) (x-2) (x-2) (x-2) ( - 2 ) (4 3 ) 3 Find g'(x): g'(x)= g'(x)= Find g''(x): ''( ) d dx −−− −− − + − −+ == = = 2 2 4 2 2 44 2 2 33 3 )(- 2 ) (- 2 ) ( -2) (2 4) ( 4 3)2( -2) 2( -2)[( -2) ( 4 3)] 2 ) 2 ) 2 [ ( - 2 ) 3 ) ] 2 [ ( - 4 4 ) 3 ) ] 2 2 ) 2 ) ( x - 2 ) ''( ) ''( ) Local Extrema: Critical Points: 1 and −− − + − − + + + + = = 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 = =− ⇒ = =⇒ = ocal minimum Inflection Points: None (x = 2 isn't in domain of g(x)) Concavity may only change at breaks in the domain (x = 2) if 2, ''(2) 0 Concave Down if > 2, ''(2) 0 Concave Up g(x) is concave xg << >⇒ down on (- ,2) and concave up on (2, ) ∞∞ (Doc #011p.44.02t)
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Math1011
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This note was uploaded on 12/06/2010 for the course MATH 1110 taught by Professor Martin,c. during the Fall '06 term at Cornell.

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

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