Explanation Given information:- The given derivative of y = ( 2 x − x 2 ) 1 / 3 . Calculations:- The given derivative of y = ( 2 x − x 2 ) 1 / 3 . To find the transition point, interval of increase and decrease, concavity, and asymptotic behavior, we will take the first and second derivative of the given equation as shown below: y = ( 2 x − x 2 ) 1 / 3 [The given derivative of the equation] y ' = d d x ( 2 x − x 2 ) 1 / 3 [Differentiate with respect to x ] y ' = 1 3 ( 2 x − x 2 ) 1 3 − 1 ⋅ d d x ( 2 x − x 2 ) [Apply power rule [ u ( x ) n ] ' = n ⋅ u ( x ) n − 1 ⋅ u ' ( x ) ] y ' = 2 ⋅ d d x [ x ] − d d x [ x 2 ] 3 ( 2 x − x 2 ) 2 3 [ ∴ , d d x [ x ] = 1 , d d x [ x 2 ] = 2 x ] y ' = 2 − 2 x 3 ( 2 x − x 2 ) 2 3 Find the second derivative by using first derivative are as shown below: y ' = 2 − 2 x 3 ( 2 x − x 2 ) 2 3 [Find the second derivative] y ' ' = d d x [ 2 − 2 x 3 ( 2 x − x 2 ) 2 3 ] [Differentiate with respect to x ] y ' ' = 1 3 ⋅ d d x [ 2 − 2 x 3 ( 2 x − x 2 ) 2 3 ] [Apply the quotient rule [ u ( x ) v ( x ) ] ' = u ' ( x ) ⋅ v ( x ) − u ( x ) ⋅ v ' ( x ) v ( x ) 2 ] y ' ' = d d x [ 2 − 2 x ] ⋅ ( 2 x − x 2 ) 2 3 − ( 2 − 2 x ) ⋅ d d x [ ( 2 x − x 2 ) 2 3 ] ( ( 2 x − x 2 ) 2 3 ) 2 3 y " = ( − 2 ⋅ d d x [ x ] + d d x [ 2 ] ) ( 2 x − x 2 ) 2 3 − 2 3 ( 2 x − x 2 ) 2 3 − 1 ⋅ d d x [ 2 x − x 2 ] ⋅ ( 2 − 2 x ) 3 ( 2 x − x 2 ) 4 3 y " = ( 0 ⋅ − 2 ⋅ 1 ) ( 2 x − x 2 )

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 4 Solutions

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Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.1 - Prob. 13E Ch. 4.1 - Prob. 14E Ch. 4.1 - Prob. 15E Ch. 4.1 - Prob. 16E Ch. 4.1 - Prob. 17E Ch. 4.1 - Prob. 18E Ch. 4.1 - Prob. 19E Ch. 4.1 - Prob. 20E Ch. 4.1 - Prob. 21E Ch. 4.1 - Prob. 22E Ch. 4.1 - Prob. 23E Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E Ch. 4.1 - Prob. 30E Ch. 4.1 - Prob. 31E Ch. 4.1 - Prob. 32E Ch. 4.1 - Prob. 33E Ch. 4.1 - Prob. 34E Ch. 4.1 - Prob. 35E Ch. 4.1 - Prob. 36E Ch. 4.1 - Prob. 37E Ch. 4.1 - Prob. 38E Ch. 4.1 - Prob. 39E Ch. 4.1 - Prob. 40E Ch. 4.1 - Prob. 41E Ch. 4.1 - Prob. 42E Ch. 4.1 - Prob. 43E Ch. 4.1 - Prob. 44E Ch. 4.1 - Prob. 45E Ch. 4.1 - Prob. 46E Ch. 4.1 - Prob. 47E Ch. 4.1 - Prob. 48E Ch. 4.1 - Prob. 49E Ch. 4.1 - Prob. 50E Ch. 4.1 - Prob. 51E Ch. 4.1 - Prob. 52E Ch. 4.1 - Prob. 53E Ch. 4.1 - Prob. 54E Ch. 4.1 - Prob. 55E Ch. 4.1 - Prob. 56E Ch. 4.1 - 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