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solutions_09_30

# solutions_09_30 - Math 1a The derivative function(Solutions...

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Math 1a: The derivative function (Solutions) September 30, 2009 1. Let f ( x ) = x ( x - 1)( x - 2) = x 3 - 3 x 2 + 2 x. Determine f 0 ( x ) using the limit definition. Sketch f ( x ) and f 0 ( x ), and compare the graphs. Solution. Recall that f 0 ( x ) is defined to be the slope of the tangent to the graph of f at the point ( x, f ( x )). We start by choosing a nearby point on the graph, say ( x + h, f ( x + h )), compute the slope of the secant joining ( x, f ( x )) and ( x + h, f ( x + h )) and take the limit as the nearby point approaches ( x, f ( x )), or equivalently as h approaches 0. Let’s work it out! f 0 ( x ) = lim h 0 Slope of the line joining ( x, f ( x )) and ( x + h, f ( x + h )) . = lim h 0 f ( x + h ) - f ( x ) h = lim h 0 ( x + h ) 3 - 3( x + h ) 2 + 2( x + h ) - ( x 3 - 3 x 2 + 2 x ) h = lim h 0 x 3 + 3 x 2 h + 3 xh 2 + h 3 - 3 x 2 - 6 xh - h 2 + 2 x + 2 h - x 3 + 3 x 2 - 2 x h = lim h 0 3 x 2 h + 3 xh 2 + h 3 - 6 xh - h 2 + 2 h h = lim h 0 3 x 2 - 6 x + 2 + 3 xh + h = 3 x 2 - 6 x + 2 . I will leave the sketch of f 0 ( x ) to you. 2. Let h ( x ) = | x - 1 | + | x + 1 | . On what intervals is h ( x ) continuous? What about differentiable? Sketch h ( x ) and its derivative.

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solutions_09_30 - Math 1a The derivative function(Solutions...

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