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Unformatted text preview: 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 ( x ) using the limit definition. Sketch f ( x ) and f ( x ), and compare the graphs. Solution. Recall that f ( 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. Lets work it out! f ( x ) = lim h Slope of the line joining ( x,f ( x )) and ( x + h,f ( x + h )) . = lim h f ( x + h ) f ( x ) h = lim h ( x + h ) 3 3( x + h ) 2 + 2( x + h ) ( x 3 3 x 2 + 2 x ) h = lim h 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 3 x 2 h + 3 xh 2 + h 3 6 xh h 2 + 2 h h = lim h 3 x 2 6 x + 2 + 3 xh + h = 3 x 2 6 x + 2...
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
 Fall '09
 BenedictGross
 Derivative

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