# 10.14 - Hand out practice exam Paradox Write x2 = x x x(x...

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Hand out practice exam. Paradox: Write x 2 = x + x + … + x ( x added to itself x times). Differentiating both sides, we get 2 x = 1 + 1 + … + 1 (1 added to itself x times), or 2 x = x . What’s going on? ..?. . The equation x 2 = x + x + … + x is valid only when x is a whole number. But the set of whole numbers does not contain any open intervals, so there exists no interval I on which differentiating-both- sides-of-the-equation can be applied. Section 2.5: The chain rule Setting of chain rule: Suppose ( f ° g )( a ) = f ( g ( a )) = f ( b ) = c . We want to compute ( f ° g ) ( a ). Chain Rule: ( f ° g ) ( a ) = f ( g ( a )) g ( a ); that is, If f ( g ( a )) exists and g ( a ) exists, then ( f ° g ) ( a ) exists and

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( f ° g ) ( a ) = f ( g ( a )) g ( a ). Equivalently: Assume g is differentiable at x 0 and f is differentiable at y 0 = g ( x 0 ). Then f ° g is differentiable at x 0 and ( f ° g ) ( x 0 ) = f ( y 0 ) g ( x 0 ). (Here x 0 is what we called a above, and y 0 is g ( a ).) With Leibniz notation, we write dz / dx = ( dz / dy ) ( dy / dx ) , where y = g ( x ), z = f ( y ) = f ( g ( x )). More precisely, the Chain Rule says ( dz / dx at x 0 ) equals ( dz / dy at y 0 ) times ( dy / dx at x 0 ), where y 0 = g ( x 0 ) and z = f ( y
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## This note was uploaded on 02/13/2012 for the course MATH 141 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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10.14 - Hand out practice exam Paradox Write x2 = x x x(x...

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