SZptFD-06

SZptFD-06 - 14 SPRING 2012 6. Mean Value Theorem....

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14 SPRING 2012 6. Mean Value Theorem. Clairaut’s Theorem Recall the Mean Value Theorem from Calculus: if f :[ a, b ] R is continuous on [ a, b ]and di f erentiable on ( a, b ), then there exists c between a and b such that f ( b ) f ( a )= f ° ( c )( b a ) . Disappointingly, we can have no such theorem for functions f : U R n R m with m> 1, even though it makes syntactic sense to write the expression f ( z ) f ( x )= f ° ( c )( z x )for such a function. Example 6.1. For instance, consider f : R R 2 de±ned by f ( t )=(co s t, sin t ), so that f ° ( t )=( sin t, cos t )forall t . There is certainly no c between 0 and π such that f ( π ) f (0) = f ° ( c )( π 0), as this would require ( 1 , 0) (1 , 0) = π ( sin c, cos c ) , and the left-hand vector has length 2, while the right-hand vector has length π , irregardless of the value of c . In order to extend the Mean Value Theorem to functions de±ned on subsets of R n ,w e have to ensure that “between” makes sense. A set S R n is convex if (1 λ ) x + λ y S whenever x , y S and 0 λ 1. Theorem 6.2 (Mean Value Theorem) . Let U R n be open and convex, and let f : U R be di f erentiable. For each x and z in U , there exists c on the line segment from x to z such that f (
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SZptFD-06 - 14 SPRING 2012 6. Mean Value Theorem....

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