FTC2 - Math 1A Fall 2009 The Fundamental Theorem of...

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Unformatted text preview: Math 1A Fall 2009 The Fundamental Theorem of Calculus, Part II. If f is continuous on [a,b], then b a f ( x ) dx = F ( b )- F ( a ) (1) where F is any antiderivative of f, that is a function such that F=f Idea of the proof. We know that our function is continuous on [a,b] so then our function is integrable on [a,b]. By definition we can write b a f ( x ) dx = lim x n i =1 f ( x * i ) x where x = b- a n and x * i be any point on [ x i- 1 , x i ], and this limit exists. Okay so that is the left hand side of equation (1) above, what about the right hand side? How can we write F ( b )- F ( a ) as a limit of a summation to compare? Remember the telescoping series? F ( b )- F ( a ) = F ( x n )- F ( x ) = F ( x n )- F ( x n- 1 ) + F ( x n- 1 )- F ( x n- 2 ) + F ( x n- 2 ) + ...- F ( x 1 ) + F ( x 1 )- F ( x ) = n i =1 [ F ( x i )- F ( x i- 1 )] Note that adding those middle terms does not change anything because each term is immediately subtracted....
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FTC2 - Math 1A Fall 2009 The Fundamental Theorem of...

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