The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus - i n i n Left F Right-...

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The Fundamental Theorem of Calculus (Part2) dx x f b a ) ( = F(b) – F(a) where F’(x) = f (x) Proof: In the beginning, we established that integration is the sum of the area under the curve. We used Reimann sums to define integrals: dx x f b a ) ( = n lim i n i i x x f = 1 ) ( = n lim i i i x x F = ) ( ' ASIDE Since i x is an unknown coordinate between Left i and Right i , we could name it ‘c’ that satisfies the MVT for F(x) in the interval Δ i x . = n lim i n i x c F = 1 ) ( ' = i i i i n i n x x Left F Right F - = →∞ ) ( ) ( lim 1 = ) ( ) ( lim 1 i
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Unformatted text preview: i n i n Left F Right- = = ) ( ) ( ... ) ( ) ( ) ( ) ( lim 2 2 1 1 1 n n n i n Left F Right F Left F Right F Left F Right F-+ +-+- = = ) ( ) ( ... ) ( ) ( ) ( ) ( lim 2 2 1 n n Left F b F Left F Right F a F Right F-+ +-+- = ) ( ) ( b F a F +-= ) ( ) ( a F b F-The limit drops because the limit of the constant is the constant F(R n ) = F(b) a = Left 1 F(L 1 ) = F(a) b = Right n y = F(x)...
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This note was uploaded on 02/04/2011 for the course MATHEMATIC 33 taught by Professor Qian during the Winter '99 term at Hong Kong Shue Yan.

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