CalcFunTheorem23 - PART I LAYMAN’S STATEMENT g(x) =...

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THE FUNDAMENTAL THEOREM OF CALCULUS PART II LAYMAN’S STATEMENT 1 Suppose f(x) is the derivative (rate of change) of F(x), and f(x) is above the x-axis from the interval a to b . 2 3 f(x) is the slope (the derivative) of F(x), and since f(x) is always positive, F(x) is always increasing from a to b . Since the slope is the rate of change, the integral of f(x) from a to b is the sum of the rate of change of F(x) in that interval (how much F(x) is changing). Therefore, the integral of f(x) is equal to the net change of F(x) from a to b. ) ( ) ( ) ( a F b F dx x f b a - =
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THE FUNDAMENTAL THEOREM OF CALCULUS
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Unformatted text preview: PART I LAYMAN’S STATEMENT g(x) = integral from a to b of f(x) where g’(x) = f(x). g(t) = ∫ x a dt t f ) ( Take the derivative of both sides: dx d g(t) = ∫ x a dt t f dx d ) ( Since integration and differentiation are inverse operations, they cancel each other out. g’(t) = f(t)dt So, g’(t) = f(t)dt and g(t) = ∫ x a dt t f ) ( , thus demonstrating the inverse relationship between integration and differentiation. This is the first part of the Fundamental Theorem of Calculus....
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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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CalcFunTheorem23 - PART I LAYMAN’S STATEMENT g(x) =...

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