CalcFunTheorem1 - 2. The definite integral from a to b of...

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Formal Statement If f is continuous on [ a, b ], then 1. If g(x) = ) ( ) ( x u a dt t f , then g’(x) = f( u(x) ) u’(x) 2. - = b a a F b F dx x f ) ( ) ( ) ( , where F is any antiderivative of f , that is, F’ = f . If the graph of the function f is continuous on the closed interval [a, b], then 1. If the definite integral from a to u(x) of f(t) equals g(x) , the derivative of g(x) equals f(x) . However, we still need to multiply by the chain rule factor, u’(x), which is the derivative of the upper bound u(x) . This reveals how derivative and integral are inverse operations that cancel each other out. However,
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Unformatted text preview: 2. The definite integral from a to b of f(x) is the net change in F , meaning it is F(b) – F(a) , where F’ = f . So, the net change in F , the antiderivative of f , is equal to the integral of f(x) from a to b . Example 1 h(x) = ∫-3 2 1 4 sinh x tdt , find h’(x) . dx d h(x) = ∫-3 2 1 4 sinh x tdt dx d h’(x) = ∫-3 2 1 4 sinh x tdt dx d h’(x) = 3x 2 sinh-1 4(x 3 ) Example 2 ∫ 5 2 5 dx x ∫ 5 2 5 dx x = 5 2 5 ln 1 | 5 x ∫ 5 2 5 dx x = 5 ln 1 (5 5 – 5 2 ) = 5 ln 1 (3125 – 25) = 5 ln 1 (3100)...
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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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