STAT312 Lecture Notes - Taylor Series, Antiderivative, Riemann Integral

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Example: recall taylor s theorem with the re- mainder in integral form. Suppose we wish to approximate the cube root of = 9 in terms of that of = 8; we will do this by expanding ( ) = 1 3 around . How close do we get if we stop after the quadratic term? ( ) = ( ) + 0 ( ) ( 3 9 = 2 08008 with an error. With a calculator, of 00022; the estimate of the error using taylor s. Improper riemann integrals, in which one or both is unbounded, endpoints are in nite, or at which are de ned by taking appropriate limits: = lim ( ) +z for any if. An application of the fundamental theorem of. Calculus is the formula for integration by parts. 0 are integrable, are di erentiable, and 0 , then. Z [ ( ) ( )]0 hence ( ) ( ) and also.

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