lect118_32_w12

lect118_32_w12 - • Approximate the value of the integral...

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Wednesday, March 21 − Lecture 32 : Applications : Evaluation of definite integrals and limits. (Refers to Section 8.9 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : Use a Taylor series to approximate the value of a definite integral and give an error bound, use a Taylor series to find the limit of a function. 32.1 Example Taylor series can be used to help evaluate definite integrals. Let if x is not 0 and define f (0) = 1/2. Find It can be shown that lim x 0 f ( x ) = f (0) = 1/2. Hence f ( x ) is continuous on [0, 1], and so the integral is proper.
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Unformatted text preview: • Approximate the value of the integral with an error inferior to 0.00001 a convergent alternating series. Now c 4 = 1/8!7 < 0.0000036. • Thus approximating the value of the integral with S 3 = 1/2! − 1/4!3 + 1/6!5 = 0.4863889. .. produces an error which is inferior to c 4 = 1/8!7 < 0.0000036. Thus the approximation of the integral with 0.48638 is correct to 4 decimal places. Finding limits of functions with Taylor series. 32.2 Example − Find the lim x → ∞ x sin (1/ x ). Writing the function as a Maclaurin series we get...
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This note was uploaded on 04/01/2012 for the course MATH 118 taught by Professor Zhou during the Winter '08 term at Waterloo.

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lect118_32_w12 - • Approximate the value of the integral...

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