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Estimate the error if cost? = 1– 2 in the integral fcosť² dt. 4!

Estimate the error if cost? = 1– 2 in the integral fcosť² dt. 4!

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Estimate the error. Please show all steps, thank you.

**Problem 12:** Estimate the error if \( \cos t^2 = 1 - \frac{t^4}{2} + \frac{t^8}{4!} \) in the integral \( \int_0^1 \cos^2 t \, dt \). --- **Explanation:** This problem asks to estimate the error involved in approximating the integral of \( \cos^2 t \) from 0 to 1 using a given expansion for \( \cos t^2 \). The expansion provided is a truncated Taylor series approximation for \( \cos t^2 \). Evaluating this expression and comparing it to the full integral can offer insights into the accuracy of the approximation. The factorial in the denominator, \( 4! \), equals 24, which should be taken into account when performing calculations.
Transcribed Image Text:**Problem 12:** Estimate the error if \( \cos t^2 = 1 - \frac{t^4}{2} + \frac{t^8}{4!} \) in the integral \( \int_0^1 \cos^2 t \, dt \). --- **Explanation:** This problem asks to estimate the error involved in approximating the integral of \( \cos^2 t \) from 0 to 1 using a given expansion for \( \cos t^2 \). The expansion provided is a truncated Taylor series approximation for \( \cos t^2 \). Evaluating this expression and comparing it to the full integral can offer insights into the accuracy of the approximation. The factorial in the denominator, \( 4! \), equals 24, which should be taken into account when performing calculations.
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