Given a function y=3cos⁡(x), where we are working in RADIANS, answer the following: For all numerical answers, answer to within 3 decimal places. Note that adatpive marking has been used to consider the method mark. a) Calculate the approximate integral of ∫0.10?(?)??∫00.1y(x)dx when ?(?)y(x) is represented by its Maclaurin series up to, and including, the 3rd term. b) What is the absolute difference between your answer in a) and the exact integral of ∫0.10?(?)??∫00.1y(x)dx ? c) Using the Maclaurin series up to, and including, the 3rd term, calculate the approximate gradient of ?y at ?=0.3x=0.3. d) Given the answer to the integrals from parts a) and b) are not the same, which of the following are true? Reducing the upper limit of the integral will reduce the error. Whilst there are errors in the approximated integral, the relative errors will generally decrease if the Maclaurin series included more than the three terms used here. The Maclaurin series is an approximation to y, hence the integrals of y and its Taylor expansion will be different. Increasing the limits of the integral will have no effect on the accuracy of the integral. Approximating the integrals between the limits [?,32?][π,32π] using the Maclaurin series from part a) is appropriate.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

Calculate the approximate integral of

∫0.10?(?)??∫00.1y(x)dx

when ?(?)y(x) is represented by its Maclaurin series up to, and including, the 3rd term.

What is the absolute difference between your answer in a) and the exact integral of

∫0.10?(?)??∫00.1y(x)dx ?

c) Using the Maclaurin series up to, and including, the 3rd term, calculate the approximate gradient of ?y at ?=0.3x=0.3.

Given the answer to the integrals from parts a) and b) are not the same, which of the following are true?

Reducing the upper limit of the integral will reduce the error.
Whilst there are errors in the approximated integral, the relative errors will generally decrease if the Maclaurin series included more than the three terms used here.
The Maclaurin series is an approximation to y, hence the integrals of y and its Taylor expansion will be different.
Increasing the limits of the integral will have no effect on the accuracy of the integral.
Approximating the integrals between the limits [?,32?][π,32π] using the Maclaurin series from part a) is appropriate.