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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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
∫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.
Transcribed Image Text:а)
Calculate the approximate integral of
0.1
S" y(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
So y(x)dx ?
c)
Using the Maclaurin series up to, and including, the 3rd term, calculate the
approximate gradient of y at x =
0.3.
![d)
Given the answer to the integrals from parts a) and b) are not the same, which of the
following are true?
O Reducing the upper limit of the integral will reduce the error.
O 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.
O The Maclaurin series is an approximation to y, hence the integrals of y and its
Taylor expansion will be different.
O Increasing the limits of the integral will have no effect on the accuracy of the
integral.
O Approximating the integrals between the limits [a, n] using the Maclaurin
series from part a) is appropriate.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc05258e-f1d9-4a7a-9153-630176732a31%2F45f24857-f671-4f38-8c3d-af2b36da9771%2F9vw84go_processed.png&w=3840&q=75)
Transcribed Image Text:d)
Given the answer to the integrals from parts a) and b) are not the same, which of the
following are true?
O Reducing the upper limit of the integral will reduce the error.
O 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.
O The Maclaurin series is an approximation to y, hence the integrals of y and its
Taylor expansion will be different.
O Increasing the limits of the integral will have no effect on the accuracy of the
integral.
O Approximating the integrals between the limits [a, n] using the Maclaurin
series from part a) is appropriate.
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