Use taylor's THeorem to determine the accuracy

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    Accuracy Theorem
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Homework Help Overview

The discussion revolves around the application of Taylor's Theorem to assess the accuracy of the approximation for cos(0.3). Participants are examining the remainder term and the error bound associated with the approximation.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants are discussing the correct form of the remainder term in Taylor's Theorem and questioning the calculations related to the error bound. There is a focus on understanding the implications of using different derivatives and the significance of the maximum value of the derivative in determining the error.

Discussion Status

The discussion is active, with participants clarifying misunderstandings about the remainder term and exploring the implications of using different values for the derivative. Some guidance has been provided regarding the use of the maximum value of the derivative for error estimation.

Contextual Notes

There is an emphasis on the need to correctly identify the derivative and the point at which it is evaluated, as well as the assumptions regarding the interval of approximation.

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Use taylor's THeorem to determine the accuracy of the approximation: cos(.3) ~=1 - (.3)^2 / 2! + (.3)^4 / 4! when i use taylors theorem, i use (.3)^4 / 4! which gets me 2.03e-10 but the asnwer is R<=2.03e-5
 
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(.3)^4/4! is not equal to 2.03e-10. What form of the remainder term are you using?
 


Okay, it should be (.3)^5 / 5!
 
Last edited:


But i still don't get it, why did i get hte right answer and ignore the f^(n+1)(c) of the formula? (Lagrange error bound) The derivative should have been cos , so i must have picked cos 0. Why?
 


The error does NOT involve f^(n+1)(c). It involves f^(n+1)(x) for some number between 0 and c (here .3). Since you don't know that number, you use the largest possible value of f^(n+1)(x) between 0 and c to get an upper bound on the error. Since cosine is decreasing, its largest value is cos(0)= 1.
 

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