Taylor Polynomial of f(x) = x^3sin(x)

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Homework Help Overview

The discussion revolves around finding the first three non-zero terms of the Taylor polynomial for the function f(x) = x^3 sin(x) at the point a = 0. Participants are exploring the application of Taylor series and the necessary derivatives involved in the process.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the Taylor series for sin(x) and its multiplication by x^3. There are questions about the correctness of previous attempts and whether the product rule is necessary for finding derivatives. Some participants reflect on the representation of the Taylor polynomial and its relation to the Maclaurin series.

Discussion Status

The discussion is ongoing, with participants questioning their understanding of the Taylor series and derivatives. Some guidance has been offered regarding the use of the relevant equation and the nature of the series at a = 0, but no consensus has been reached on the correct approach or solution.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the information they can share or seek. There is an emphasis on understanding the derivation process rather than simply arriving at an answer.

november1992
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Homework Statement



Find the first 3 non-zero terms of the Taylor polynomial generated by f (x) = [itex]x^{3}[/itex] sin(x) at a = 0.

Homework Equations


[itex]f^{n}[/itex](x) * [itex](x-a)^{n}[/itex] / (n!)

The Attempt at a Solution



I got the question wrong: my answer was 1/3! + 1/5! + 1/7!
Here is the answer below. I was wondering how my teacher got that answer. Do you have to do the product rule to find the derivatives?
sxjbfc.png
 
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Do you know the Taylor series for ##\sin x\ ##? He just multiplied that by ##x^3##.
 
november1992 said:

Homework Statement



Find the first 3 non-zero terms of the Taylor polynomial generated by f (x) = [itex]x^{3}[/itex] sin(x) at a = 0.

Homework Equations


[itex]f^{n}[/itex](x) * [itex](x-a)^{n}[/itex] / (n!)

The Attempt at a Solution



I got the question wrong: my answer was 1/3! + 1/5! + 1/7!
Here is the answer below. I was wondering how my teacher got that answer. Do you have to do the product rule to find the derivatives?
sxjbfc.png

The Taylor polynomial is a function of x. Your answer is really just a single number.

You wrote something in the relevant equation section. Do you know what it represents and what it's used for?

Your textbook undoubtedly has some examples that are like this problem. Have you looked at any of them?
 
At a=0, it means that the function f(x) is approximated near the point x=a=0. In other words, you have a Maclaurin series. Adapt and use the relevant equation from your post #1.
 

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