Linear Approximation of Tanx at a=0: Determining Accuracy Within 0.1

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Homework Help Overview

The problem involves verifying the linear approximation of the function tan(x) at a=0 and determining the range of x values for which this approximation is accurate within 0.1. The subject area is calculus, specifically focusing on linear approximations and Taylor series.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the initial attempt to establish an inequality involving tan(x) and question how to approach the problem without a clear method. Some suggest estimating a remainder term related to the Taylor series, while others express uncertainty about the Taylor series itself and suggest exploring numerical methods or calculator experimentation.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some guidance has been offered regarding numerical approaches and the use of calculators, but there is no consensus on a specific method or solution path yet.

Contextual Notes

There is a mention of the original poster's unfamiliarity with the Taylor series, which may limit their ability to engage with certain suggested approaches. Additionally, the problem's requirements regarding the nature of the answer (exact vs. proof) are being questioned.

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Homework Statement


Verify the linear approximation tanx = x at a=0.
Then determine the values of x for which the linear approximation is accurate to within 0.1.


Homework Equations


L(x)=f(a) + f'(a) (x-a)


The Attempt at a Solution


Besides writing down that tanx - 0.1 < x < tanx + 0.1
I have no idea how to approach this!
 
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If you want an 'exact' answer (to some number of signficant digits) to that inequality you would have to solve it numerically. I suspect they want you to estimate a remainder term to the Taylor series. What forms of that do you know?
 
I don't know what the Taylor series is. I don't think they're looking for a numerical answer though. I think they're looking for some sort of proof? I just started learning linear approximations.
 
If you don't know what a Taylor series is, forget that suggestion. Just goof around with your calculator to figure out how big x can be and still keep |x-tan(x)|<0.1. There's not really a nice way to do it.
 

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