Requirements for a Tangent at the Origin: Function Analysis

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Homework Help Overview

The discussion revolves around the conditions necessary for a function to possess a tangent at the origin, specifically examining functions such as f(x) = 0 and f(x) = xsin(1/x) for x ≠ 0. Participants are exploring the implications of these conditions in the context of calculus.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to define what constitutes a finite derivative and its significance for having a tangent at the origin. Questions about the nature of finite derivatives and their geometric interpretation are being raised.

Discussion Status

Some participants have provided guidance on analyzing the difference quotient to determine the existence of a finite derivative at the origin. There is an ongoing exploration of the definitions and implications of these concepts without a clear consensus yet.

Contextual Notes

There is an emphasis on understanding the behavior of the function as it approaches the origin, particularly in relation to the limit of the difference quotient. The discussion is framed within the context of calculus, suggesting that participants may have varying levels of familiarity with the subject.

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Homework Statement



What must hold true for a function to have a tangent at the origin.

Eg. Given f(x) = 0, x = 0

and f(x0 = xsin (1/x) x does not equal 0

will the graph have a tangent at the origin?

Homework Equations





The Attempt at a Solution

 
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Jan Hill said:

Homework Statement



What must hold true for a function to have a tangent at the origin.

Eg. Given f(x) = 0, x = 0

and f(x0 = xsin (1/x) x does not equal 0

will the graph have a tangent at the origin?

Homework Equations


It must have a finite derivative when x = 0. Check its difference quotient.
 
What is a finite derivative?
 
Jan Hill said:
What is a finite derivative?

Have you had or are you taking calculus? If so then you should know what a derivative is. A function has a finite derivative at a point if its derivative at that point exists and is finite.

Geometrically this means that the graph of the function is smooth enough at the given point that it has a tangent line that is not vertical.

To work this problem you need to analyze

\lim_{h\rightarrow 0}\frac{f(0+h)-f(0)} h

for your function

f(x) = x\sin\frac 1 x,\ f(0)=0
 

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