Is the School Definition of a Tangent Line Always Accurate in Calculus?

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Discussion Overview

The discussion revolves around the accuracy of the school definition of a tangent line in calculus, particularly whether it applies universally to all curves. Participants explore the implications of this definition in various contexts, including specific functions like y=|x| and y=sin x.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions the school definition of a tangent line, suggesting it may not hold true in all cases, particularly for functions like y=|x| and y=sin x.
  • Another participant supports the new concept of tangent, stating that the curve y=|x| does not have a tangent at x=0 due to the lack of a unique slope at that point.
  • A different viewpoint suggests that the school definition may be accurate in the context of circles, where a tangent line is defined as touching the circle at exactly one point.
  • One participant proposes that in calculus, the definition of a tangent should be understood as a local concept, indicating that a tangent can "barely touch" the curve at a point while potentially crossing it elsewhere.
  • Another participant humorously points out the circular nature of the definition of "touch," questioning its clarity and applicability.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the school definition of a tangent line, with some supporting the new concept while others defend the traditional definition, indicating that the discussion remains unresolved.

Contextual Notes

Participants highlight limitations in the school definition, particularly its applicability to various types of curves and the implications of local versus global definitions of tangents.

Inhsdehkc
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When i was at school i used to think that any line that touches a curve at only one point is called as the tangent at that point to the curve! But after reading derivative i think this definition of tangent is not correct at all conditions!
For example,
x-axis cannot be called as tangent at origin to the function y=|x| though it touches the curve at origin only!
second example,
tangent at a point on a curve y=sin x may also touch any other point on the same curve(thus there are two points of intersection)

IS this new concept of tangent i have is right OR the definition of tangent at school is right?

please help!
 
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Inhsdehkc said:
IS this new concept of tangent i have is right OR the definition of tangent at school is right?

The new concept of tangent that you have is correct.

In terms of the new concept, the curve y = |x| does not have a tangent at x = 0 since no unique slope for the curve is defined at that point.
 
Thanks Stephen Tashi!
 
Inhsdehkc said:
When i was at school i used to think that any line that touches a curve at only one point is called as the tangent at that point to the curve!

that would be a circular definition anyway …

"touch" means "tangent", doesn't it? :wink:
​
(unless it means "meets but does not cut", in which case the x-axis would not be tangent to y= x3 !)
 
In defense of "school":
A line is tangent to a circle if and only if it touchs the circle in exactly one point.
I suspect that is what you are remembering and your memory is trying to extend it to all curves.
 
In Calculus, we have to redefine a tangent to only have a definition locally, sort of "barely touching" the curve in question at the point, though it can just cross the curve in other locations. Does this make sense?
 

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