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

[Inhsdehkc]

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!

 

[Stephen Tashi]

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.

 

[Inhsdehkc]

Thanks Stephen Tashi!

 

[tiny-tim]

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?
​

(unless it means "meets but does not cut", in which case the x-axis would not be tangent to y= x³ !)

 

[HallsofIvy]

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.

 

[Whovian]

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?