Tangent at a point on a curve

1. May 20, 2012

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?

2. May 20, 2012

Stephen Tashi

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.

3. May 21, 2012

Inhsdehkc

Thanks Stephen Tashi!

4. May 21, 2012

tiny-tim

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 !)

5. May 22, 2012

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.

6. May 22, 2012

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?