1. ## identify tangent line

there are two lines AB and CD which one is tangent line and why?

I know they say AB is tangent line but how to Mathematically( Geometrically(Trigonometrically )) prove it?

we can take tangent theta too and tangent psi too then both are through tangent of some angle so aren't they both tangent lines?

2. ## Re: identify tangent line

The curve is smooth and there is only one line that is (locally) entirely on one side of the curve.

3. ## Re: identify tangent line

Good answer to some extent but why isn't CD a tangent line?

4. ## Re: identify tangent line

CD is not going in the same direction as the curve at the point where they cross.

5. ## Re: identify tangent line

Good answer to some extent but why isn't CD a tangent line?
I'd say, if you were to extend $\displaystyle y= f(x)$ and also the line segment CD. Then the line segment p to D becomes the secant, from which will give the average change until $\displaystyle \lim_{x \to 0}$ the secant becomes the tangent at point p.

6. ## Re: identify tangent line

Ah well, let's see what wiki says:

Originally Posted by wiki
In geometry, the tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point—that is, coincides with the curve at that point and, near that point, is closer to the curve that any other line passing through that point. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f'(c) where f' is the derivative of f.
As you can see, CD does not "just touch".

And if we look at the derivative, we'd be looking at a line through P and a second point on the curve.
When we let the second point approach P, we get the tangent line.
This is AB and not CD.

7. ## Re: identify tangent line

we can take tangent theta too and tangent psi too then both are through tangent of some angle so aren't they both tangent lines?
You need to review the definition of "tangent line". It has nothing to do with "tangent of an angle".