Does tangent line of a point on a curve really contain only one point?

Question

If we consider one function, and take the derivative of that function in a given point, we can easily write up the tangent line of that point. But the definition says that a tangent line contains only one point of a curve, but i think that the tangent line will contain the next point of that curve also that is infinitely close to the previous point.(i assume this from the fact that in the previous original point the derivative value of the original function will be equal to the derivative value of a tangent line) I hope this is understandable, i am not english man

Answer 1

The general equation for a line in $\mathbb{R}^2$ at the point $(x_0,y_0)$ is $y-y_0=\frac{dy}{dx}(x-x_0)$. Now, it is useful to get rid of the assumption that a line or a curve has a "thickness". For sure, when we physically draw a line or a curve, it must have some thickness for us to see it.

However, the mathematical concept of a line (or a curve in $\mathbb{R}^2$, for that matters) has no "thickness". Thus, the tangent line to a point generated by the derivative of the curve contains only the point at which the derivative is taken. This concept is sometimes hard to conceptualize at first, because when one thinks of a curve, one (generally) thinks of it graphically, and thus having a certain thickness, where as it really does not have any.

An interesting consequence of this is that in the neighbourhood of $(x_0,y_0)$, the tangent line yields a linear approximation of the curve, which can be useful in multivariable calculus to approximate the value of a function in a neighbourhood of a point of $\mathbb{R}^n$