Linear approximation for one variable

Most textbooks explain the idea of finding the tangent line at a certain point of a function. The geometric idea is that if you consider a very small interval, the function can be approximated by a linear function. Linear means a constant rate of change. Some textbooks give the idea of zooming in a function's graph. If we take a parabola and zoom in enough, a small piece of it should be rendered as a straight line on a computer's screen.

With calculus we are always plotting graphs over an euclidean space. In euclidean geometry the shortest distance between two points is always a straight line. This is one reason to explain why we have the problem of finding a tangent line. Between two points we have infinitely many paths, but among all of them there is one that is a straight line and it happens to minimize the distance travelled between the two points. Not every teacher mentions this and there is also a problem of schedule. Time is often too short to teach this.