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. 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. That's the whole geometric idea of the derivative.

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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.

It's clear that the tangent line is a good approximation of the function if we consider a certain margin of error. The graph clearly shows that beyond a certain margin the error is too great. One way to think about it is to consider how hard it is to calculate the value of a function. It may be feasible to consider that between two points we can disregard the precision and use a function that is easier or faster to calculate.

In analytical geometry we learn that the equation of a line is [math]\displaystyle{ r = x_0 + \bold{v}t }[/math] in vectorial form. There is an starting point, a vector and a parameter. We can obtain the function that is the tangent line with the same idea: [math]\displaystyle{ (p, \ L(p)) }[/math] is the first point. [math]\displaystyle{ (x, \ L(x)) }[/math] the second point.