Understanding the Purpose of Tangent Lines in Calculus

[peps]

I'm starting my first lesson in Calc I this week and I have a question regarding limits and the tangent line.

I understand how to calculate the limit and determine the slope as Q approaches P. I also understand in connecting the two points with the secant line but I don't seem to understand the purpose of the tangent line.

Given point P, I can determine the slope and the limit by choosing other points along the curve. I choose a point Q and connect it to P with a secant line but what is the purpose of the tangent line? Is the point that the tangent and the graph crosses supposed to indicate the limit?

 

[christianjb]

I'm not sure I have the answer you want, but

The secant line is equal to the tangent line in the limit that the two points come infinitesimally close.

Thus, the derivative, which involves infinitesimal displacements is simply the slope (change in y / change in x) of the tangent line.

 

[HallsofIvy]

One major use of Calculus is to approximate a complicated function by a much simpler, linear function- that is, replacing a function by the function corresponding to the tangent line.

In fact, the problem of finding a tangent line to a graph was studied by Fermat and Pascal long before Newton and Leibniz created the "Calculus".