Tangent Lines

Tangents have many important uses in the world of calculus and mathematics. They are the geometric representation of a derivative, or rate of change, at a specified point. Before starting into the world of differential calculus, we need to examine the tangent line and thoroughly understand its importance.
The name tangent is derived from the Latin word 'to touch'.
Roughly speaking, a tangent is a line that just touches a curve at one given point (it is important to note that a tangent line can touch a curve at many point globally, but just one point locally). We can visualize this as a straight line that can only be pointed in one specific direction, so that it when brought close to a curve, it only barely touches it at one point on the curve. The tangent slope has a very important meaning in the world of calculus.

Essentially if you draw a tangent line to a curve, then its slope is the 'rate of change' of the curve. Take for instance a position graph for an automobile, plotted as distance v.
s time.

The slope of the tangent line at any given point will give you the velocity of the car at that point (in this case a value in kilometres/hour or miles/hour).

The slope of a tangent line at any given point is the instantaneous rate of change, which is not necessarily easy to calculate algebraically. In simple cases, the slope of the line can be read from the graph, though in most cases it will need to be calculated algebraically.

This calculation is done using the formal definition of a limit.

This is beyond the scope of this article, but will be examined in your differential calculus class.
Overall the tangent is a simple concept that has some very powerful applications.
Learn to use and draw them well, as they will always be used in future calculus studies.