Slope of a line

Slope of a Line

The slope of a line is denoted by m and can be determined with two points: A(x1,y1) and B(x2, y2).


It can also be determined by:


in which α is the smallest positive angle from the x axis to the line. When the line is horizontal, m = 0 and when the line is vertical, m is infinity or not defined.




Slope of a Curve

Suppose curve C (f(x)) is a function of x, the tangent line of the curve C at point A is desired.

Now, consider B point where x2 is not equal to x1, the slope of AB is:



Tangent of a Curve

Let point B approaches A along the curve C by letting x2 approaches x1, when x2 - x1 is small enough, the slope of AB will overlap with its tangent line at point A.Thus, the tangent at point A can be written as:


    Tangent Example - Velocity


Velocity is a good example of a tangent line. Normally, the distance s is a function of time t. The velocity of a time interval is



Velocity of any Function

On a graph of distance plotted as a function of time, the tangent (instantaneous velocity) is:


Therefore, the instantaneous velocity at time t, is equal to the slope of the tangent line at point A.