Increment. The basic property of a linear function

Increment. The basic property of a linear function

increment of x,y

The difference between the final and initial values of the independent variable x on transition from the initial value   to the final value  is called the increment of x, equal to  The difference between the final and initial values of the function y = f(x) is called the corresponding increment of the function:

These increments are often denoted by:

It must be pointed out that the symbol x has to be regarded as a single entity in denoting the increment of x. We shall consider the case of a linear function, when

Subtracting term by term, we have:

This equality shows that the linear function y = ax -b has the property that the increment of the function (y2 - y1) is proportional to the increment of the independent variable (x2- x1), the coefficient of proportionality being equal to a, i.e. to the slope of the graph of the function. Turning to the graph itself (Fig. 8), corresponding to the increment of the independent variable we have the segment X1P = x = x2 - x1  and corresponding to the increment of the function, PM2 = y = y2-y1 ; and formula(4) follows at once from considering the triangle M1pM2.

 the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek