22 



GEOMETRICAL ILLUSTRATION. 



35. Suppose we have given the equation y = <f) (x), and 

 that we attribute to the independent variable x all possible 

 values between oo and + oo and notice the corresponding 

 values of y. Geometry gives us the means of representing 

 distinctly this succession of values. We can take x for an 

 abscissa measured from a y 



fixed origin along a certain 



axis, and y for the corre- 



sponding ordinate measured 



along an axis at right angles 



to the first. The values of 



y corresponding to those of 



x in the equation y = <j>(x) 



will belong to a curve 



AMN, the form of which 



will indicate the series of 



values we are considering. It is necessary to have always 



present in our mind not merely any particular value of x 



and the corresponding value of y, but the whole series of 



corresponding values of these two variables. 



36. Among the properties which the function < (x}, or the 

 line which represents it, possesses, the most remarkable, that 

 in fact which is the object of the differential calculus and the 

 consideration of which is perpetually occurring in all applica- 

 tions of this calculus, is the degree of rapidity with which the 

 function varies when the variable begins to vary from any 

 assigned value. The degree of rapidity of increase of the 

 function when the variable is made to increase may differ not 

 only in different functions but also in the same function 

 according to the value attributed to the variable from which 

 the increase is supposed to commence. Suppose we give to x 

 a particular value denoted by OP, to which corresponds a 

 determinate value of y or tf> (x) represented by MP. Let x, 

 starting from the value assigned, increase by a quantity which 

 we denote by A#, and which is represented by PQ. The 

 function y will vary in consequence by a certain quantity 

 which we denote by Ay, so that 



therefore 



A?/ = < (x + Ax) $ (a?J. 



