16 THE SYSTEMS CONCEPT 



expression which describes the instantaneous rate of change-the "daugh- 

 ter " or derived, function. A system of "operations" has also been devel- 

 oped by which the same thing is accomplished. In this sense d/dx is an 

 "operator," operating on y in a specific manner which accomplishes the 

 same result as the method of increments gave us in Example (3) . 



Conversely, if the rate of change is given (most often directly from the 

 experiment), it is possible from the daughter equation to reverse the method 

 of increments, and establish and examine the mother equation (Figure 1-6). 

 The process is simply to sum the increments, under special conditions, when 

 they are infinitesimally small. A system of operations has also been worked 

 out for this process. The operator is symbolized as an elongated S , called 

 the "integral sign," f, contrasted against the operator, "d", for the inverse 



process. 



.-. ^Pj^ e_ntio t_i £n 



rate of change 



Figure 1-6. Definition of Differentiation and of 

 Integration. 



Described in the previous Sections 1 to 5 are the basic ideas of the calcu- 

 lus The process of finding from the mother function, F{x), the daughter 

 function, F'(x), which expresses rate of change, is called differentiation, or 

 obtaining the derivative or derived function; the reverse process of summation 

 of an infinite number of values of the derived function, F'(x), to give the 

 mother function, F(x), is called integration or obtaining the integral. 



Two more definitions in shorthand will prove to be useful, the second order 

 derivative and the partial derivative. Both are actually quite simple concepts. 

 We often run into a situation in which we wish to express how fast the speed 

 is changing. (Consider the automobile example, given in Section 4, in which 

 we are now interested in acceleration.) Since speed is dS/dt, the rate of 

 change of speed is d/d«dS/dt), which is abbreviated d 2 S/dt 2 with the 

 operator "d," in the numerator squared and the whole differential in the 

 denominator squared. It is obvious that the rate of change of acceleration 

 would be expressed as d'S/dt\ and that higher orders exist, although they 

 are not of common interest to us here. 



