12 THE SYSTEMS CONCEPT 



Increments may be as large or as small as we like. If we reduce the dis- 

 tance between A and B, the value of Ax is reduced; this can continue until 

 Ax is infinitesimally small (so small that we cannot think of anything 

 smaller). Infinitely small increments are called infinitesimals, and are written 

 in shorthand with the Arabic letter "d", i.e., d.v. 



Combining the ideas of Sections 3 and 4, it is seen that as A and B ap- 

 proach P, Ax gets smaller and smaller until, at the limit, Ax — ► dx, and it can 

 be made infinitely small. This means that if we view the point, P, from B, we 

 can move B in on P as closely as we please — in fact to an infinitely small 

 distance away — and observe Pfrom as closely as we please. At the limit we 

 observe Pfrom an infinitely small distance away, i.e., as A.v — > 0. 



With the concepts of increments and limits we have implicitly intro- 

 duced the concept of continuous number, as opposed to the discrete number 

 which is familiar to us in our unitary, decimal, and fraction systems. Con- 

 tinuous number admits of the possibility of continuous variation of x be- 

 tween A and B; the number of steps can be infinite. Continuous number 

 is involved when a car accelerates from to 40 mph: the car passes through 

 every conceivable velocity between and 40, and not in the discrete jumps 

 which our decimal and/or fraction systems would describe. At best, these 

 latter are but very useful approximations, and can be considered as con- 

 venient, regular stop-off points, or stations, along the path of continuously 

 increasing number. 



5. Instantaneous Rate of Change 



Any living being is a complex system of interrelated physical and chemi- 

 cal processes. Each of these processes in the "well" being is characterized 

 by a particularly critical rate (speed or velocity) which enables it to fit into 

 the complex system without either being too slow and holding all the other 

 subsequent processes back, or too fast and allowing a runaway of certain 

 subsequent processes. The study of the factors affecting the rates of 

 processes is called "kinetics," and is discussed in detail for some biological 

 processes, in Chapter 8. 



Average rate or speed, over some time interval, is often useful; but it is the 

 instantaneous rate, or the speed at any instant, that is most useful for an 

 understanding of these complex, interrelated reactions. 



If j; = f(x) and the function is continuous, we may be interested in how- 

 fast y changes at any value of .v. In a diffusion process, for example, y would 

 be a concentration and v the time. The question is: How much is the concen- 

 tration in some particular volume changing per second at some particular 

 second in time? The following three examples, one experimental, one graph- 

 ical, and one analytical, illustrate the use of limits and increments to de- 

 scribe this situation. 



