14 



THE SYSTEMS CONCEPT 



RADIUS, r 



Figure 1-4. Volume of a Spherical Cell as a Function of its Radius. 

 Determination of rate of change of V as r changes, i.e., dV/dr. 



This case is now similar to (1) and need not be discussed in detail. A 

 point, P', is chosen; a straight line joining P and P' is drawn, and the value 

 of A V/Ar determined from the graph. At successive points closer and closer 

 to .Pthe same thing is done, until it is more or less evident what will be the 

 limiting value of A V/ A r as A r approaches zero. Once again, Lim A V/Ar = 



d V/dr, the slope at P. It turns out that for this case d V/dr = Airr 2 . 



(3) Analytical: A simple example* will illustrate one way in which this 

 can be done algebraically. 



The law established by Galileo at Padua governing the free fall of a body 

 (Figure 1-5) toward earth, is expressed as S = 1/2 gt 2 , where S is the dis- 

 tance fallen, t is the time of fall, and g is the value of acceleration due to 

 gravity (32 ft per sec per sec.) This example is chosen not because of its 

 specific relation to medical physics but because of its simplicity as an illus- 

 tration of the algebraic determination of instantaneous rate of change by 

 means of the method of increments. The experimental and graphical ex- 

 amples, (1) and (2), are limited in that an extrapolation of incremental pro- 

 portions is always necessary. In the algebraic method this is not necessary, 

 but the limit still can be examined from as close in as it is possible to 

 imagine. 



*As an alternative one could have considered a child blowing up a balloon, and asked the 

 question: How fast does the area of the balloon change as the radius changes? The area is 

 given by A --= 4irr , also a parabolic function. Less easily conceived examples appear later. 



