. Biophysics: concepts and mechanisms. Biophysics. 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, , 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/Ar as Ar approaches zero. Once again, Lim A V/Ar = d V/d
. Biophysics: concepts and mechanisms. Biophysics. 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, , 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/Ar as Ar 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 = Airr2. (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 gt2, 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 Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly re
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