THE TEN PILLARS 13 



(1) Experimental : To measure the instantaneous velocity of an automobile 

 (refer again to Figure 1-3) requires measurements of distance and time be- 

 tween two stations, A and B. Two observers with stop watches and a tape 

 measure can easily do this. They measure a value of Ax/ At, which is the in- 

 crement of distance covered in an increment of time. But the car is acceler- 

 ating between A and B, and hence Ax/ At is only an average value between 

 A and B, and may be quite different from the velocity as the car passes P. 

 Better values can be obtained the closer the observers are to P, but of course 

 no value can be obtained if both observers are at P because Ax = and 

 At = 0, and 0/0 is indeterminate, or can have any value from — « to + °°. 

 The best value is obtained by taking observations at several values of A 

 and B, at smaller and smaller values of Ax, until a good extrapolation to 

 Ax = can be made. Hence the limit of Ax /At as At approaches zero is 

 the instantaneous velocity at the point, P. In shorthand notation, the instan- 

 taneous velocity at PisLim Ax/ At. 



A/— 



This symbolic description is further simplified by use of the infinitesimal 

 symbols: Lim Ax/ At = dx/dt. Conversely the previous statement is actu- 



ally the definition of dx/dt. In other words, dx/dt is the instantaneous rate 

 of change of x as t changes. A very simple experimental check on the method 

 is to ride in a car and note the speedometer reading at point P. 



Both of these methods of determining instantaneous rate are exemplified 

 in biological processes. 



(2) Graphical: A graph of the function which expresses the volume of the 

 spherical cell, V = 4/37rr 3 , is shown in Figure 1-4. The question arises: 

 How fast does the volume of the cell change with change in radius at a par- 

 ticular value of the radius, r,? In other words, how "steep" is the slope of 

 the curve, V vs r, at the point, r,? 



Slope or gradient is defined by surveyors as "rise"/"run," where "rise" is 

 the vertical height from the base to the top and "run" is the level, or hori- 

 zontal distance from the foot of the hill to the top. The ratio "rise/run" de- 

 fines the value (trigonometric function) of the tangent of the angle enclosed by 

 the level direction and the direction toward the top. 



The same is true in analytic geometry, the slope of the straight line join- 

 ing P and P' being given by the ratio of the distances between P and P' as 

 measured along the ordinate and along the abscissa. For example, slope 



V 2 - F, 



= AV/Ar. 



r~, — 



What we want to know is the value of the slope of the straight line which 

 cuts the curve, V vs r, only once and at point P, that is, the slope of the 

 tangent (geometrical figure) at P. This will give the instantaneous rate of 

 change of Fas r changes, at P, or d V/dr at r,. 



