Introduction to First Derivative Test

A main idea of this section involves what we can learn from the derivative of 
a function. Why does knowing a derivative matter in biology aside from telling 
us whether something we can measure is increasing or decreasing? 

Much of biology deals with life systems that are in some sense stable - you 
have all heard of homeostasis with examples such as fairly constant body 
temperature in "homeotherms (those organisms which maintain fairly constant 
temperatures as compared to ectotherms in which the organism's temperature is 
greatly affected by external temperature conditions and varies with this. 
Stability here refers to the idea that certain processes not only stay near 
the same value (98.6 degress Celcius) but also returns to this value if 
perturbed (due to a fevber say, assuming the individual doesn't die). Much of 
the science of disease deals with cvases in which the body's physiological 
capacity to maintain a stable condition is lost. Heart arythmia's, fevers 
arising from infection, many disorders of the nervous system are all examples 
of this and there is a field of dynamical disease that studies how this 
arises (a great exposition is given in Glass, L., M.C. Mackey. From Clocks 
to Chaos: The Rhythms of Life, Princeton University Press, Princeton, l988). 

So we'd like to have a way to tell when a physiological process is homeostatic, 
and one objective of this course is to provide you with the mathematical 
machinery to understand this. One part of this is understanding that when a 
derivative is equal to zero, so f'(x) = 0, then the quantity being measured 
is stationary (not changing). Consider the example of f(x) giving the cell 
division rate in healthy tissue as a function of some enzyme which affects 
this. Then  f'(x) = 0 implies that the division rate is staying constant. Now 
suppose that there is a genetic mutation which modifies the enzyme, thus 
affecting the function f(x). If we had a way to quantify how this occured, 
we could potentially investigate the instability in the cell division rate. 
This is what leads to sometimes tumor growth - an instability in cell division 
rates. So in this sectoion of the course, we will investigate conditions for 
changes in function's derivatives as an indication of the potential for 
instability to arise in the processes measured by the function.