Polynomial and rational functions if observed from a graphic

Polynomial and rational functions, if observed from a graphical point of view, seem to have lots of \"ups and downs\" (i.e., they increase, decrease, or remain constant). Take a time period in your personal or professional life and model it based on any of these functions, or a combination thereof. Discuss the critical points of change and what led to them. Try to go through a time period long enough where several of these changes took place.

Solution

A rational function can be used to determine the concentration of a certain drug after a fixed amount of hours.The equation used to model this real world topic is given by:

C(t) = (4t)/(t²+2)

Now here 4t and t²+2 are two polynomials in t and so C(t) is combination of both rational and polynomial functions

The critical points are given by the equation C\'(t) = 0

=> C\'(t) = [(t²+2)4 - 4t(2t)]/(t²+2)² = 0

=> C\'(t) = [(-4t²+8)]/(t²+2)² = 0

=> -4t²+8 = 0 => t² = 2

=> t = 2^1/2 = 1.414

So now it seems t=1.414 is critical point here

Now For t=0, C\'(0) = 8/4 = 2>0

For t=2, C\'(2) = -8/36 <0

Hence t=1.414 gives minimum value of C(t)

This means that the function C(t) increases for [0,1.414) and decreases for (1.414,infinity)

This is because the concentration of medicine increase and gets maximum at 1.414 hours and then later the concentartion decreases from thereafter

Now as t approaches infinity, C(t) = 4t/t²(1+2/t²) = 4/t(1+2/t²) approaches 0 (becasue for large t , denominator becomes large and so concentration assumes value 0