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Mathematics is a fundamental subject that cannot be ignored in the field of science and engineering. The rates of change are exhibited in almost all the works in these areas in different forms. These forms include exponential, logarithmic, trigonometric rational and polynomial functions. It should be noted that interpretation of the rates of change may be calculated at a point or over a given period. The former is referred as the instantaneous rate while the latter the average rate of functions.
Real-World Rate of Change
The growth of bacteria takes place through a process called binary fission. In this process, a cell divides forming two daughter cells that are identical to the parent cell provided mutation does not take place. For instance, the small intestine bacteria while inhabiting areas optimal for growth have a doubling time of about 10 hours. In this regard, we are to model the growth of these small intestine bacteria with a starting population of 5,000 bacteria per ml of fluid. Therefore, the growth of the small intestine bacteria is modeled by the exponential function below,
Where is time in hours,
b is the number of bacteria per ml,
k is a constant term representing a growth factor,
is the time taken for the initial amount to double.
From the above exponential model, it is possible to determine the average rate of change between the 30 hours and 40
hours and the instantaneous rate of change at the 34 hours as shown below,
The exponential function is differentiated to obtain the instantaneous rate at any time as follows,
Therefore, at 34 hours, the rate is given by,
In this regard, the growth rate of bacteria is 3658 bacteria per hour in one ml of fluid at 34 hours.
Population after 30 hours
Population after 40 hours
Average rate of change
Change in time;
Change in the population;
Implying that between the 30 hours and 40 hours; the growth rate of the small intestine bacteria is 400 bacteria per hour in one ml of fluid.
The study of bacteria is essential in determining the population of the bacteria that may cause life-threatening diseases like cholera and typhoid fever. On the other hand, the small intestine bacteria are responsible for abdominal pains and bloating, diarrhea, and chronic excess gas. In the field of medicine, a patient diagnosed with such symptoms is likely to be affected by the increase in the population of the small intestine bacteria.
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