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Figure 1. Plot of a Michaelis-Menten function. This function is always increasing and concave down. It has a horizontal asymptote, y=4. |
For this part, we will cover all the theories and techniques that are covered in the traditional calculus-I course. Unlike in the traditional calculus-I course where most of application problems taught are physics problems, we will carefully choose a mixed set of examples and homework problems to demonstrate the importance of calculus in biology, chemistry and physics, but emphasizing the biology applications.
Example 1. Traditionally, the first application discussed in Calculus I is the distance/velocity/acceleration problem for moving objects including the free-fall problem. For our Bio-enriched Calculus I, we will consider the Michaelis-Menten kinetics function [4][9]:
This function has many applications in biological fields. For example, it can be used for modeling in enzyme reaction or population growth. Here
n could be the nutrient concentration and
f be the growth rate function for bacteria;
Kmax and
Kn are positive constant parameters standing for maximum growth rate and the nutrient density at which the bacteria growth rate reaches
Kmax /2. This example can be used to introduce the dependence on nutrient as the first derivative and the acceleration (deceleration) of it as the second derivative. In the later discussions of related rates, we can revisit this example for the relationship of two time dependent functions,
u(t) and
n(t):