Rate = Distance = f(d) - f(c)
Time d – c
Many applications of rates of change are for describing the motion of an object moving in a straight line, or, as in the case of my Biology experiment, describing the rate in which the enzyme accelerated the substrate’s reaction.
The rate of a reaction of an object, seen as its speed, is the amount of product formed or the amount of reactant used up per unit of time. The rate also depends on the variables of the reactants as well as the conditions in which the reaction is taking place. For example, the rate can be affected by the concentrations of the substances used, as well as the temperature at which the reaction occurs, or the surface area of the reactants and catalysts.
Mathematical Applications
In order to find the rate of change of an object, one would, essentially, be required to find the slope of a graph. Differential Calculus can be applied in this situation, in that by finding the derivative. For example, consider a man who is 6 feet tall and walking toward a lamppost 20 feet high at a rate of 5 feet per second. The light at the top of the lamppost (20 feet above the ground) is casting a shadow of the man. At what rate is the tip of his shadow moving and at what rate is the length of his shadow changing when he is 10 feet from the base of the lamppost?
If z were to equal the distance from the tip of the shadow to the base of the
lamppost, y equaled the length of the shadow, and x equaled the distance from the
man to the base of the lamppost, we would also know that dx
dt = -5 ft/sec
From that, we could say that y = x
6 14
y = (3/7)x
dy/dt = (3/7)dx/dt = -15/7 ft/sec
z / 20 = x / 14
z = (10/7)x
dz/dt = (10/7)dx/dt = -50/7 ft/sec
Knowing how to take the derivatives of certain equations, and finding the rates of change, can be extremely useful to practitioners in the field, as well. For example, a scientist or biologist may know everything that there is to know about their science-related subject, but would not be able to interpret or analyze the results of an experiment without at least a working knowledge of mathematics, including calculus. The equation above demonstrated a method that can be used to determine the rate of change of the man’s shadow as he walks, using Differential Calculus.
Procedure
The example that was mentioned earlier made reference to a Biology experiment that was performed in which the effect of substrate concentration vs. time was measured. Nine beakers with different concentrations of enzymes were utilized. Small paper discs were soaked in each of the different enzyme solutions of different concentrations, after which they were placed on the bottom of a hydrogen peroxide solution (the substrate) and timed in order to see how long it would take for the enzyme to react with the substrate and cause the paper disc to float to the top of the hydrogen peroxide solution.
Conclusion
Differential calculus has many practical uses outside of the classroom in the ‘real’ world, and is a valuable tool in determining the rates of changes that are occurring, whether they be chemical reactions, such as in the case of the enzyme and substrate concentration experiment, or in other cases, such as rates of changes in the acceleration or deceleration of a car, or in the rate a rock falls off of a cliff.
