Differential Calculus – Applications

Applications of derivatives

All of you are well familiar with what we mean by derivative in your basic differential calculus course. I am an engineer by profession so I am more interested in applied aspect of derivative rather than theoretical. Derivative is a powerful tool in many real life engineering as well as in business applications. Trick is to not fear with derivative formulas instead try to understand how beautifully its helping us in solving many complex problems. Let me give some examples below:

Physics

Physics is nothing but explanation of various natural phenomenon like heat, sound, optics, kinematics etc. Calculus has helped a lot in simplifying the mathematical part for putting the theoretical concept in the form of formula. Let me give one simple example of newton’s law of cooling:

Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings).

Above law is very intuitive and simple to understand. Suppose you have a pot of hot milk then it cools faster in the refrigerator than outside, why? Reason is difference in temperature is higher when milk is kept in a fridge than outside. Now let’s try to use simple calculus to put the theory into concept:

rate of change of the temperature of an object = dT/dt  [T- Temperature, t – time]

Now as per the Newton’s law:

dT/dt  proportional to (T-Ta) where Ta is an ambience temperature or

dT/dt  = -k*(T-Ta)

Why the sign of K is negative? Because temperature change with time is decreasing. Its solution is also very simple. It’s a simple first order differential equation or a mere integration of right side of expression over time period will provide the solution.

Chemistry

Now let’s see how calculus helps us in formulating the concept of chemistry. Let’s take the simple example of radioactive decay of nuclear material.  The theoretical law states that:

“Rate of radioactive decay is proportional to the amount of radioactive material present.”

Above concept is very simple, higher the amount of radioactive material, faster is the decay and vice versa. Now let’s try to put above statement in the form of mathematical formula. Let’s assume that at given time amount of radioactive material is N then as the above law:

dN/dt proportional to N or in other word dN/dt = -KN [K is negative as material decays with time]

So again above equation is same as of Physics example. At this point, role of chemistry stops and calculus takes over! Its solution is simple, just take the integration on right side over time which will give the answer N = N0 exp(-Kt).

Business application

Enough of physics and chemistry now, let’s see how derivative helps us to handle the real life business applications. There is a filed in investment banking known as quantitative finance. All smart guys from IIT are hired by investment banks to build the mathematical models for high frequency trading of stocks in the stock market. These financial models are based on advance mathematics hence there’s a need to people who are good in math. Needless to say, these maths geeks are paid in tons for their interest and talent in mathematics. I will cover few examples how calculus plays a vital role in building such models with few simple examples in next post, meanwhile stay tuned…..