Calculus is the study of change and motion, in the same way that geometry is the study of shape and algebra is the study of rules of operations and relations. It is the culmination of algebra, geometry, and trigonometry, which makes it the next step in a logical progression of mathematics.

Calculus defines and deals with limits, derivatives, and integrals of functions. The key ingredient in calculus is the notion of infinity. The essential link to completing calculus and satisfying concerns about infinite behavior is the concept of the limit, which lays the foundation for both derivatives and integrals.

Calculus is often divided into two sections:

**Differential calculus** (dealing with derivatives, ex: rates of change and tangents)

**Integral calculus** (dealing with integrals, ex: areas and volumes)

Differential calculus and Integral calculus are closely related as we will see in subsequent pages. It is important to have a conceptual idea of what calculus is and why it is important in order to understand how calculus works.

To study the interaction of different species within an ecosystem calculus is used. The concentration of drugs in a living organism can be answered with calculus. Physical attributes, such as backbone length and skull length are measured by calculus. Calculus is used for measuring growth rate of bacteria and certain species.

Calculus used in robotics to know how robotic parts work on given command and also used by engineers for building skyscrapers, bridges. Calculus was initially developed for better navigation system. Electrical and Computer engineers use calculus for system design. Calculus is used to improve safety of vehicles.

In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue. In analytic geometry and the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, concavity and inflection points.

Calculus makes it much easier to visualize graphs. We may already have a good grasp of linear functions and how to visualize their graphs easily, but what about the graph of something like y = x^{3} + 2x^{2} – x + 1 ? Elementary calculus tells us exactly where that graph will be increasing, decreasing, and twisting. We can even find the highest and lowest points on the graph without plotting a single point.

One of the most common use of calculus in our daily life is the PID controller or the Proportional-Integral-Derivative Controller. The concept is explained with simple analogies to be understood by a layman.

It has a wide range of applications:

1. Coffee Machines

2. Automatic Air-conditioners

3. Cruise Control in Cars

4. Water Mixers

5. Online Advertisements

6. Industrial Control Systems, Ships, Missiles, Rockets, UAV, etc.

PID is a kind of continuous machine learning loop where you need to control the flow of the output without having any prior information of the input (which might be variable and highly volatile).

This is by far the most common use. You set the desired room temperature at 15. Assume that the current room temperature is at 35. The air conditioner will start to cool at a small constant rate first. It then rapidly ramps up the cooling rate in a short period of time. At a certain point of time, the room temperature will overshoot 15 and reach around 12. It then decreases the cooling rate down significantly so that the room temperature can go up. The temperature might overshoot again and reach 16. It increases the cooling again and keeps oscillating around the target temperature till the error gets infinitesimally small. If you change the desired temperature to 20, it will automatically race towards the new temperature and oscillate till the error becomes infinitesimally small again.

The best part of a PID Controller is that it will make sure that the temperature goes to the desired point independent of external factors. If you keep a heater in the room, it will work harder to cool the room.

Consider a mixing instrument which is supposed to mix hot and cold water in the right proportion so that the mixture flows out at exactly 35 degrees. This is very easy if you know the exact temperature of the input water flow and other constants like the thickness of pipe. If you don't know the input temperatures or if the input temperatures of hot and cold waters are variable and keep changing every hour or every minute, the problem gets complex. Industrial applications face even more volatility.

The PID Controller opens the hot valve slightly to measure the change in the output temperature. It will then keep changing it frequently by a constant value. If the desired temperature is far away, it automatically increases the rate of change. So it will first open at 'x', then 2x, 10x, etc. till it overshoots the desired temperature and say reaches 50 degrees. It will then turn the process back and decrease it till it reaches 35. In case it overshoots again to 30, it goes up again oscillating around the target temperature till the error goes infinitesimally small (infinitesimally close to desired temperature).

You launch an ad campaign on Facebook (or any site). You target all the UK users in the age group 18 - 22 who are females. You have a spend budget of 200$ per day. There are 100 other advertisers with similar complex targeting criteria with different budgets. Whenever you load a page, the delivery system is supposed to make a decision on which ad to serve. It is at the same time supposed to make sure that budgets of all clients are spent evenly throughout the day. The usage pattern of users throughout the day is unpredictable. It becomes a very complex problem if you consider all the dynamics involved.

The controller in the ad delivery system will learn and set a pacing value for every campaign. This is the rate at which that campaign is supposed to be delivered. The pacing is set at a certain value at midnight. Post that it will be adjusted every minute by the PID algorithm so that the delivery for all campaigns remain smooth. If you change your campaign's budget in the afternoon, it will automatically change the delivery pacing to meet the goal.