# Practical Applications of Calculus

Instructor: Sudha Aravindan

Sudha is currently an Information Technology Specialist and a EdD student at the University of Delaware.

We know that calculus, the study of how things change, is an important branch of mathematics. In fact, it's got some amazing applications outside the classroom. In this lesson we will learn about some the applications of calculus.

## What is Calculus?

Calculus is the branch of mathematics that studies how things change, and what the effects of changes are on a system. Calculus is derived from the Latin word 'calculus', which means 'a small pebble used for counting'.

Calculus has two main branches: differential calculus and integral calculus. Differential calculus studies how things change when considering the whole to be made up of small quantities. In other words, differential calculus deals with all the small components or parts that make up the whole system. Integral calculus is complementary to differential calculus: it studies the whole system as an accumulation of small quantities or components.

In simple terms, differential calculus breaks things up into smaller quantities to determine how small changes affects the whole. Integral calculus puts together small quantities to determine how the whole is formed from the small quantities and is affected by the small changes.

Mathematicians use calculus to find information about changes to a system or whole, in addition to many other uses. These findings are useful in all sorts of different areas; that means calculus has many applications in real life.

## Calculus in Everyday Life

### The Swimming Pool

Consider a swimming pool that is filled by water dripping from a tap. At any point in time, you can theoretically add up all small drops of find the total volume of water in the tank. This is an application of integral calculus, because it uses small droplets of water to determine the whole volume of water at any point in time.

On the other hand, if you would like to know the rate at which water is filling the pool, you could count the number of drops per second that drip from the tap. This is differential calculus, since it considers how small droplets fills the tank without actually determining the total volume of water in the tank.

### The Volume of a Box

Imagine you have a piece of cardboard, of which you already know the length and width. You want to build a box with the cardboard, and you're trying to determine what dimensions will give you the maximum possible volume for the box.

This can be solved using differential calculus. Start with a small box with a length, width, and height of 1 unit (1x1x1). Keep increasing the height, width and length in small increments until you find the ideal dimensions using the piece of cardboard where the volume will be the maximum. Here you are considering small changes in dimension to identify the ideal dimension.

## Calculus in the Business World

### The Grocery Store

A grocery store owner wants to find the optimal time to display berries. She knows that berries will sell better if they're on display at a certain time, but since they're perishable she doesn't want to leave them out all day to ensure they're on display at the ideal moment. How can the owner find the right time to display her product?

This can be solved using differential calculus. The storekeeper could record the quantity of the product sold at different times during the day, then identify the time at which the product sold the most. Here, she is considering small increments of time to identify the rate of sales.

### The Assembly Line

You run a factory which manufactures a product in 50 steps; that is, there are 50 tasks to perform to create one unit. Using integral calculus, you can find out how long it takes to build each unit. Just find the time for each task, then add the individual times to calculate the total time for all 50 tasks.

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