## What is Calculus?

Calculus is the study of “continuous change”, dealing with tangent lines to a curve, or function, and the area under a curve. It deals with change at a specific point, with the instantaneous rate of change at a point being known as the *derivative*, and the area under a curve (between the function and the x-axis) being known as the *integral*. It is one of the first courses of undergraduate higher-level mathematics, and it is split into two different categories, integral and differential calculus.

## Differential and Integral Calculus

Differential calculus is the typical first semester, or half, of an introductory calculus course. This deals with the derivative and introduces the concept of limits, which approximate the value as a function approaches a certain point or infinity. This naturally ties in with the concept of finding a tangent line at a point. This concept of the derivative has applications in many other areas and fields of mathematics.

Integrals are the second half of calculus, the complement of the derivative. Integrals represent the area under a curve, which also has many useful applications in fields like engineering, optimization in computer science, and machine learning.

## Applications

Calculus is known as the gateway to higher mathematics, and it has applications in a wide variety of fields. Calculus is used in every branch of the sciences, including computer science, statistics, and engineering. One particularly interesting application is in data science or machine learning. The fundamental concept of optimization, or how a machine learning model learns, is through calculus. It uses calculus to minimize the errors made by the computer until an almost-optimal model is reached. It also has many applications in engineering as well as physics, particularly anything that involves motion, speed, and acceleration, which can be represented in terms of functions and their derivatives.

## Prerequisites/What you need to know

To understand calculus, you will need to go through the standard high school math curriculum. This is typically the sequence Algebra I -> Algebra 2 -> Trignometry -> Precalculus -> Single-variable Calculus -> Multivariable Calculus

## Resources

- Calculus course on Khanacademy
- MIT Open Courseware – Single Variable Calculus
- The essence of Calculus by 3Blue1Brown on YouTube
- Supplemental Calculus Video by Vihart
- Supplemental Numberphile podcast
- Calculus, Vol. 1 by Tom M. Apostol
- Calculus by Michael Spivak