An Introduction to the Study of Calculus

The branch of mathematics studies rates of change

It's all in the formula...
Calculus studies rates of change. PeopleImages / Getty Images

Calculus is the study of rates of change. The principals behind calculus date back centuries to the ancient Greeks, as well as to ancient China, India and even medieval Europe. Before calculus was invented, all math was static: It could only help calculate objects that were perfectly still. But, the universe is constantly moving and changing. No objects—from the stars in space to subatomic particles or cells in the body—are always at rest.

Indeed, just about everything in the universe is constantly moving. Calculus helped to determine how particles, stars, and matter, actually move and change in real time.


Calculus was developed in the latter half of the 17th century by two mathematicians, Gottfried Leibniz and Isaac Newton. Newton first developed calculus and applied it directly to the understanding of physical systems. Independently, Leibniz developed the notations used in calculus. Put simply, while basic math uses operations such as plus, minus, times, and division (+, -, x, and ÷), calculus uses operations that employ functions and integrals to calculate rates of change.

The Story of Mathematics explains the importance of Newton's fundamental theorem of the calculus:

"Unlike the static geometry of the Greeks, calculus allowed mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us, such as the orbits of planets, the motion of fluids, etc."

Using calculus, scientists, astronomers, physicists, mathematicians, and chemists could now chart the orbit of the planets and stars, as well as the path of electrons and protons at the atomic level. Economists to this day use calculus to determine the price elasticity of demand.

Two Types of Calculus

There are two main branches of calculus: differential and integral calculus.

Differential calculus determines the rate of change of a quantity, while integral calculus finds the quantity where the rate of change is known. Differential calculus examines the rates of change of slopes and curves, while integral calculus determines the areas of those curves.

Practical Applications

Calculus has many practical applications in real life, as the website, teachnology explains:

"Among the physical concepts that use concepts of calculus include motion, electricity, heat, light, harmonics, acoustics, astronomy, and dynamics. In fact, even advanced physics concepts including electromagnetism and Einstein's theory of relativity use calculus."

Calculus is also used to calculate the rates of radioactive decay in chemistry, and even to predict birth and death rates, the science website notes. Economists use calculus to predict supply, demand, and maximum potential profits. Supply and demand are, after all, essentially charted on a curve—and an ever-changing curve at that.

Economists refer to this ever-changing curve as "elastic," and the actions of the curve as "elasticity." To calculate an exact measure of elasticity at a particular point on a supply or demand curve, you need to think about infinitesimally small changes in price and, as a result, incorporate mathematical derivatives into your elasticity formulas.

Calculus allows you to determine specific points on that ever-changing supply-and-demand curve.