What is Calculus?

Summary: 

• What is calculus?: Calculus is a branch of mathematics that studies and analyzes continuous change and motion through two fundamental principles—differentiation and integration .
• Definition of calculus: The definition of calculus encompasses the process of taking differences (the measure of change) and taking sums (how things accumulate)
• What is calculus used for?: Calculus is used across numerous fields: studying motion and forces in physics and engineering; model population dynamics in biology; optimizing algorithms in computer science and machine learning; calculating probabilities and expected values in statistics; and analyzing marginal costs and revenues in economics.
• What jobs use calculus? Jobs that use calculus include physicists, engineers, biologists, computer scientists, data scientists, statisticians, economists, and professionals in technology fields involving optimization and machine learning.

Why Study Calculus?

As an advanced branch of mathematics, calculus can be an intimidating subject. But its core tenet is pretty simple: Calculus is a branch of mathematics that studies and analyzes continuous change and motion. 

It’s easy to see, then, that calculus is a fundamental tool, easily applied to real-world situations. Calculus is also an essential framework for a variety of careers, from physics and engineering to economics and data science. 

For some students, calculus can be an instrumental course in future success: 89% of college admissions officers agree that students who have taken calculus in high school are more likely to succeed in college. And 47% agreed that not taking calculus may limit a student’s options for college.¹

Whether you’re a student grasping its basics or an educator seeking effective educational tools, understanding what calculus is is essential. Ahead, we’ll define calculus, covering its history and fundamental principles; take a look at the different types of calculus and their key terminologies; and explore what calculus is used for and what jobs use calculus. 

Definition of Calculus

Broadly, calculus centers on the study of change and motion. But more specifically, the definition of calculus encompasses the process of taking differences—the measure of change—and taking sums—how things accumulate.

Calculus has two major branches: differential calculus and integral calculus. Within those branches lie calculus’s basic principles: limits, derivatives, and integrals. Limits explore how functions behave as they approach certain points; derivatives offer insights into the rate of change of a function; and integrals help quantify accumulations.

In this way, calculus provides a framework for predicting rates of change. Differential calculus, for example, allows us to calculate rocket trajectories for launching spacecraft into orbit. Integral calculus, on the other hand, can be applied to calculating the center of mass of irregular structures, like bridges.² Understanding what calculus is used for, then, is crucial for both students and professionals, as a way to tackle complex problems and predict future outcomes. 

History of Calculus

Calculus evolved over many centuries in various parts of the world, including ancient Greece, the Middle East, India, China, Japan, and Western Europe. Today, we typically credit the development of modern calculus to two 17th-century figures: Sir Isaac Newton and Gottfried Wilhelm Leibniz.

Newton and Leibniz developed calculus independently from each other, publishing their first findings around the same time in two different countries (England and Germany, respectively). Newton’s focus was limits and motion; Leibniz introduced notations—symbols for derivatives, integrals, limits, and functions. Together, their contributions laid the groundwork for applying calculus in various fields, including physics and engineering.

In the 18th and 19th centuries, mathematicians like Leonard Euler and Augustin-Louis Cauchy further refined and expanded the principles of calculus. Calculus has since become an essential tool in numerous scientific disciplines, extending to fields as diverse as economics, biology, and computer science. And its applications continue to grow, underscoring the importance of calculus in modern science and technology.

Two Fundamental Principles of Calculus

As mentioned above, calculus is built on two fundamental principles: differences and sums. Differences are known as differentiation, or how quantities change. Differentiation is useful for calculating instantaneous rates, like an object's speed at a specific moment. Sums are known as integration, and focus on how quantities accumulate. Integration is useful for calculating the cumulative effect of continuous change. 

The fundamental theorem connects the concepts of differentiation and integration, illustrating the two as inverse processes. This theorem is what makes calculus so powerful, as it shows that the process of finding rates of change (differentiation) and the process of finding accumulation (integration) are fundamentally connected.

Types of Calculus

We have a solid definition of calculus and its two fundamental principles. But what are the types of calculus?

The two traditional types of calculus align with its fundamental principles: differential calculus and integral calculus. Differential calculus deals with the rates at which things change and the slopes of curves. Integral calculus focuses on the accumulation of quantities, calculating the area under curves. 

Multivariable calculus focuses on functions that involve multiple variables instead of a single variable. And numerical calculus, also known as numerical analysis, studies algorithms that use approximation methods. 

Key Terminology in Calculus

As an advanced branch of mathematics, calculus has many of its own unique terms. These include: 

• Variables and constants: Variables represent changing quantities, with “x” being the most common variable. Alternatively, constants are values that remain fixed. 
• Functions: Functions describe relationships between variables, typically denoted as “f(x).” 
• Symbols: An oft-used symbol in calculus is the integral symbol, or “∫” for integration, and the derivative symbol, or “d/dx” for differentiation.
• Limits and continuity: Limits relate to how functions behave as they approach certain points. Continuity refers to a function that isn’t broken or interrupted. 

Understanding these key terms is the first step in understanding not only what calculus is, but also in creating a base-level understanding of its fundamentals. 

What is Calculus Used For and What Jobs Use Calculus

Outside of the classroom, what is calculus used for? And what jobs use calculus?

The good news: Calculus has far-reaching applications across a variety of fields. Its most obvious application is in physics and engineering, where calculus is used to study the motion of objects under various forces and conditions, optimize design aspects such as shape, size, force, stress, heat, and motion, and calculate rates of change. 

Calculus may also be used in biology, helping to model population dynamics and reaction rates. It is also useful in economics, where it can be used to analyze marginal costs and revenues.

Outside of the above career paths, calculus is also useful for jobs in technology and computer science, where algorithms are crucial for optimization. And although it may be a bit of a surprise, the principles of calculus often also contribute to many other unexpected fields: statistics, economics, and even art and music theory. 

Calculus, at its core, is the mathematical study of continuous change and motion—a framework built on the fundamental principles of differentiation and integration. From its history in Ancient Greece, to its independent development by Newton and Leibniz in the 17th century, to its refinement by mathematicians like Euler and Cauchy, calculus has evolved into an indispensable tool across scientific, technological, and analytical disciplines. 

Understanding what calculus is and what calculus is used for reveals its remarkable versatility. As the applications of calculus continue to expand into new fields and technologies, its importance in modern science and innovation only grows stronger.