Calculus, originally called infinitesimal calculus, is the mathematical study of continuous change. It has two major branches: **Differential Calculus**, which concerns instantaneous rates of change and the slopes of curves, and **Integral Calculus**, which concerns the accumulation of quantities and the areas under or between curves. These two branches are linked by the **Fundamental Theorem of Calculus**, which demonstrates that differentiation and integration are inverse operations. Calculus is built on the fundamental concepts of limits, functions, derivatives, integrals, and infinite series.
While ancient mathematicians developed ideas that were foundational to calculus (such as Archimedes' method of exhaustion for calculating areas), modern calculus was developed independently in the late 17th century by Sir Isaac Newton and Gottfried Wilhelm Leibniz. Newton developed his 'method of fluxions' to solve problems in physics, particularly mechanics, while Leibniz focused on developing a more formal and versatile notation. The ensuing Newton–Leibniz calculus controversy over who deserved credit was a famous intellectual dispute. Ultimately, both are credited as co-inventors, as their work provided the systematic foundation upon which the field is built.
Calculus is a gateway to higher mathematics and a foundational tool in nearly every branch of the physical sciences, computer science, statistics, engineering, economics, business, and medicine. It is indispensable in physics for describing motion, electricity, and magnetism. In engineering, it's used for designing structures and understanding dynamic systems. In economics, it's used to find optimal solutions for maximizing profit or minimizing cost. Computer graphics, machine learning, and data analysis all rely heavily on calculus for optimization and modeling. Its ability to model and solve problems involving continuous change makes it one of the most significant achievements in the history of science.