Calculus is the mathematical study of continuous change. It has two main branches: differential calculus, which studies rates of change like the slope of a curve at any point, and integral calculus, which studies accumulation of quantities.
Differential calculus finds instantaneous rates of change. For example, it can tell us the exact speed of a car at any specific moment. The derivative gives us the slope of a curve at any point, which represents how fast something is changing at that instant.
Integral calculus finds the accumulation of quantities. For example, it can calculate the total distance traveled when speed varies over time. The integral gives us the area under a curve, which represents the total accumulation.
The Fundamental Theorem of Calculus connects both branches. It shows that differentiation and integration are inverse operations. When you take the derivative of an integral, you get back the original function. This beautiful relationship unifies all of calculus.
To summarize: Calculus is the study of continuous change. Derivatives find instantaneous rates of change, while integrals calculate accumulation. These two branches are connected as inverse operations through the Fundamental Theorem. Calculus is an essential tool for understanding and modeling any dynamic system where things are constantly changing.