Calculus is the mathematical study of continuous change and motion. It provides powerful tools to analyze rates of change, areas under curves, and solve optimization problems. Here we see a parabola with its tangent line, demonstrating how calculus helps us understand instantaneous rates of change.
Differential calculus studies rates of change and slopes of curves. The derivative measures the instantaneous rate of change at any point. Here we see how a secant line approaches a tangent line as the distance between two points gets smaller, illustrating the concept of a limit in the derivative definition.
Integral calculus studies accumulation and areas under curves. The integral measures total accumulation over an interval. Here we see how Riemann rectangles approximate the area under a curve, and as we use more rectangles, the approximation becomes more accurate, approaching the true integral value.
The Fundamental Theorem of Calculus is the bridge connecting differential and integral calculus. It shows that differentiation and integration are inverse operations. As we move the boundary of integration, the rate at which the area changes equals the height of the function at that point.
To summarize what we have learned: Calculus is the mathematical study of continuous change, divided into differential and integral calculus. Differential calculus analyzes rates of change, while integral calculus measures accumulation. The Fundamental Theorem beautifully connects these two branches, making calculus an essential tool for science, engineering, and economics.