Calculus is a fundamental branch of mathematics that studies change and motion. It provides powerful tools to analyze rates of change, find areas under curves, solve optimization problems, and understand dynamic systems. The curve you see demonstrates how calculus helps us find tangent lines that show instantaneous rates of change at any point.
Differential calculus focuses on rates of change and slopes of curves. The derivative measures the instantaneous rate of change at any point. We can visualize this by starting with a secant line between two points, then making the distance smaller and smaller until we get the tangent line, which represents the derivative.
Integral calculus deals with accumulation and finding areas under curves. We can approximate the area using rectangles, and as we use more and more rectangles, we get closer to the exact area. The integral represents this limiting process, giving us the precise area under the curve.
The Fundamental Theorem of Calculus is the bridge that connects differential and integral calculus. It shows that differentiation and integration are inverse operations. As we change the upper limit of integration, the rate of change of the area equals the height of the function at that point.