Calculus begins with the fundamental concept of limits. A limit tells us what value a function approaches as the input gets arbitrarily close to a specific point. For example, consider the function f(x) equals x squared minus 1, divided by x minus 1. Although this function is undefined at x equals 1, we can see what happens as x approaches 1 from both sides.
Differentiation finds the instantaneous rate of change of a function. The derivative is defined as the limit of the difference quotient as h approaches zero. Geometrically, this gives us the slope of the tangent line to the curve at any point. Watch as the secant line approaches the tangent line when h gets smaller.
Integration is the reverse of differentiation and finds the area under a curve. We approximate this area using rectangles, and as we use more and more rectangles, the approximation becomes exact. The definite integral represents the limit of this sum as the number of rectangles approaches infinity.
The Fundamental Theorem of Calculus is the bridge that connects differentiation and integration. It states that if F is an antiderivative of f, then the derivative of the integral equals the original function, and the definite integral can be computed using antiderivatives. This shows that differentiation and integration are inverse operations.
In summary, calculus rests on four fundamental principles that work together as a unified whole. Limits provide the mathematical foundation for precise definitions. Derivatives use limits to measure instantaneous rates of change. Integrals use limits to find exact areas and accumulations. The Fundamental Theorem elegantly connects derivatives and integrals, showing they are inverse operations. Together, these concepts give us powerful tools to understand and describe change and accumulation throughout mathematics, science, and engineering.