Welcome to Calculus! Calculus is the mathematical study of continuous change and motion. It has two main branches: Differential Calculus, which studies rates of change through derivatives, and Integral Calculus, which studies accumulation through integrals. Here we see a simple function and its derivative represented by the tangent line. Calculus has countless applications in physics, engineering, economics, biology, and many other fields.
Limits are the foundation of calculus. They describe the behavior of functions as inputs approach specific values. The limit of f of x as x approaches a equals L means that as x gets arbitrarily close to a, f of x gets arbitrarily close to L. Here we see how as x approaches 2, the function x squared minus 1 approaches the value 3. This concept of limits is essential for understanding continuity, derivatives, and integrals.
Derivatives measure instantaneous rates of change. The derivative is defined as the limit of the difference quotient as h approaches zero. Geometrically, the derivative represents the slope of the tangent line to the curve at a given point. We can see how the secant line approaches the tangent line as h gets smaller. The power rule is a fundamental tool for finding derivatives: the derivative of x to the n equals n times x to the n minus 1. Derivatives have many applications including finding velocity and acceleration, and solving optimization problems.
Integrals measure accumulation and the area under curves. The definite integral is defined as the limit of Riemann sums as the number of rectangles approaches infinity. We can see how the rectangles approximate the area under the curve, and as we use more rectangles, the approximation becomes more accurate. The Fundamental Theorem of Calculus connects derivatives and integrals, showing that integration and differentiation are inverse operations. To find antiderivatives, we use rules like the power rule for integration.
To summarize what we have learned about calculus: Limits form the foundation of all calculus concepts, providing the mathematical framework for understanding continuous change. Derivatives measure instantaneous rates of change and give us the slope of tangent lines. Integrals calculate accumulation and the area under curves through Riemann sums. The Fundamental Theorem of Calculus elegantly connects derivatives and integrals as inverse operations. Finally, calculus has vast applications across science, engineering, economics, and many other fields, making it one of the most powerful tools in mathematics.