Welcome to calculus! Calculus is the mathematics of change and accumulation. It has two fundamental operations: differentiation, which finds the rate of change, and integration, which finds accumulation. These operations are inverse to each other. Let me show you a simple example with the function f of x equals one half x squared.
Differentiation finds the instantaneous rate of change of a function. Geometrically, this means finding the slope of the tangent line at any point on the curve. The mathematical definition uses limits: f prime of x equals the limit as h approaches zero of f of x plus h minus f of x, all divided by h. For example, if f of x equals x squared, then f prime of x equals 2x. Watch how the slope changes as we move along the curve.
Integration finds the accumulation of quantities over an interval. Geometrically, this means finding the area under a curve. We write this as the integral from a to b of f of x dx. Integration is the inverse operation of differentiation. We can approximate the area using rectangles - the more rectangles we use, the better our approximation becomes. For example, the integral of x squared dx equals x cubed over 3 plus a constant C.
The Fundamental Theorem of Calculus connects differentiation and integration as inverse operations. Part one states that the derivative of an integral equals the original function. Part two states that the integral of a derivative gives us the change in the original function. This powerful theorem shows that these two operations undo each other. Watch how as we change the upper limit of integration, the area changes, and its rate of change equals the function value at that point.
To summarize what we have learned about calculus: Differentiation finds the instantaneous rate of change of a function, while integration finds the accumulation of quantities over an interval. These two operations are inverse to each other, connected by the Fundamental Theorem of Calculus. Understanding these concepts is essential for analyzing change and motion in mathematics, physics, engineering, and economics.