Welcome to the world of calculus. Calculus is a branch of mathematics that studies change and accumulation. It focuses on concepts like rates of change, slopes of curves, areas, volumes, and infinite sequences and series. The graph shows a parabola representing a function. The red line is a tangent to the curve, showing the instantaneous rate of change at that point - this is what differential calculus studies. The green shaded area represents the area under the curve, which is a fundamental concept in integral calculus.
Differential calculus focuses on the rate of change of functions. The derivative represents the instantaneous rate of change of a function at a specific point. Geometrically, it's the slope of the tangent line to the curve at that point. In physics, velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity. The formal definition of a derivative is the limit of the difference quotient as h approaches zero. On the graph, we can see how a secant line approaches the tangent line as the interval gets smaller, illustrating the concept of the derivative as a limit.
Integral calculus focuses on accumulation and area. The definite integral represents the area under a curve between two points. The indefinite integral is the inverse operation of differentiation. In physics, displacement is the integral of velocity with respect to time, and work is the integral of force over displacement. The formal definition of a definite integral is the limit of Riemann sums as the number of partitions approaches infinity. On the graph, we can see how increasing the number of rectangles gives a better approximation of the area under the curve, illustrating the concept of the integral as a limit of sums.
The Fundamental Theorem of Calculus establishes the connection between differentiation and integration. The first part states that if F is an antiderivative of f, then the definite integral of f from a to b equals F(b) minus F(a). The second part states that if f is a continuous function, then the derivative of the integral of f from a to x with respect to x equals f(x). This theorem shows that differentiation and integration are inverse operations. On the graph, as we move the upper limit of integration, the area under the curve changes, and the rate of change of this area is precisely the value of the function at that point.
To summarize what we've learned about calculus: Calculus is a branch of mathematics that studies rates of change and accumulation. Differential calculus focuses on instantaneous rates of change, with the derivative as its central concept. Integral calculus studies areas under curves and accumulation, with the integral as its key concept. The Fundamental Theorem of Calculus reveals that differentiation and integration are inverse operations. Calculus has widespread applications in physics, engineering, economics, and many other fields. It provides powerful tools for modeling and solving real-world problems involving change and accumulation.