Solved problems in Calculus

In this short course some calculus exercises are solved, in particular on: derivatives, integrals, limits, calculation of areas, arc length, volumes of revolution.

The problems are solved step by step. The prior knowledge requirements are pretty basic. Previous knowledge of the concepts: functions, trigonometry, simple high school algebra would be useful.

In this course Calculus is explained by focusing on understanding the key concepts rather than resorting to rote learning. The process of reasoning by using mathematics is the primary objective of the course, and not simply being able to do computations.

Let’s summarize here in the following the two fundamental concepts: differential and integral calculus.

Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function.

Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration.

The indefinite integral, also known as the antiderivative, is the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F.

The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral involves the limit of a sum of areas of rectangles, called a Riemann sum.