Differential and Integral Calculus Theory and Cases

The purpose of this chapter is to introduce different numeric sets that form the real numbers ℝ, their properties and their geometrical representation in an oriented number line. Some specific properties are addressed, such as irreducible fractions and divisibility.

This section will review the functions and maps operators only using the regularities and differences found in their graphics. A broader description will be presented in (Chap. 5), once the concepts of limit (Chap. 3), continuity (Chap. 4), and differentiability (Chap. 5) are studied.

This chapter introduces the limit operator of a function and a map, its properties and procedure. This operator has an important role in the derivative and integral operators.

This chapter introduces the continuity of functions and maps with the limit operator, it also reviews the concept of continuity geometrically and analytically.

This chapter introduces the concept of differentiability of functions and maps, the properties and rules using the limit operator, the Implicit function theorem, the Inverse function theorem, and L’Hospital’s rule.

This chapter defines the concept of arithmetic sequences or sequences. Here, we will review different types of sequences, their properties, limits, and convergence, as well as the important Cauchy sequences, we will also see an extension of them whose elements are functions, the Sequences of functions.

This chapter focuses on the definition of convergence, its classification and the arithmetic series, particularly Fourier and Taylor series.

This chapter reviews two operators the sequence of functions (ƒκ) and the series of functions Σnκ=1ƒκ In both cases, we will see the concepts of uniform and pointwise convergence to the function they converge and the persistence of the properties these functions have with respect to the function ƒ they converge to.

This chapter studies the antiderivative function and its relation with the integral operator. The Fundamental Theorem of Calculus is reviewed and different integration techniques are exemplified. The series of functions named Fourier series and their application to geometrically approximate a function is also studied here.