MOOC
The study of functions is the discussion of some of their properties. To do this, we need certain theorems that allow us, for example, to find the variations of the function under study. We've already seen the theorem of finite increments. In this chapter, we study its generalization. When we study a function, we also want to know how it behaves at infinity. Unfortunately, the limits of some functions can be quite complicated to study, so we introduce the Bernoulli-l'Hospital rule and demonstrate it. This rule uses the derivative to determine the hard-to-calculate limits of most quotients. Having studied the behavior of a function at infinity, we now turn our attention to its graphical representation. We ask ourselves the following questions: does the function have a local maximum or minimum? Does the function have a global maximum or minimum? Is the function convex or concave? Are there vertical, horizontal or oblique asymptotes? To answer these questions, we define criteria that will enable us to study the function in detail. Finally, we apply this theory to a function using an example. The limited expansion of a function at a point is a polynomial approximation of this function in the vicinity of this point. Among other things, it makes it easier to find function limits, calculate derivatives or study function properties. We begin with a precise definition of the limited expansion of a function. We also define functions of class Cn. At first, this formula may seem abstract, so we give a graphical interpretation and examples for well-known functions. We continue to manipulate bounded developments by calculating the composition of two bounded developments using two examples. This discussion of bounded developments leads us on to the study of integer series and their radius of convergence. Integer series have remarkable convergence properties. Conversely, certain indefinitely differentiable functions can be written as an integer series in the vicinity of one of their points. This is the Taylor series. We study an example: the geometric series. We study the Taylor series of a function in detail, as well as some counterexamples. Finally, we conclude this discussion with Euler's formula. This formula is based on integer series developments of the exponential function with one complex variable and of the sin and cos functions.