Exercises of Function Study E-Book

Simone Malacrida

• Herausgeber: Simone Malacrida
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2022

Beschreibung

In this book, exercises are carried out regarding the following mathematical topics:
study of real variable functions
finding domains, asymptotes, points of discontinuity, stationary points, and inflections
Initial theoretical hints are also presented to make the performance of the exercises understood

Leseprobe

Table of Contents

"Exercises of Function Study"

INTRODUCTION

THEORETICAL OUTLINE

EXERCISES

"Exercises of Function Study"

SIMONE MALACRIDA

In this book, exercises are carried out regarding the following mathematical topics:

study of real variable functions

finding domains, asymptotes, points of discontinuity, stationary points, and inflections

Initial theoretical hints are also presented to make the performance of the exercises understood

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

ANALYTICAL INDEX

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INTRODUCTION

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I – THEORETICAL OUTLINE

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II – EXERCISES

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

INTRODUCTION

In this workbook some examples of calculations related to the study of functions of real variable are carried out.

One of the major results of the mathematical analysis is given precisely by a detailed study of each type of function, being able to determine not only the local characteristics of the topology at the basis of the analysis itself, but also the global ones.

We are thus able to graphically visualize, in the Cartesian plane, each function regardless of its formalization.

As we will see, the concepts of limit and derivative are necessary, as well as what has been learned from the foundations of mathematical analysis.

It goes without saying that the study of functions with real variables represents a significant leap forward also from an application point of view.

Each mathematical function, in fact, can be put in correspondence with various technological and physical problems, as well as applications.

What is presented in this workbook is generally addressed during the last year of scientific high schools and, in a more in-depth way, in the first course of mathematical analysis at university level.

I

THEORETICAL OUTLINE

The study of a function with a real variable can be represented by an algorithm, divided into three distinct phases.

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Phase 1: Information from the starting function

a) Definition set : the domain of the function is calculated by remembering that the denominators must be different from zero, theroots of roots with even index must be greater than or equal to zero, the argument of the logarithms must be positive, the argument of tangents must be different from multiples of 90°, the arcsine and arccosine functions are between -1 and +1.

b) Determination of symmetries : a function is symmetrical with respect to the ordinate axis if it does not change its value by replacing the variable with its opposite. If, on the other hand, it assumes the opposite value, it is symmetrical with respect to the origin. A function can be symmetrical with respect to any point of the Cartesian plane or to any axis parallel to the ordinate axis. In this case, a coordinate change is applied by translating the Cartesian axes.

c) Determination of periodicity : a function is periodic if it repeats itself identically after a certain period. Typical periodic functions are the trigonometric functions.

d) Intersections with the coordinate axes : the intersection with the ordinate axis is obtained by canceling the independent variable. The intersection with the abscissa axis is obtained by solving the equation f(x)=0.

e) Sign of the function : within the definition set, the positivity set of the function is obtained by solving the inequality f(x)>0.

f) Calculation of the limits on the frontier : the limits at the frontier of the definition set are calculated. If the domain is unbounded above or below this translates into calculating limits at infinity. If it happens that these limits are finite, it means that the function has horizontal asymptotes at infinity given by horizontal lines with ordinate equal to the value of the limit. In the case in which these limits are infinite, if the function divided by x has a limit at infinity which is finite, there are oblique asymptotes whose angular coefficient is given by the value of this limit. The horizontal or oblique asymptotes can be present in a maximum number of two or not be present. In the figure there are two examples: