Exercises of Derivatives 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:
derivatives and differential calculus
geometric and physical applications of derivatives
remarkable theorems of differential calculus
Initial theoretical hints are also presented to make the performance of the exercises understood

Leseprobe

Table of Contents

“Exercises of Derivatives”

INTRODUCTION

DIFFERENTIAL CALCULATION

REMARKABLE THEOREMS

GEOMETRIC AND PHYSICAL APPLICATIONS

“Exercises of Derivatives”

SIMONE MALACRIDA

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

derivatives and differential calculus

geometric and physical applications of derivatives

remarkable theorems of differential calculus

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

––––––––

INTRODUCTION

––––––––

I – DIFFERENTIAL CALCULATION

Exercise1

Exercise 2

Exercise 3

Exercise 4

Exercises or 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Exercise 12

Exercise 13

Exercise 14

Exercise 1 5

Exercise 16

Exercise 1 7

Exercise 18

Exercise 19

Exercise 20

––––––––

II – REMARKABLE THEOREMS

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

––––––––

III - GEOMETRIC AND PHYSICAL APPLICATIONS

Exercise 1

Exercise 2

Exercise 3

INTRODUCTION

In this workbook, some examples of calculus relating to derivatives and differential calculus are carried out.

Furthermore, the main theorems used in differential calculus and the geometrical and physical implications of such calculus are presented.

Derivatives and differential calculus play a primary role in mathematical analysis, not only for their physical and geometrical applications, but for the very nature of these operations.

With the derivatives it is possible to carry out an in-depth study of the functions and to construct equations (differentials in fact) which are the basis of the description of many physical and natural phenomena.

In order to understand in more detail what is explained in the resolution of the exercises, the reference theoretical context is recalled at the beginning of each chapter.

What is presented in this workbook is generally addressed during the last year of scientific high schools and in the first courses of mathematical analysis at university level.

I

DIFFERENTIAL CALCULATION

Given a real function of real variable, we call the increment of the function around a given point, the following quantity:

While the increment of the independent variable is given by h.

The incremental ratio is defined as the ratio of the increments:

If h is positive, we speak of right incremental ratio, if it is negative, of left incremental ratio.

The limit as h tends to zero of the incremental ratio is called the derivative and is indicated in various ways.

The first notation is that of Lagrange, the second is used in physics, the third is the notation of Cauchy-Euler, the fourth is that of Leibnitz, the last is that of Newton.

––––––––

The derivative calculated in the right neighborhood is called the right derivative and the one calculated in the left neighborhood is called the left derivative.

A function is differentiable at a point if and only if there are finite left and right limits of the incremental ratio and these limits are equal.

A function is differentiable everywhere, or in an interval, if it is differentiable at any point, or at any point in the interval.

The function that assumes at each point the value of the derivative at that point is called a derivative function , precisely because it derives from the starting function.

The derivative of the derivative is called the second derivative and so on up to the n-th derivative which is indicated as follows:

Having used the previous notations to indicate the nth derivative.

A necessary condition for the derivability of a function is its continuity.