Mathematical Analysis 1 E-Book

Alessio Mangoni

• Herausgeber: Alessio Mangoni
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2020

Beschreibung

This book on mathematical analysis is intended for both high school and college students to prepare for math exams. The main topics covered are trigonometry, limits, sequences and series, derivatives, integrals. The text contains graphs, figures and examples of application of the theory with various recall to physics. In the second part of the book we propose and solve various original exercises.

Leseprobe

Contents

Contents

Introduction

Trigonometry

Trigonometric functions

Fundamental relations

Law of sines

Law of cosines

Addition formulas

Prosthaphaeresis formulas

Prosthaphaeresis formulas for the sine

Prosthaphaeresis formulas for the cosine

Prosthaphaeresis formulas for the tangent

Prosthaphaeresis formulas for the cotangent

Werner formulas

First Werner formula

Second Werner formula

Third Werner formula

Chord theorem

Definitions

Relation between angle at the centre and angle at circumference

Area of a generic triangle

Application examples

Limits

Introduction

Accumulation point

Definition of limit

Limit from the right and left

Continuity of a function

Uniqueness of the limit

Limit of a sum or product

Limit of a sum

Limit of a product

Theorem of the permanence of the sign

Squeeze theorem

Notable limits

Notable limits 1

Notable limits 2

Notable limits 3

Notable limits 4

Notable limits 5

Notable limits 6

Sequences and series

Introduction

Definition of sequence

Limit of sequences

Definition of series

Algebraic and geometric sequences

Term n-th

Partial sum n-th

A particular geometric series

Theorems

Comparison test

Asymptotic comparison test

Ratio test

Asymptotic ratio test

Absolute convergence test

Root test

Leibniz's test

Derivatives

Incremental ratio and derivative

Definizione di derivata

Stationary point

Properties of the derivative

Derivatives of elementary functions

Chain rule

Weierstrass theorem

Fermat's theorem on stationary points

Rolle's theorem

Lagrange's theorem

Cauchy's theorem

De L'Hopital's theorem

From mathematics to physics

Integrals

Introduction

Definition of integral

Linearity of the integral

Linearity of the integral

Additivity of the integral

Absolute value theorem

Mean value theorem

Fundamental theorem

Definition

Primitives of elementary functions

Methods of integration

Integration by parts

Integration by substitution

From mathematics to physics

Exercises

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Exercise 12

Exercise 13

Exercise 14

Exercise 15

Exercise 16

Exercise 17

Exercise 18

Exercise 19

Exercise 20

Exercise 21

Exercise 22

Exercise 23

Exercise 24

Exercise 25

Exercise 26

Exercise 27

Exercise 28

Exercise 29

Solutions

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Exercise 12

Exercise 13

Exercise 14

Exercise 15

Exercise 16

Exercise 17

Exercise 18

Exercise 19

Exercise 20

Exercise 21

Exercise 22

Exercise 23

Exercise 24

Exercise 25

Exercise 26

Exercise 27

Exercise 28

Exercise 29

 

Trigonometry

Trigonometric functions

The basic trigonometric functions are the sine and cosine of an angle. Consider the circumference, of unit radius, shown in the figure



Given a point P on the circumference, consider the angle whose measure, in radians, is indicated by x in the figure. We define sine of the angle x, denoted by



the length, without unit of measure, of the ordinate of the point P. This function, has domain and range respectively



and



and is periodic of period



In the figure is shown the plot of the function.



Similarly we define cosine of the angle x, and denoted by



the length, always without unit of measure, of the abscissa of the point P. This function has, like sin x, domain and range



and



and is periodic of period



In the figure is shown the plot of the function.



We observe that the plots of the sine and cosine functions are one coincident with the other translated by an amount of



i.e.



If the circumference in the figure were of not unitary radius R, since the angle x is invariant, the abscissa and the ordinate of the point P would be



and



respectively.

We now define the tangent of the angle x. The tangent of the angle x is defined as follows



This function has domain and range



due to zeroes in the denominator and



is periodic of period



In the figure is shown the plot of the function.



There are also other functions, of secondary use, related to those introduced so far. The secant of the angle x is defined as



This function has domain and range





and is periodic of period



In the figure is shown the plot of the function.



The cosecant of the angle x is defined as



This function has domain and range





and is periodic of period



In the figure is shown the plot of the function.



The cotangent of the angle x is defined as



This function has domain and range





and is periodic of period



In the figure is shown the plot of the function.



It is also possible to define the inverse functions of all these trigonometric functions, however we must restrict their domain to make them bijective, that is a necessary condition for inverting them. Usually the domain is restricted to the ranges



or



We now define the inverse functions of sin x ,cos x and tan x which are called arccosine, arccosine and arctangent.

We define arcsine of x in



denoted by



the angle (unique thanks to the restriction) in the range



such that



This function is the inverse of sin x and has domain and range





In the figure is shown the plot of the function.



We define arccosine of



denoted by



the angle (unique thanks to the restriction) in the range



such that



This function is the inverse of cos x and has domain and range





In the figure is shown the plot of the function.



We define arctangent of x, denoted by



the angle (unique thanks to the restriction) in the range



such that



This function is the inverse of tan x and has domain and range





In the figure is shown the plot of the function.



We now define the inverse functions of sec x and csc x, called arcsecant and arccosecant, respectively. We define arcsecant of



denoted by



the angle (unique thanks to the restriction) in the range



such that



This function is the inverse of sec x and has domain and range





In the figure is shown the plot of the function.



We define arccosecant of