Mathematical Analysis 1: theory and solved exercises
5/5
()
About this ebook
Read more from Alessio Mangoni
Fundamentals of physics Rating: 2 out of 5 stars2/5Complex numbers Rating: 0 out of 5 stars0 ratingsThe mathematics of quantum mechanics Rating: 0 out of 5 stars0 ratingsSpecial relativity Rating: 0 out of 5 stars0 ratingsRelativity, decays and electromagnetic fields Rating: 0 out of 5 stars0 ratingsThe Dirac equation Rating: 0 out of 5 stars0 ratings
Related to Mathematical Analysis 1
Related ebooks
Analytic Geometry Rating: 4 out of 5 stars4/5Theory of Approximation Rating: 0 out of 5 stars0 ratingsAnalytic Inequalities Rating: 5 out of 5 stars5/5Differential Geometry Rating: 0 out of 5 stars0 ratingsLinear Algebra Rating: 3 out of 5 stars3/5Matrices and Linear Algebra Rating: 4 out of 5 stars4/5Algebraic Geometry Rating: 0 out of 5 stars0 ratingsFamous Problems of Geometry and How to Solve Them Rating: 5 out of 5 stars5/5Concise Vector Analysis Rating: 0 out of 5 stars0 ratingsDeterminants and Matrices Rating: 3 out of 5 stars3/5Modern Multidimensional Calculus Rating: 0 out of 5 stars0 ratingsAn Introduction to Lebesgue Integration and Fourier Series Rating: 0 out of 5 stars0 ratingsTopology and Geometry for Physicists Rating: 4 out of 5 stars4/5Advanced Calculus Rating: 0 out of 5 stars0 ratingsInfinite Sequences and Series Rating: 3 out of 5 stars3/5Understanding Proof: Explanation, Examples and Solutions of Mathematical Proof Rating: 0 out of 5 stars0 ratingsElementary Functional Analysis Rating: 4 out of 5 stars4/5Topology Essentials Rating: 5 out of 5 stars5/5Complex Integration and Cauchy's Theorem Rating: 0 out of 5 stars0 ratingsEntire Functions Rating: 0 out of 5 stars0 ratingsA First Course in Functional Analysis Rating: 0 out of 5 stars0 ratingsFundamentals of Modern Mathematics: A Practical Review Rating: 0 out of 5 stars0 ratingsAn Introduction to Differential Geometry - With the Use of Tensor Calculus Rating: 4 out of 5 stars4/5Introduction to Algebraic Geometry Rating: 4 out of 5 stars4/5Elementary Vectors: Pergamon International Library Rating: 5 out of 5 stars5/5Constructive Real Analysis Rating: 0 out of 5 stars0 ratingsTable of Integrals, Series, and Products Rating: 4 out of 5 stars4/5Introduction to Matrices and Vectors Rating: 0 out of 5 stars0 ratingsExploring University Mathematics: Lectures Given at Bedford College, London Rating: 0 out of 5 stars0 ratingsComplex Variable Methods in Elasticity Rating: 0 out of 5 stars0 ratings
Mathematics For You
Basic Math & Pre-Algebra For Dummies Rating: 4 out of 5 stars4/5Algebra - The Very Basics Rating: 5 out of 5 stars5/5The Golden Ratio: The Divine Beauty of Mathematics Rating: 5 out of 5 stars5/5Calculus Made Easy Rating: 4 out of 5 stars4/5The Little Book of Mathematical Principles, Theories & Things Rating: 3 out of 5 stars3/5Algebra I Workbook For Dummies Rating: 3 out of 5 stars3/5Geometry For Dummies Rating: 5 out of 5 stars5/5Mental Math Secrets - How To Be a Human Calculator Rating: 5 out of 5 stars5/5Quantum Physics for Beginners Rating: 4 out of 5 stars4/5Basic Math & Pre-Algebra Workbook For Dummies with Online Practice Rating: 4 out of 5 stars4/5The Everything Guide to Algebra: A Step-by-Step Guide to the Basics of Algebra - in Plain English! Rating: 4 out of 5 stars4/5Painless Algebra Rating: 0 out of 5 stars0 ratingsCalculus Essentials For Dummies Rating: 5 out of 5 stars5/5Flatland Rating: 4 out of 5 stars4/5The Everything Everyday Math Book: From Tipping to Taxes, All the Real-World, Everyday Math Skills You Need Rating: 5 out of 5 stars5/5Precalculus: A Self-Teaching Guide Rating: 4 out of 5 stars4/5Mental Math: Tricks To Become A Human Calculator Rating: 5 out of 5 stars5/5Is God a Mathematician? Rating: 4 out of 5 stars4/5The Thirteen Books of the Elements, Vol. 1 Rating: 0 out of 5 stars0 ratingsIntroducing Game Theory: A Graphic Guide Rating: 4 out of 5 stars4/5Game Theory: A Simple Introduction Rating: 4 out of 5 stars4/5Summary of The Black Swan: by Nassim Nicholas Taleb | Includes Analysis Rating: 5 out of 5 stars5/5Relativity: The special and the general theory Rating: 5 out of 5 stars5/5A Mind for Numbers | Summary Rating: 4 out of 5 stars4/5My Best Mathematical and Logic Puzzles Rating: 5 out of 5 stars5/5
Reviews for Mathematical Analysis 1
1 rating0 reviews
Book preview
Mathematical Analysis 1 - Alessio Mangoni
2020
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
Introduction
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.
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