Esericizi OFA di Analisi matematica 1

First and second degree disequations

First degree equations

First degree equations solutions

Second degree equations

Second degree equations solutions

• Two solutions (distinct)
• Two solutions (coincident)
• Two solutions (coincident)
• Two solutions (coincident)
• No solutions
• No solutions
• Two solutions (distinct)
• Two solutions (coincident)
• Two solutions (coincident)
• Two solutions (distinct)

Irrational equations

Solutions

Equations with exponential terms

Equations with logarithms

Equations with logarithms: solutions

Absolute values

Goniometric disequations

(Formulas: https://en.wikipedia.org/wiki/Trigonometry, https://www.math.it/formulario/goniometria.htm)

• Cos(x) > 0
• Sin(x) > 1
• Tg(2x) <= 0
• (1-2sin(x)) (2cos(x) + √3) <= 0
• 23 sen(x) cos(x) - √3 cos (x) < 3 sen(x) - √3cos(x)

(Solutions in the notes) https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(OpenStax)/07%3A_Trigonometric_Identities_and_Equations/7.E%3A_Trigonometric_Identities_and_Equations_(Exercises)

OFA recovery

Lesson 1

Academic year 2020-2021

Giorgia Marcellino

Email: giorgia.marcellino@studenti.unipd.it

Translated by Niccolò Turcato (niccolo.turcato@studenti.unipd.it)

Informazioni

• Check if you have the OFA and try to recover it as soon as possible (November or December) to be able to take the exams from the curriculum
• For any problems write to the engineering secretariat or your tutor.

Set theory notation

A set can be defined as a collection of objects, called elements of the set. A = { a, b, c }.

• Brace Insieme – lettera maiuscola
• Elemento – lettera minuscola

Finite and infinite sets

• Finite sets: contain a finite number of elements
• Infinite sets: contain an infinite number of elements

N = { 0, 1, 2, … }

D = { n ∈ N : n = 2k + 1, k ∈ N }

Insiemi simbologia

a ∈ A → belongs to A

d ∉ A → d does not belong to A

A = B → A is equal to B (they contain the same elements)

C ⊂ A → C is a proper subset of / is properly contained in A

D ⊆ A → D is a subset of / is contained in A (can be equal)

∅ → empty set

U → Environment set or universe, contains all possible elements

Insiemi matches

• Unique correspondence
• Two-way correspondence

There is a unique correspondence between two sets A and B if each element a of A can be associated with one and only one element b of B through a specific relation.

Between two sets A and B there is a two-way correspondence if each element a of A can be associated with one and only one element b of B and vice versa.

Insiemi operations

A = { 0, 1, 2, 6, 8, 9 } B = { 0, 1, 3, 4, 7, 9 } U = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Union A ∪ B → elements that belong to at least one of the two sets (or both)

A ∪ B = { 0, 1, 2, 3, 4, 6, 7, 8, 9 }

Intersection A ∩ B → elements that belong to both A and B (simultaneously)

A ∩ B = { 0, 1, 9 }

If A ∩ B = ∅, the two sets are said to be disjointed.

Insiemi operations

A = { 0, 1, 2, 6, 8, 9 } B = { 0, 1, 3, 4, 7, 9 } U = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Difference A\B or A-B → elements of A that do not belong to B

A \ B = { 2, 6, 8 }

B \ A = { 3, 4, 7 }

Complementary B or B^c → given a universe set U, we define E as a complementary set of a set B (B^c), the set given by all elements of U that do not belong to B, that is, U-B

B^c = U – B = { 2, 5, 6, 8 }

Insiemi operations

A = { 0, 1, 2, 6, 8, 9 } B = { 0, 1, 3, 4, 7, 9 } U = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Cartesian product → The Cartesian product of A and B (not necessarily different) is given by the set of pairs of points (a,b) such that a ∈ A and b ∈ B. If A = ∅ and B = ∅ then A × B = ∅

Numerical sets

Meaning and Symbols

• Natural Numbers
• Integers
• Positive integers (not including zero)
• Negative integers (not including zero)
• Rational Numbers
• Positive Rational Numbers (not including zero)
• Negative Rational Numbers (not including zero)
• Real Numbers
• Positive Real Numbers (not including zero)
• Negative Real Numbers (not including zero)
• Complex Numbers

Numerical Sets

(Natural) = { 0, 1, 2, 3 ….. }

• Ordered set of numbers
• Contains all positive integers
• Only the following operations are defined: + x → (m,n) → m+n addition ∙ x → (m,n) → m ∙ n multiplication ^ x → (m,n) → m elevation to power

Insiemi numerici

(Integers) = { … -3, -2, -1, 0, 1, 2, 3 … }

• Ordered set of numbers
• Contains all positive and negative integers
• The following operations are defined: - + x → (m,n) → m+n addition - x → (m,n) → m-n subtraction ∙ x → (m,n) → m ∙ n multiplication ^ x → (m,n) → "m" ^ "n" elevation to power

Numerical Sets

(Rational) = { , ≠ 0 }

Rational numbers always have a finite or periodic decimal representation: 3 11 = 0.75 = 0,244 45

• The following operations are defined: + x → (m,n) → m+n addition - x → (m,n) → m-n subtraction ∙ x → (m,n) → m ∙ n multiplication m/ x → (m,n) → division n m n ^ x → (m,n) → elevation to power

Numerical Sets

(Irrationals) - unlimited non-periodic decimal numbers that cannot be written as a fraction between two integers

• Examples of irrational numbers are the roots of non-perfect squares and the value of π, π

Numerical Sets

(Reals) - The union of rational and irrational numbers is the set of real numbers.

• With the real numbers it is possible to carry out the operations of: 1. Addition and Subtraction, 2. Multiplication and Division, 3. Power Elevation and Root Extraction (odd index of all numbers and even index of only positive numbers)

Numbers

Divisibility Criteria

Divisibility is the property of an integer to be divisible by another. Divisibility criteria are used to determine whether a number n is divisible by another number m without dividing. Prime numbers are defined as those numbers that are divisible only by 1 and by themselves.

Decomposition into Prime Factors

Decompose a number n into prime factors means determining the prime numbers that multiplied between them damage as result the number itself. You start by dividing the number by its smaller divider and proceed gradually with the numbers obtained up to 1. 108 = 2 x 3³

MCD e mcm

MCD = maximum common divisor

It is the largest common divisor of the numbers considered → by breaking down the two numbers n and m into prime factors and taking the product of the common prime factors (each taken only once), with the minimum exponent with which they appear.

mcm = minimum common multiple

It is the smallest multiple of the numbers considered → by breaking down the two numbers n and m into prime factors and taking the product of the common and uncommon prime factors (each taken only once), with the maximum exponent with which they appear.

MCD(60,75,210) and mcm(60,75,210)

Powers

Powers n N exponent

a R , n N the power of a number is the product of the number for itself as many times as the exponent indicates.

Properties of Powers

Remarkable products with powers

Powers

Integer powers n Z

Properties of Powers

Powers with Rational Exponent

Properties of Powers

Roots

To give a correct root definition of a real number a, we must distinguish two cases:

• If n is odd: we say n-th root of a that number b that elevated to the natural number n gives us back a, that is: if n is odd n√a = b means to find a number b that elevated to n gives a, that is "b"^"n" = a
• If n is even: we say n-th root of a that positive number b that elevated to n gives us back a, that is: if n is equal n √ a = b means to find a positive number b that elevated to n gives a, that is "b"^"n" = a and such that a ≥ 0 and b ≥ 0 . In any case, the root of even order of a number is always considered positive by convention.

In both cases a is defined as rooting, n root and √ is the root symbol.

Roots of Positive Reals

Properties of Roots

Equations

An equation is an equality of literals verified only for particular values attributed to the letter(s) in it. 8x + 5x = 4x + 13 2

The letter in the equation is called unknown. Terms that do not contain an unknown are called known terms. The degree of an equation is given by the degree of the monomial of maximum degree. The values that make the equation true are called solutions or roots of the equation. An equation is defined in integer terms when only coefficients or known integer terms appear; fractional terms when fractional coefficients or known terms appear in them.

Equations

Principles of Equivalence

Equations of the First Degree to an Unknown

ax = b

An equation written in the form is said to be in normal form. To solve a reduced equation in normal form it is enough to divide both members of the equation by the coefficient of x. x = x = If the equation is not reduced to normal form, the principles of equivalence must be applied appropriately so as to reduce the equation in normal form.

Equations

Second Degree Equations

An equation of the 2nd degree is said to be written in normal or canonical form if it is in the form ax² + bx + c = 0 with real a, b and c and a ≠ 0.

It is called a discriminant of an equation of 2nd degree, and is denoted by Δ, the number b² - 4ac.

The solutions are derived from the formula   b x_{1, 2} 2 a

Equations

• If Δ > 0 the solutions are 2 and distinct – S = {(-b + √Δ) / 2a, (-b - √Δ) / 2a}
• If Δ = 0 the solutions are 2 coincident – S = {-b / 2a}
• If Δ < 0 the solutions do not exist – S = {∅}

Second Degree Equations – Special Cases

Case I

Given the equation ax² = b, if a ≠ 0, then x = b/a, x = a

Case II

Given the equation (x - a) ∙ (x - b) = 0 for the law of cancellation of the product you have x a = 0 → x = a x b = 0 → x = b

Solutions are the numbers in the set { a, b }

Consider sets A={0,2,4}, B={1,2,3}, C={x:x , x is odd}

(A U B) ∩ B = ? = ?

The number 308 can be divided by …?

MCD and mcm between 2205 and 525 are …?

Simplify with Compute the addition and the product of the solutions of equation: Which of the following equations is true for any value of x and y in the Real set of numbers? = …?

Compute the value of the following expression: Compute the decomposition in prime factors of 30¹³

A is the set of positive odd or prime integer numbers, which of the following is true?

In a group of 100 people, 51 speak English, 36 speak French and 12 speak both English and French. How many of them do not speak neither English or French?

Given real number x, = …?

Sort this sequence of numbers in ascending order:

OFA recovery

Lesson 2

Academic year 2020-2021

Giorgia Marcellino

Email: giorgia.marcellino@studenti.unipd.it

Translated by Niccolò Turcato (niccolo.turcato@studenti.unipd.it)

Informazioni

• Check if you have the OFA and try to recover it as soon as possible (November or December) to be able to take the exams from the curriculum
• If you do not recover you cannot register for the exams in January!
• For any problems write to the engineering secretariat or your tutor.

Inequalities

Definitions

An inequality is an inequality between literal expressions verified only for particular values attributed to the letter(s) in it.

The values that make inequality