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References

Teaching Language

Course Content

Suggested readings

Learning Objectives

Prerequisites

Teaching Methods

Further information

Type of Assessment

Course program

Belonging Department

Scienze per l'Economia e l'Impresa

Course Type

Attività formativa monodisciplinare

Scientific Area

SECS-S/06 - METODI MATEMATICI DELL'ECONOMIA E DELLE SCIENZE ATTUARIALI E FINANZIARIE

Course year

First year - First Term

Teaching Term

dal 13/09/2018 al 07/12/2018

Attendance required

No

Credits

9

Type of Evaluation

Voto Finale

Teaching Hours

72

Course Content

Course program

Lectureship

- Last name/s A-BO MENICUCCI DOMENICO
- Last name/s BP-C GORI MICHELE
- Last name/s D-GE MENICUCCI DOMENICO
- Last name/s GF-L BIANCALANI FRANCESCO
- Last name/s GF-L MANCINO MARIA ELVIRA
- Last name/s M-P QUARTIERI FEDERICO
- Last name/s M-P VILLANACCI ANTONIO
- Last name/s Q-Z COLUCCI DOMENICO

Italian

Italian

Italian

Italian

Italian

Italian

Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential calculus. Introduction to functions of several variables.

Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential calculus. Introduction to functions of several variables.

Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential calculus. Introduction to functions of several variables.

Enrico Giusti, Elementi di Analisi Matematica, 2008, Bollati Boringhieri.

On the part "Introduction to functions of several variables", instructors will provide reading material.

On the part "Introduction to functions of several variables", instructors will provide reading material.

Enrico Giusti, Elementi di Analisi Matematica, 2008, Bollati Boringhieri.

On the part "Introduction to functions of several variables", instructors will provide reading material.

On the part "Introduction to functions of several variables", instructors will provide reading material.

Enrico Giusti, Elementi di Analisi Matematica, 2008, Bollati Boringhieri.

On the part "Introduction to functions of several variables", instructors will provide reading material.

On the part "Introduction to functions of several variables", instructors will provide reading material.

On the part "Introduction to functions of several variables", instructors will provide reading material.

On the part "Introduction to functions of several variables", instructors will provide reading material.

On the part "Introduction to functions of several variables", instructors will provide reading material.

The goal of this course is to provide mathematical tools which allow to build and understand simple economic models.

The goal of this course is to provide mathematical tools which allow to build and understand simple economic models.

The goal of this course is to provide mathematical tools which allow to build and understand simple economic models.

Natural numbers, whole numbers and rational numbers. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Percentages. Real numbers (intuitive idea). Absolute value. Powers and roots. Polynomials. Sum and product of polynomials. Square and cube of a binomial. Notable products. Factorization of simple polynomials. Rational expressions. Sum and product of rational expressions. Identities. Equations and solutions / roots of an equation. Inequalities and solutions of an inequality. First and second degree equations and inequalities. Equations and inequalities of a higher degree. Equations and inequalities with rational expressions. Irrational equations and inequalities. Equation systems and inequalities. Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems of two equations in two unknowns. Parallelism and perpendicularity of two lines. Equation of the parabola. Circumference equation. Set theory: inclusion, intersection, union, complements and empty set.

Natural numbers, whole numbers and rational numbers. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Percentages. Real numbers (intuitive idea). Absolute value. Powers and roots. Polynomials. Sum and product of polynomials. Square and cube of a binomial. Notable products. Factorization of simple polynomials. Rational expressions. Sum and product of rational expressions. Identity. Equations and solutions/roots of an equation. Inequalities and solutions of an inequality. First and second degree equations and inequalities. Equations and inequalities of a higher degree. Equations and inequalities with rational expressions. Irrational equations and inequalities. Equation systems and inequalities. Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems of two equations in two unknowns. Parallelism and perpendicularity of two lines. Equation of the parabola. Circumference equation. Set theory: inclusion, intersection, union, complements and empty set.

Natural numbers, whole numbers and rational numbers. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Percentages. Real numbers (intuitive idea). Absolute value. Powers and roots. Polynomials. Sum and product of polynomials. Square and cube of a binomial. Notable products. Factorization of simple polynomials. Rational expressions. Sum and product of rational expressions. Identities. Equations and solutions / roots of an equation. Inequalities and solutions of an inequality. First and second degree equations and inequalities. Equations and inequalities of a higher degree. Equations and inequalities with rational expressions. Irrational equations and inequalities. Equation systems and inequalities. Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems of two equations in two unknowns. Parallelism and perpendicularity of two lines. Equation of the parabola. Circumference equation. Set theory: inclusion, intersection, union, complements and empty set.

Natural numbers, whole numbers and rational numbers. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Percentages. Real numbers (intuitive idea). Absolute value. Powers and roots. Polynomials. Sum and product of polynomials. Square and cube of a binomial. Notable products. Factorization of simple polynomials. Rational expressions. Sum and product of rational expressions. Identities. Equations and solutions / roots of an equation. Inequalities and solutions of an inequality. First and second degree equations and inequalities. Equations and inequalities of a higher degree. Equations and inequalities with rational expressions. Irrational equations and inequalities. Equation systems and inequalities. Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems of two equations in two unknowns. Parallelism and perpendicularity of two lines. Equation of the parabola. Circumference equation. Set theory: inclusion, intersection, union, complements and empty set.

Class lectures. The course length is 12 weeks with three classes per week.

Class lectures. The course length is 12 weeks with three classes per week.

Class lectures. The course length is 12 weeks with three classes per week.

Class lectures. The course length is 12 weeks with three classes per week.

Class lectures. The course length is 12 weeks with three classes per week.

Class lectures. The course length is 12 weeks with three classes per week.

The course has an internet page on the platform Moodle, which provides further information on the course.

The course has an internet page on the platform Moodle, which provides further information on the course.

The course has an internet page on the platform Moodle, which provides further information on the course.

Each student must take a written exam. If the students gets a failing grade smaller than 18) in the written exam, then the student fails the exam.

If the students gets a passing grade (greater or equal than 18) in the written exam, then the student may be asked, at the discretion of the teacher, to take an oral test, and the student can request, at his own discretion, to take an oral test. In both such cases the final outcome of the examination is determined on the basis of the evaluations of the written exam and the oral exam.

If the students gets a passing grade in the written exam and an oral exam is not required by the teacher or by the student, then the final grade on the exam coincides with the grade obtained in the written exam.

To obtain the "30 cum Lode" evaluation, the student must take the oral exam.

If the students gets a passing grade (greater or equal than 18) in the written exam, then the student may be asked, at the discretion of the teacher, to take an oral test, and the student can request, at his own discretion, to take an oral test. In both such cases the final outcome of the examination is determined on the basis of the evaluations of the written exam and the oral exam.

If the students gets a passing grade in the written exam and an oral exam is not required by the teacher or by the student, then the final grade on the exam coincides with the grade obtained in the written exam.

To obtain the "30 cum Lode" evaluation, the student must take the oral exam.

Each student must take a written exam. If the student gets a failing grade (smaller than 18) in the written exam, then the student fails the exam.

If the student gets a passing grade (greater or equal than 18) in the written exam, then the student may be asked, at the discretion of the teacher, to take an oral test, and the student can request, at his own discretion, to take an oral test. In both such cases the final outcome of the examination is determined on the basis of the evaluations of the written exam and the oral exam.

If the students gets a passing grade in the written exam and an oral exam is not required by the teacher or by the student, then the final grade of the exam coincides with the grade obtained in the written exam.

To obtain the "30 cum Lode" evaluation, the student must take the oral exam.

If the student gets a passing grade (greater or equal than 18) in the written exam, then the student may be asked, at the discretion of the teacher, to take an oral test, and the student can request, at his own discretion, to take an oral test. In both such cases the final outcome of the examination is determined on the basis of the evaluations of the written exam and the oral exam.

If the students gets a passing grade in the written exam and an oral exam is not required by the teacher or by the student, then the final grade of the exam coincides with the grade obtained in the written exam.

To obtain the "30 cum Lode" evaluation, the student must take the oral exam.

Each student must take a written exam. If the students gets a failing grade smaller than 18) in the written exam, then the student fails the exam.

If the students gets a passing grade (greater or equal than 18) in the written exam, then the student may be asked, at the discretion of the teacher, to take an oral test, and the student can request, at his own discretion, to take an oral test. In both such cases the final outcome of the examination is determined on the basis of the evaluations of the written exam and the oral exam.

If the students gets a passing grade in the written exam and an oral exam is not required by the teacher or by the student, then the final grade on the exam coincides with the grade obtained in the written exam.

To obtain the "30 cum Lode" evaluation, the student must take the oral exam.

If the students gets a passing grade (greater or equal than 18) in the written exam, then the student may be asked, at the discretion of the teacher, to take an oral test, and the student can request, at his own discretion, to take an oral test. In both such cases the final outcome of the examination is determined on the basis of the evaluations of the written exam and the oral exam.

If the students gets a passing grade in the written exam and an oral exam is not required by the teacher or by the student, then the final grade on the exam coincides with the grade obtained in the written exam.

To obtain the "30 cum Lode" evaluation, the student must take the oral exam.

Each student must take a written exam. If the students gets a failing grade smaller than 18) in the written exam, then the student fails the exam.

If the students gets a passing grade (greater or equal than 18) in the written exam, then the student may be asked, at the discretion of the teacher, to take an oral test, and the student can request, at his own discretion, to take an oral test. In both such cases the final outcome of the examination is determined on the basis of the evaluations of the written exam and the oral exam.

If the students gets a passing grade in the written exam and an oral exam is not required by the teacher or by the student, then the final grade on the exam coincides with the grade obtained in the written exam.

To obtain the "30 cum Lode" evaluation, the student must take the oral exam.

If the students gets a passing grade (greater or equal than 18) in the written exam, then the student may be asked, at the discretion of the teacher, to take an oral test, and the student can request, at his own discretion, to take an oral test. In both such cases the final outcome of the examination is determined on the basis of the evaluations of the written exam and the oral exam.

If the students gets a passing grade in the written exam and an oral exam is not required by the teacher or by the student, then the final grade on the exam coincides with the grade obtained in the written exam.

To obtain the "30 cum Lode" evaluation, the student must take the oral exam.

If the students gets a passing grade (greater or equal than 18) in the written exam, then the student may be asked, at the discretion of the teacher, to take an oral test, and the student can request, at his own discretion, to take an oral test. In both such cases the final outcome of the examination is determined on the basis of the evaluations of the written exam and the oral exam.

If the students gets a passing grade in the written exam and an oral exam is not required by the teacher or by the student, then the final grade on the exam coincides with the grade obtained in the written exam.

To obtain the "30 cum Lode" evaluation, the student must take the oral exam.

If the students gets a passing grade (greater or equal than 18) in the written exam, then the student may be asked, at the discretion of the teacher, to take an oral test, and the student can request, at his own discretion, to take an oral test. In both such cases the final outcome of the examination is determined on the basis of the evaluations of the written exam and the oral exam.

If the students gets a passing grade in the written exam and an oral exam is not required by the teacher or by the student, then the final grade on the exam coincides with the grade obtained in the written exam.

To obtain the "30 cum Lode" evaluation, the student must take the oral exam.

Students can avoid reading proofs of theorems with *.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem * on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions on functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem * on the limit of the sum of functions. Theorem * on the limit of the product of functions. Theorem * on the limit of the quotient of functions. Theorems * on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem * on the continuity of the sum of functions. Theorem * on the continuity of the product of functions. Theorem * on the continuity of the quotient of functions. Theorem * on the continuity of the composition of functions. Theorem * of the zeros. Intermediate value theorem for continuous functions. Theorem * of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem * on the derivative of the sum of functions. Theorem * on the derivative of the product of functions. Theorem * on the derivative of the quotient of functions. Theorem * on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum / minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems * of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem * on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem * on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem * of the total differential.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem * on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions on functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem * on the limit of the sum of functions. Theorem * on the limit of the product of functions. Theorem * on the limit of the quotient of functions. Theorems * on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem * on the continuity of the sum of functions. Theorem * on the continuity of the product of functions. Theorem * on the continuity of the quotient of functions. Theorem * on the continuity of the composition of functions. Theorem * of the zeros. Intermediate value theorem for continuous functions. Theorem * of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem * on the derivative of the sum of functions. Theorem * on the derivative of the product of functions. Theorem * on the derivative of the quotient of functions. Theorem * on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum / minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems * of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem * on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem * on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem * of the total differential.

Students can avoid reading proofs of theorems with *.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem* on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions of functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem* on the limit of the sum of functions. Theorem* on the limit of the product of functions. Theorem* on the limit of the quotient of functions. Theorem* on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem* on the continuity of the sum of functions. Theorem* on the continuity of the product of functions. Theorem* on the continuity of the quotient of functions. Theorem* on the continuity of the composition of functions. Theorem* of the zeros. Intermediate value theorem for continuous functions. Theorem* of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem* on the derivative of the sum of functions. Theorem* on the derivative of the product of functions. Theorem* on the derivative of the quotient of functions. Theorem* on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum/minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems* of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem* on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem* on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem* of the total differential.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem* on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions of functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem* on the limit of the sum of functions. Theorem* on the limit of the product of functions. Theorem* on the limit of the quotient of functions. Theorem* on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem* on the continuity of the sum of functions. Theorem* on the continuity of the product of functions. Theorem* on the continuity of the quotient of functions. Theorem* on the continuity of the composition of functions. Theorem* of the zeros. Intermediate value theorem for continuous functions. Theorem* of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem* on the derivative of the sum of functions. Theorem* on the derivative of the product of functions. Theorem* on the derivative of the quotient of functions. Theorem* on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum/minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems* of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem* on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem* on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem* of the total differential.

Students can avoid reading proofs of theorems with *.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem * on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions on functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem * on the limit of the sum of functions. Theorem * on the limit of the product of functions. Theorem * on the limit of the quotient of functions. Theorems * on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem * on the continuity of the sum of functions. Theorem * on the continuity of the product of functions. Theorem * on the continuity of the quotient of functions. Theorem * on the continuity of the composition of functions. Theorem * of the zeros. Intermediate value theorem for continuous functions. Theorem * of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem * on the derivative of the sum of functions. Theorem * on the derivative of the product of functions. Theorem * on the derivative of the quotient of functions. Theorem * on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum / minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems * of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem * on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem * on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem * of the total differential.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem * on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions on functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem * on the limit of the sum of functions. Theorem * on the limit of the product of functions. Theorem * on the limit of the quotient of functions. Theorems * on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem * on the continuity of the sum of functions. Theorem * on the continuity of the product of functions. Theorem * on the continuity of the quotient of functions. Theorem * on the continuity of the composition of functions. Theorem * of the zeros. Intermediate value theorem for continuous functions. Theorem * of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem * on the derivative of the sum of functions. Theorem * on the derivative of the product of functions. Theorem * on the derivative of the quotient of functions. Theorem * on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum / minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems * of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem * on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem * on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem * of the total differential.

Students can avoid reading proofs of theorems with *.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem * on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions on functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem * on the limit of the sum of functions. Theorem * on the limit of the product of functions. Theorem * on the limit of the quotient of functions. Theorems * on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem * on the continuity of the sum of functions. Theorem * on the continuity of the product of functions. Theorem * on the continuity of the quotient of functions. Theorem * on the continuity of the composition of functions. Theorem * of the zeros. Intermediate value theorem for continuous functions. Theorem * of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem * on the derivative of the sum of functions. Theorem * on the derivative of the product of functions. Theorem * on the derivative of the quotient of functions. Theorem * on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum / minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems * of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem * on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem * on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem * of the total differential.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem * on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions on functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem * on the limit of the sum of functions. Theorem * on the limit of the product of functions. Theorem * on the limit of the quotient of functions. Theorems * on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem * on the continuity of the sum of functions. Theorem * on the continuity of the product of functions. Theorem * on the continuity of the quotient of functions. Theorem * on the continuity of the composition of functions. Theorem * of the zeros. Intermediate value theorem for continuous functions. Theorem * of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem * on the derivative of the sum of functions. Theorem * on the derivative of the product of functions. Theorem * on the derivative of the quotient of functions. Theorem * on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum / minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems * of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem * on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem * on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem * of the total differential.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem * on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions on functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem * on the limit of the sum of functions. Theorem * on the limit of the product of functions. Theorem * on the limit of the quotient of functions. Theorems * on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem * on the continuity of the sum of functions. Theorem * on the continuity of the product of functions. Theorem * on the continuity of the quotient of functions. Theorem * on the continuity of the composition of functions. Theorem * of the zeros. Intermediate value theorem for continuous functions. Theorem * of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem * on the derivative of the sum of functions. Theorem * on the derivative of the product of functions. Theorem * on the derivative of the quotient of functions. Theorem * on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum / minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems * of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem * on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem * on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem * of the total differential.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem * on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions on functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem * on the limit of the sum of functions. Theorem * on the limit of the product of functions. Theorem * on the limit of the quotient of functions. Theorems * on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem * on the continuity of the sum of functions. Theorem * on the continuity of the product of functions. Theorem * on the continuity of the quotient of functions. Theorem * on the continuity of the composition of functions. Theorem * of the zeros. Intermediate value theorem for continuous functions. Theorem * of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem * on the derivative of the sum of functions. Theorem * on the derivative of the product of functions. Theorem * on the derivative of the quotient of functions. Theorem * on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum / minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems * of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem * on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem * on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem * of the total differential.