Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential calculus. Introduction to integral calculus. Introduction to functions of several variables.
Course Content - Last names BP-C
Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential calculus. Introduction to integral calculus. Introduction to functions of several variables.
Course Content - Last names D-GE
Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential calculus. Introduction to integral calculus. Introduction to functions of several variables.
Course Content - Last names GF-L
Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential calculus. Introduction to integral calculus. Introduction to functions of several variables.
Course Content - Last names M-P
Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential calculus. Introduction to integral calculus. Introduction to functions of several variables.
Course Content - Last names Q-Z
Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential calculus. Introduction to integral calculus. Introduction to functions of several variables.
Enrico Giusti, Elementi di Analisi Matematica, 2008, Bollati Boringhieri.
Instructors will provide reading material about functions of two variables.
Learning Objectives - Last names A-BO
The goal of this course is to provide mathematical tools that allow to build and understand simple economic models. At the end of the course, students will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, students will also need to be able to understand the formalism and the syntax, solve exercises and problems and perform simple deductive reasoning.
Learning Objectives - Last names BP-C
The goal of this course is to provide mathematical tools that allow to build and understand simple economic models. At the end of the course, students will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, students will also need to be able to understand the formalism and the syntax, solve exercises and problems and perform simple deductive reasoning.
Learning Objectives - Last names D-GE
The goal of this course is to provide mathematical tools that allow to build and understand simple economic models. At the end of the course, students will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, students will also need to be able to understand the formalism and the syntax, solve exercises and problems and perform simple deductive reasoning.
Learning Objectives - Last names GF-L
The goal of this course is to provide mathematical tools that allow to build and understand simple economic models. At the end of the course, students will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, students will also need to be able to understand the formalism and the syntax, solve exercises and problems and perform simple deductive reasoning.
Learning Objectives - Last names M-P
The goal of this course is to provide mathematical tools that allow to build and understand simple economic models. At the end of the course, students will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, students will also need to be able to understand the formalism and the syntax, solve exercises and problems and perform simple deductive reasoning.
Learning Objectives - Last names Q-Z
The goal of this course is to provide mathematical tools that allow to build and understand simple economic models. At the end of the course, students will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, students will also need to be able to understand the formalism and the syntax, solve exercises and problems and perform simple deductive reasoning.
Prerequisites - Last names A-BO
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 to an equation. Inequalities and solutions to 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. Systems of equations 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. Equation of the circumference. Set theory: inclusion, intersection, union, complement of a set and empty set. Linear functions, quadratic functions and their graph.
Prerequisites - Last names BP-C
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 to an equation. Inequalities and solutions to 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. Systems of equations 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. Equation of the circumference. Set theory: inclusion, intersection, union, complement of a set and empty set. Linear functions, quadratic functions and their graph.
Prerequisites - Last names D-GE
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 to an equation. Inequalities and solutions to 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. Systems of equations 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. Equation of the circumference. Set theory: inclusion, intersection, union, complement of a set and empty set. Linear functions, quadratic functions and their graph.
Prerequisites - Last names GF-L
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 to an equation. Inequalities and solutions to 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. Systems of equations 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. Equation of the circumference. Set theory: inclusion, intersection, union, complement of a set and empty set. Linear functions, quadratic functions and their graph.
Prerequisites - Last names M-P
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 to an equation. Inequalities and solutions to 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. Systems of equations 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. Equation of the circumference. Set theory: inclusion, intersection, union, complement of a set and empty set. Linear functions, quadratic functions and their graph.
Prerequisites - Last names Q-Z
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 to an equation. Inequalities and solutions to 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. Systems of equations 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. Equation of the circumference. Set theory: inclusion, intersection, union, complement of a set and empty set. Linear functions, quadratic functions and their graph.
Teaching Methods - Last names A-BO
Lectures and tutorials in classroom.
Teaching Methods - Last names BP-C
Lectures and tutorials in classroom.
Teaching Methods - Last names D-GE
Lectures and tutorials in classroom.
Teaching Methods - Last names GF-L
Lectures and tutorials in classroom.
Teaching Methods - Last names M-P
Lectures and tutorials in classroom.
Teaching Methods - Last names Q-Z
Lectures and tutorials in classroom.
Further information - Last names A-BO
The course has an internet page on the Moodle platform, which provides further information on the course and teaching material.
Further information - Last names BP-C
The course has an internet page on the Moodle platform, which provides further information on the course and teaching material.
Further information - Last names D-GE
The course has an internet page on the Moodle platform, which provides further information on the course and teaching material.
Further information - Last names GF-L
The course has an internet page on the Moodle platform, which provides further information on the course and teaching material.
Further information - Last names M-P
The course has an internet page on the Moodle platform, which provides further information on the course and teaching material.
Further information - Last names Q-Z
The course has an internet page on the Moodle platform, which provides further information on the course and teaching material.
Type of Assessment - Last names A-BO
The exam takes place through a written test consisting of multiple choice questions. Details can be found on the Moodle page of the course.
If the written test is sufficient, students may request, at their discretion, to take an oral test. Similarly, instructors may ask a student to take an oral test. The oral test will consist of theoretical questions and exercises.
If a student does not take the oral test, the exam grade will coincide with the grade obtained in the written test. If a student takes the oral test, the exam grade will be determined on the basis of the grade of the written test and the evaluation of the oral test. Therefore, the final grade can be higher, lower or equal to the grade obtained in the written test. In particular, the final grade may be insufficient.
The purpose of the examination is to ascertain:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding of the formalism and the syntax related to the concepts studied,
-the ability to apply the acquired knowledge to solve exercises and problems,
-the ability to perform simple deductive reasoning.
IMPORTANT INFORMATION. Due to the emergency related to COVID-19, the exam might be modified. In such a case, students will be promptly informed.
Further details on the exam are available on the Moodle page of the course.
Type of Assessment - Last names BP-C
The exam takes place through a written test consisting of multiple choice questions. Details can be found on the Moodle page of the course.
If the written test is sufficient, students may request, at their discretion, to take an oral test. Similarly, instructors may ask a student to take an oral test. The oral test will consist of theoretical questions and exercises.
If a student does not take the oral test, the exam grade will coincide with the grade obtained in the written test. If a student takes the oral test, the exam grade will be determined on the basis of the grade of the written test and the evaluation of the oral test. Therefore, the final grade can be higher, lower or equal to the grade obtained in the written test. In particular, the final grade may be insufficient.
The purpose of the examination is to ascertain:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding of the formalism and the syntax related to the concepts studied,
-the ability to apply the acquired knowledge to solve exercises and problems,
-the ability to perform simple deductive reasoning.
Further details on the exam are available on the Moodle page of the course.
IMPORTANT INFORMATION. Due to the emergency related to COVID-19, the exam might be modified. In such a case, students will be promptly informed.
Type of Assessment - Last names D-GE
The exam takes place through a written test consisting of multiple choice questions. Details can be found on the Moodle page of the course.
If the written test is sufficient, students may request, at their discretion, to take an oral test. Similarly, instructors may ask a student to take an oral test. The oral test will consist of theoretical questions and exercises.
If a student does not take the oral test, the exam grade will coincide with the grade obtained in the written test. If a student takes the oral test, the exam grade will be determined on the basis of the grade of the written test and the evaluation of the oral test. Therefore, the final grade can be higher, lower or equal to the grade obtained in the written test. In particular, the final grade may be insufficient.
The purpose of the examination is to ascertain:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding of the formalism and the syntax related to the concepts studied,
-the ability to apply the acquired knowledge to solve exercises and problems,
-the ability to perform simple deductive reasoning.
Further details on the exam are available on the Moodle page of the course.
IMPORTANT INFORMATION. Due to the emergency related to COVID-19, the exam might be modified. In such a case, students will be promptly informed.
Type of Assessment - Last names GF-L
The exam takes place through a written test consisting of multiple choice questions. Details can be found on the Moodle page of the course.
If the written test is sufficient, students may request, at their discretion, to take an oral test. Similarly, instructors may ask a student to take an oral test. The oral test will consist of theoretical questions and exercises.
If a student does not take the oral test, the exam grade will coincide with the grade obtained in the written test. If a student takes the oral test, the exam grade will be determined on the basis of the grade of the written test and the evaluation of the oral test. Therefore, the final grade can be higher, lower or equal to the grade obtained in the written test. In particular, the final grade may be insufficient.
The purpose of the examination is to ascertain:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding of the formalism and the syntax related to the concepts studied,
-the ability to apply the acquired knowledge to solve exercises and problems,
-the ability to perform simple deductive reasoning.
IMPORTANT INFORMATION. Due to the emergency related to COVID-19, the exam might be modified. In such a case, students will be promptly informed.
Further details on the exam are available on the Moodle page of the course.
Type of Assessment - Last names M-P
The exam takes place through a written test consisting of multiple choice questions. Details can be found on the Moodle page of the course.
If the written test is sufficient, students may request, at their discretion, to take an oral test. Similarly, instructors may ask a student to take an oral test. The oral test will consist of theoretical questions and exercises.
If a student does not take the oral test, the exam grade will coincide with the grade obtained in the written test. If a student takes the oral test, the exam grade will be determined on the basis of the grade of the written test and the evaluation of the oral test. Therefore, the final grade can be higher, lower or equal to the grade obtained in the written test. In particular, the final grade may be insufficient.
The purpose of the examination is to ascertain:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding of the formalism and the syntax related to the concepts studied,
-the ability to apply the acquired knowledge to solve exercises and problems,
-the ability to perform simple deductive reasoning.
IMPORTANT INFORMATION. Due to the emergency related to COVID-19, the exam might be modified. In such a case, students will be promptly informed.
Further details on the exam are available on the Moodle page of the course.
Type of Assessment - Last names Q-Z
The exam takes place through a written test consisting of multiple choice questions. Details can be found on the Moodle page of the course.
If the written test is sufficient, students may request, at their discretion, to take an oral test. Similarly, instructors may ask a student to take an oral test. The oral test will consist of theoretical questions and exercises.
If a student does not take the oral test, the exam grade will coincide with the grade obtained in the written test. If a student takes the oral test, the exam grade will be determined on the basis of the grade of the written test and the evaluation of the oral test. Therefore, the final grade can be higher, lower or equal to the grade obtained in the written test. In particular, the final grade may be insufficient.
The purpose of the examination is to ascertain:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding of the formalism and the syntax related to the concepts studied,
-the ability to apply the acquired knowledge to solve exercises and problems,
-the ability to perform simple deductive reasoning.
IMPORTANT INFORMATION. Due to the emergency related to COVID-19, the exam might be modified. In such a case, students will be promptly informed.
Further details on the exam are available on the Moodle page of the course.
Course program - Last names A-BO
Students can skip proofs of theorems with *.
Real numbers. Operations and orders. Geometric representation of real numbers. Absolute value. 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.
The concept of function. Real functions of real variable. Domain and graph of a function. Image of a function. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions of functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and bounded below functions, upper bound and lower bound of a function, maximum points and minimum points of a function, maximum value and minimum value of a function. Elementary functions: exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.
Limit of a function at a point. Theorem of uniqueness of the limit. Theorem on persistence of 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.
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.
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.
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. Study of the graph of a function.
Integrable functions on an interval and integral of a function. Theorem* on the integrability of continuous and bounded functions. Theorem* on the integrability of monotone and bounded functions. Fundamental theorem* of calculus. Primitive and indefinite integral. Integration by substitution. Integration by parts.
Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Partial derivatives. Partial derivatives and monotonicity. Theorem* of the total differential.
Course program - Last names BP-C
Students can skip proofs of theorems with *.
Real numbers. Operations and orders. Geometric representation of real numbers. Absolute value. 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.
The concept of function. Real functions of real variable. Domain and graph of a function. Image of a function. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions of functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and bounded below functions, upper bound and lower bound of a function, maximum points and minimum points of a function, maximum value and minimum value of a function. Elementary functions: exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.
Limit of a function at a point. Theorem of uniqueness of the limit. Theorem on persistence of 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.
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.
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.
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. Study of the graph of a function.
Integrable functions on an interval and integral of a function. Theorem* on the integrability of continuous and bounded functions. Theorem* on the integrability of monotone and bounded functions. Primitive and indefinite integral. Integration by substitution. Integration by parts.
Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Partial derivatives. Partial derivatives and monotonicity. Theorem* of the total differential.
Course program - Last names D-GE
Students can skip proofs of theorems with *.
Real numbers. Operations and orders. Geometric representation of real numbers. Absolute value. 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.
The concept of function. Real functions of real variable. Domain and graph of a function. Image of a function. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions of functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and bounded below functions, upper bound and lower bound of a function, maximum points and minimum points of a function, maximum value and minimum value of a function. Elementary functions: exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.
Limit of a function at a point. Theorem of uniqueness of the limit. Theorem on persistence of 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.
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.
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.
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. Study of the graph of a function.
Integrable functions on an interval and integral of a function. Theorem* on the integrability of continuous and bounded functions. Theorem* on the integrability of monotone and bounded functions. Primitive and indefinite integral. Integration by substitution. Integration by parts.
Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Partial derivatives. Partial derivatives and monotonicity. Theorem* of the total differential.
Course program - Last names GF-L
Students can skip proofs of theorems with *.
Real numbers. Operations and orders. Geometric representation of real numbers. Absolute value. 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.
The concept of function. Real functions of real variable. Domain and graph of a function. Image of a function. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions of functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and bounded below functions, upper bound and lower bound of a function, maximum points and minimum points of a function, maximum value and minimum value of a function. Elementary functions: exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.
Limit of a function at a point. Theorem of uniqueness of the limit. Theorem on persistence of 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.
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.
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.
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. Study of the graph of a function.
Integrable functions on an interval and integral of a function. Theorem* on the integrability of continuous and bounded functions. Theorem* on the integrability of monotone and bounded functions. Primitive and indefinite integral. Integration by substitution. Integration by parts.
Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Partial derivatives. Partial derivatives and monotonicity. Theorem* of the total differential.
Course program - Last names M-P
Students can skip proofs of theorems with *.
Real numbers. Operations and orders. Geometric representation of real numbers. Absolute value. 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.
The concept of function. Real functions of real variable. Domain and graph of a function. Image of a function. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions of functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and bounded below functions, upper bound and lower bound of a function, maximum points and minimum points of a function, maximum value and minimum value of a function. Elementary functions: exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.
Limit of a function at a point. Theorem of uniqueness of the limit. Theorem on persistence of 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.
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.
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.
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. Study of the graph of a function.
Integrable functions on an interval and integral of a function. Theorem* on the integrability of continuous and bounded functions. Theorem* on the integrability of monotone and bounded functions. Primitive and indefinite integral. Integration by substitution. Integration by parts.
Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Partial derivatives. Partial derivatives and monotonicity. Theorem* of the total differential.
Course program - Last names Q-Z
Students can skip proofs of theorems with *.
Real numbers. Operations and orders. Geometric representation of real numbers. Absolute value. 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.
The concept of function. Real functions of real variable. Domain and graph of a function. Image of a function. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions of functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and bounded below functions, upper bound and lower bound of a function, maximum points and minimum points of a function, maximum value and minimum value of a function. Elementary functions: exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.
Limit of a function at a point. Theorem of uniqueness of the limit. Theorem on persistence of 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.
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.
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.
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. Study of the graph of a function.
Integrable functions on an interval and integral of a function. Theorem* on the integrability of continuous and bounded functions. Theorem* on the integrability of monotone and bounded functions. Primitive and indefinite integral. Integration by substitution. Integration by parts.
Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Partial derivatives. Partial derivatives and monotonicity. Theorem* of the total differential.