Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential 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 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 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 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 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 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.
Enrico Giusti, Elementi di Analisi Matematica, 2008, Bollati Boringhieri.
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.
Enrico Giusti, Elementi di Analisi Matematica, 2008, Bollati Boringhieri.
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.
Learning Objectives - Last names A-BO
The goal of this course is to provide mathematical tools which allow to build and understand simple economic models. At the end of the course the student will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, the student will also need to be able to understand and appropriately use 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 which allow to build and understand simple economic models. At the end of the course the student will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, the student will also need to be able to understand and appropriately use 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 which allow to build and understand simple economic models. At the end of the course the student will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, the student will also need to be able to understand and appropriately use 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 which allow to build and understand simple economic models. At the end of the course the student will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, the student will also need to be able to understand and appropriately use 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 which allow to build and understand simple economic models. At the end of the course the student will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, the student will also need to be able to understand and appropriately use 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 which allow to build and understand simple economic models. At the end of the course the student will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, the student will also need to be able to understand and appropriately use 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 / 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.
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. 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.
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 / 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.
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 / 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.
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 / 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.
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 / 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.
Teaching Methods - Last names A-BO
Class lectures. The course length is 12 weeks with three classes per week.
Teaching Methods - Last names BP-C
Class lectures. The course length is 12 weeks with three classes per week.
Teaching Methods - Last names D-GE
Class lectures. The course length is 12 weeks with three classes per week.
Teaching Methods - Last names GF-L
Class lectures. The course length is 12 weeks with three classes per week.
Teaching Methods - Last names M-P
Class lectures. The course length is 12 weeks with three classes per week.
Teaching Methods - Last names Q-Z
Class lectures. The course length is 12 weeks with three classes per week.
Further information - Last names A-BO
The course has an internet page on the platform Moodle, which provides further information on the course.
Further information - Last names BP-C
The course has an internet page on the platform Moodle, which provides further information on the course.
Further information - Last names D-GE
The course has an internet page on the platform Moodle, which provides further information on the course.
Further information - Last names GF-L
The course has an internet page on the platform Moodle, which provides further information on the course.
Further information - Last names M-P
The course has an internet page on the platform Moodle, which provides further information on the course.
Further information - Last names Q-Z
The course has an internet page on the platform Moodle, which provides further information on the course.
Type of Assessment - Last names A-BO
The learning assessment takes place through a written test. The written test is aimed to verify:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding and use 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.
The written test has a duration of 105 minutes and includes:
- exercises aimed to verify the knowledge of the concepts and theorems presented during the course and the ability to perform simple deductive reasoning;
- exercises aimed to verify the ability to solve exercises and problems.
If the student gets a failing grade (smaller than 18) in the written test, then the student fails the exam.
If the student gets a passing grade (greater or equal than 18) in the written test, then the student may request, at his own discretion, to take an oral test. If the student does not take the oral test the final grade will coincide with the grade obtained in the written test. If the student takes the oral test the final grade is determined on the basis of the grade of the written test and the evaluation of the oral test. To achieve the "30 cum laude" grade it is necessary for the student to take the oral test.
IMPORTANT INFORMATION For the exam of September 8, 2020, due to the virus COVID-19, the learning assessment takes place through a test the student takes through the website e-val.unifi.it Please see the moodle website for this course for futher details.
Type of Assessment - Last names BP-C
The learning assessment takes place through a written test. The written test is aimed to verify:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding and use 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.
The written test has a duration of 105 minutes and includes:
- exercises aimed to verify the knowledge of the concepts and theorems presented during the course and the ability to perform simple deductive reasoning;
- exercises aimed to verify the ability to solve exercises and problems.
If the student gets a failing grade (smaller than 18) in the written test, then the student fails the exam.
If the student gets a passing grade (greater or equal than 18) in the written test, then the student may request, at his own discretion, to take an oral test. If the student does not take the oral test the final grade will coincide with the grade obtained in the written test. If the student takes the oral test the final grade is determined on the basis of the grade of the written test and the evaluation of the oral test. To achieve the "30 cum laude" grade it is necessary for the student to take the oral test.
IMPORTANT INFORMATION. For the exam of September 8, 2020, due to the virus COVID-19, the learning assessment takes place through a test the student takes through the website e-val.unifi.it. Please see the moodle website of this course for futher details.
Type of Assessment - Last names D-GE
The learning assessment takes place through a written test. The written test is aimed to verify:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding and use 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.
The written test has a duration of 105 minutes and includes:
- exercises aimed to verify the knowledge of the concepts and theorems presented during the course and the ability to perform simple deductive reasoning;
- exercises aimed to verify the ability to solve exercises and problems.
If the student gets a failing grade (smaller than 18) in the written test, then the student fails the exam.
If the student gets a passing grade (greater or equal than 18) in the written test, then the student may request, at his own discretion, to take an oral test. If the student does not take the oral test the final grade will coincide with the grade obtained in the written test. If the student takes the oral test the final grade is determined on the basis of the grade of the written test and the evaluation of the oral test. To achieve the "30 cum laude" grade it is necessary for the student to take the oral test.
IMPORTANT INFORMATION For the exam of September 8, 2020, due to the virus COVID-19, the learning assessment takes place through a test the student takes through the website e-val.unifi.it Please see the moodle website for this course for futher details.
Type of Assessment - Last names GF-L
The learning assessment takes place through a written test. The written test is aimed to verify:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding and use 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.
The written test has a duration of 105 minutes and includes:
- exercises aimed to verify the knowledge of the concepts and theorems presented during the course and the ability to perform simple deductive reasoning;
- exercises aimed to verify the ability to solve exercises and problems.
If the student gets a failing grade (smaller than 18) in the written test, then the student fails the exam.
If the student gets a passing grade (greater or equal than 18) in the written test, then the student may request, at his own discretion, to take an oral test. If the student does not take the oral test the final grade will coincide with the grade obtained in the written test. If the student takes the oral test the final grade is determined on the basis of the grade of the written test and the evaluation of the oral test. To achieve the "30 cum laude" grade it is necessary for the student to take the oral test.
Type of Assessment - Last names M-P
The learning assessment takes place through a written test. The written test is aimed to verify:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding and use 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.
The written test has a duration of 105 minutes and includes:
- exercises aimed to verify the knowledge of the concepts and theorems presented during the course and the ability to perform simple deductive reasoning;
- exercises aimed to verify the ability to solve exercises and problems.
If the student gets a failing grade (smaller than 18) in the written test, then the student fails the exam.
If the student gets a passing grade (greater or equal than 18) in the written test, then the student may request, at his own discretion, to take an oral test. If the student does not take the oral test the final grade will coincide with the grade obtained in the written test. If the student takes the oral test the final grade is determined on the basis of the grade of the written test and the evaluation of the oral test. To achieve the "30 cum laude" grade it is necessary for the student to take the oral test.
Type of Assessment - Last names Q-Z
The learning assessment takes place through a written test. The written test is aimed to verify:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding and use 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.
The written test has a duration of 105 minutes and includes:
- exercises aimed to verify the knowledge of the concepts and theorems presented during the course and the ability to perform simple deductive reasoning;
- exercises aimed to verify the ability to solve exercises and problems.
If the student gets a failing grade (smaller than 18) in the written test, then the student fails the exam.
If the student gets a passing grade (greater or equal than 18) in the written test, then the student may request, at his own discretion, to take an oral test. If the student does not take the oral test the final grade will coincide with the grade obtained in the written test. If the student takes the oral test the final grade is determined on the basis of the grade of the written test and the evaluation of the oral test. To achieve the "30 cum laude" grade it is necessary for the student to take the oral test.
Course program - Last names A-BO
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.
Course program - Last names BP-C
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.
Course program - Last names D-GE
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.
Course program - Last names GF-L
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.
Course program - Last names M-P
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.
Course program - Last names Q-Z
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.