The course has both an educational goal, introducing reasoning and mathematical symbolism, and calculus training. We propose the conceptual and methodological tools to understand the basic scientific language, providing the fundamentals of mathematical analysis (in one variable) and linear algebra (in two and three dimensions), oriented towards the understanding of geometric and mathematical physics models. The topics of differential and integral calculus in one variable and linear algebra are presented from both a geometric and analytical point of view and a modeling description is provided. The course aims to enable the student to understand basic mathematical concepts and use them autonomously to formulate or critically read problems and check their applications.
teacher profile teaching materials
Cartesian coordinates in the plane. Distance between points on the line, in the plane. Equation of a circle. Absolute value as distance from the origin of a point on the real line.
Linear algebra (in 2 and 3 dimensions): points and vectors; slope of a segment; sum and difference of vectors, product by a scalar, parallelism conditions; scalar product, orthogonality conditions; vector product, equivalence of the geometric and coordinate formulation for both products.
Introduction to the functions of a variable, relationships between quantities. Graph of a function. Algebra of graphs.
Examples and definition of limit: to infinity, and then to a point. Operations with limits, Squeeze theorem. Limits of quotients of polynomials. Asymptotes. Some important limits.
Continuous functions; continuity at a point and an interval. Theorems on continuous functions: existence of the maximum and minimum, intermediate values. Discontinuity.
Exponential and logarithm functions.
Derivatives: geometric meaning, definition. Operations with derivatives: sum, product, quotient, multiplication by a constant. Derivation techniques, derivatives of the main functions. Derivation of composite functions and the inverse of a function. Equation of the tangent line at a point on the graph. Stationary points.
Fermat's theorem. Rolle and the mean value or Lagrange theorems. Monotonicity and sign of the first derivative. Linear approximation, or first-order Taylor formula. Second derivatives, concavities, inflections. Graph sketching. Related changes, growth rates.
Introduction to integrals: indefinite and definite integrals, their meaning. The problem of calculating the area of a region in the plane. The mean value theorem. The fundamental theorem of integral calculus. Integration for parts and replacement.
Introduction to Differential Equations: growth models, logistic equation. Separation of variables method; Cauchy problems. Exponential growth and decay.
Programme
Quantifiers. Numbers: natural, integer, rational, real. Axioms of real numbers; density of Q in R. Irrationality of 2.Cartesian coordinates in the plane. Distance between points on the line, in the plane. Equation of a circle. Absolute value as distance from the origin of a point on the real line.
Linear algebra (in 2 and 3 dimensions): points and vectors; slope of a segment; sum and difference of vectors, product by a scalar, parallelism conditions; scalar product, orthogonality conditions; vector product, equivalence of the geometric and coordinate formulation for both products.
Introduction to the functions of a variable, relationships between quantities. Graph of a function. Algebra of graphs.
Examples and definition of limit: to infinity, and then to a point. Operations with limits, Squeeze theorem. Limits of quotients of polynomials. Asymptotes. Some important limits.
Continuous functions; continuity at a point and an interval. Theorems on continuous functions: existence of the maximum and minimum, intermediate values. Discontinuity.
Exponential and logarithm functions.
Derivatives: geometric meaning, definition. Operations with derivatives: sum, product, quotient, multiplication by a constant. Derivation techniques, derivatives of the main functions. Derivation of composite functions and the inverse of a function. Equation of the tangent line at a point on the graph. Stationary points.
Fermat's theorem. Rolle and the mean value or Lagrange theorems. Monotonicity and sign of the first derivative. Linear approximation, or first-order Taylor formula. Second derivatives, concavities, inflections. Graph sketching. Related changes, growth rates.
Introduction to integrals: indefinite and definite integrals, their meaning. The problem of calculating the area of a region in the plane. The mean value theorem. The fundamental theorem of integral calculus. Integration for parts and replacement.
Introduction to Differential Equations: growth models, logistic equation. Separation of variables method; Cauchy problems. Exponential growth and decay.
Attendance
The teaching regulations of the Course are followed; in any case, attendance is highly recommended.Type of evaluation
Written test and oral test. Mid term tests teacher profile teaching materials
Linear Algebra (in 2 and 3 dimensions): Points and vectors; slope of a line segment; vector addition and subtraction, scalar multiplication, conditions for parallelism; dot product, conditions for orthogonality; cross product, equivalence of the geometric and coordinate formulations for both products.
Functions & Limits: Introduction to functions of one variable, relations between quantities. Graph of a function. Algebra of graphs.Examples and definition of a limit: at infinity, and at a point. Operations with limits, Squeeze Theorem. Limits of rational functions. Asymptotes. Standard limits.ContinuityContinuous functions; continuity at a point and on an interval. Theorems on continuous functions: Extreme Value Theorem (existence of maximum and minimum), Intermediate Value Theorem. Discontinuities.Exponential and logarithmic functions.
Derivatives: geometric meaning, definition. Operations with derivatives: sum, product, quotient, constant multiple rules. Differentiation techniques, derivatives of elementary functions. The Chain Rule and the derivative of an inverse function. Equation of the tangent line to a graph at a point. Stationary points.Fermat's Theorem. Rolle's Theorem and the Mean Value Theorem. Monotonicity and the sign of the first derivative. Linear approximation, or first-order Taylor formula. Second derivatives, concavity, points of inflection. Curve sketching. Related rates, growth rates.
Integrals & Differential EquationsIntroduction to integrals: indefinite and definite integrals, their meaning. The problem of finding the area of a plane region. The Mean Value Theorem for integrals. The Fundamental Theorem of Calculus. Integration by parts and by substitution.Introduction to differential equations: growth models, logistic equation. Separation of variables; Cauchy problems (initial value problems). Exponential growth and decay.
Dario Benedetto, Mirko Degli Esposti, Carlotta Maffei Matematica per le scienze della vita, Terza edizione, Casa Editrice Ambrosiana. Zanichelli, 2015
Programme
Numbers & Analytic GeometryQuantifiers. Numbers: natural, integer, rational, real. Axioms of real numbers; density of Q in R. Irrationality of the square root of 2.Cartesian coordinates in the plane. Distance between points on the line and in the plane. Equation of a circle. Absolute value as the distance from the origin of a point on the real line.Linear Algebra (in 2 and 3 dimensions): Points and vectors; slope of a line segment; vector addition and subtraction, scalar multiplication, conditions for parallelism; dot product, conditions for orthogonality; cross product, equivalence of the geometric and coordinate formulations for both products.
Functions & Limits: Introduction to functions of one variable, relations between quantities. Graph of a function. Algebra of graphs.Examples and definition of a limit: at infinity, and at a point. Operations with limits, Squeeze Theorem. Limits of rational functions. Asymptotes. Standard limits.ContinuityContinuous functions; continuity at a point and on an interval. Theorems on continuous functions: Extreme Value Theorem (existence of maximum and minimum), Intermediate Value Theorem. Discontinuities.Exponential and logarithmic functions.
Derivatives: geometric meaning, definition. Operations with derivatives: sum, product, quotient, constant multiple rules. Differentiation techniques, derivatives of elementary functions. The Chain Rule and the derivative of an inverse function. Equation of the tangent line to a graph at a point. Stationary points.Fermat's Theorem. Rolle's Theorem and the Mean Value Theorem. Monotonicity and the sign of the first derivative. Linear approximation, or first-order Taylor formula. Second derivatives, concavity, points of inflection. Curve sketching. Related rates, growth rates.
Integrals & Differential EquationsIntroduction to integrals: indefinite and definite integrals, their meaning. The problem of finding the area of a plane region. The Mean Value Theorem for integrals. The Fundamental Theorem of Calculus. Integration by parts and by substitution.Introduction to differential equations: growth models, logistic equation. Separation of variables; Cauchy problems (initial value problems). Exponential growth and decay.
Core Documentation
James Stewart, Calcolo. Funzioni di una variabile. Apogeo Education - Maggioli Editore (più i capitoli del secondo volume, sull’algebra lineare e sulle equazioni differenziali, che verranno forniti in pdf)Dario Benedetto, Mirko Degli Esposti, Carlotta Maffei Matematica per le scienze della vita, Terza edizione, Casa Editrice Ambrosiana. Zanichelli, 2015
Reference Bibliography
Giorgio Israel, La Matematica e la realtà. Capire il mondo con i numeri. Carocci, 2015. Capire il mondo con i numeri. Carocci, 2015. Richard Courant, Herbert Robbins, Che cos’è la matematica, Torino, Bollati Boringhieri.Attendance
The course follows the academic regulations of the Degree Program; in any case, attendance is highly recommended.Type of evaluation
There is a written exam, an oral exam, and two midterm evaluations