The objective of this course is to give students an understanding of basic calculus as well as to enable them to approach problems from a mathematical perspective.
scheda docente
materiale didattico
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.
Harmonic oscillator, its solution, modeling discussion of the solution.
Dario Benedetto, Mirko Degli Esposti, Carlotta Maffei Matematica per le scienze della vita
Terza edizione, Casa Editrice Ambrosiana. Zanichelli, 2015
Paolo Marcellini, Carlo Sbordone. Elementi di calcolo (versione semplificata per i nuovi corsi di laurea). Liguori Ed.
Programma
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.
Harmonic oscillator, its solution, modeling discussion of the solution.
Testi Adottati
James Stewart, Calculus: single variable calculus (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
Paolo Marcellini, Carlo Sbordone. Elementi di calcolo (versione semplificata per i nuovi corsi di laurea). Liguori Ed.
Bibliografia Di Riferimento
Giorgio Israel, La Matematica e la realtà. Capire il mondo con i numeri. Carocci, 2015. Richard Courant, Herbert Robbins, Che cos’è la matematica, Torino, Bollati Boringhieri.Modalità Valutazione
Written test and oral test. Ongoing tests. The written test consists of exercises similar to those carried out during the lessons.