Università Roma Tre

Quantifiers. Natural, integers, rational and real numbers. Axioms of real numbers. The square root of 2 is irrational.

Cartesian coordinates in the plane and in the space; coordinate planes. Points and vectors. Distance: formal definition. Absolute value. Density of Q in R. Distance in the plane and in the space; equations of circumference and sphere.

Linear algebra (in 2 and 3 dimensions): slope of a segment, vector sum, scalar and vector product. Equivalence of geometric and coordinate definitions of vectors. Orthogonality and parallelism conditions.

Introduction to functions. Graphic of a function in the coordinate planes. Operations with graphics.

Open and closed sets, accumulation points, definitions and examples. Limits. Operations with limits, examples of limits of quotients of polynomials. Asymptotes. Comparison theorem. List of relevant limits.

Continuous functions (continuity in a point and in an interval). Theorems on continuous functions: existence of maximum and minimum values, intermediate values. Exponential and logarithmic functions.

Derivatives: definition, geometric meaning. Operations with derivatives: sum, product, quotient, multiplication by a number. Main rules of derivation, derivatives of relevant functions. Equation of the tangent line at a point to the graph. Derivative of a composite function and inverse functions. Stationary points.

Fermat's theorem. Theorems of Rolle and Lagrange. Monotony and sign of the first derivative. Linear approximation.

Second derivatives, concavity, inflections. Plotting graphs of functions. Theorems of Cauchy and De l'Hopital. Applicative problems and optimizations. Taylor polynomial. Formula of the rest of Lagrange for n=2.

Introduction to indefinite and definite integrals. The problem of calculating the area of a flat region. The theorem of the integral average. The fundamental theorem of integral calculus. Integration by parts and substitution.

Introduction to the use of computer software for plotting functions.

Testi

James Stewart, Calcolo. Funzioni di una variabile, Apogeo Education - Maggioli Editore.

Robert A. Adams Calcolo Differenziale I ed. CEA (Casa Editrice Ambrosiana)

Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli



Bibliografia

James Stewart, Calcolo. Funzioni di una variabile, Apogeo Education - Maggioli Editore.

Robert A. Adams Calcolo Differenziale I ed. CEA (Casa Editrice Ambrosiana)

Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli

Courant, Robbins "Che Cos' è La Matematica?" Ed. Boringhieri

Quantifiers. Natural, integers, rational and real numbers. Axioms of real numbers. The square root of 2 is irrational.

Cartesian coordinates in the plane and in the space; coordinate planes. Points and vectors. Distance: formal definition. Absolute value. Density of Q in R. Distance in the plane and in the space; equations of circumference and sphere.

Linear algebra (in 2 and 3 dimensions): slope of a segment, vector sum, scalar and vector product. Equivalence of geometric and coordinate definitions of vectors. Orthogonality and parallelism conditions.

Introduction to functions. Graphic of a function in the coordinate planes. Operations with graphics.

Open and closed sets, accumulation points, definitions and examples. Limits. Operations with limits, examples of limits of quotients of polynomials. Asymptotes. Comparison theorem. List of relevant limits.

Continuous functions (continuity in a point and in an interval). Theorems on continuous functions: existence of maximum and minimum values, intermediate values. Exponential and logarithmic functions.

Derivatives: definition, geometric meaning. Operations with derivatives: sum, product, quotient, multiplication by a number. Main rules of derivation, derivatives of relevant functions. Equation of the tangent line at a point to the graph. Derivative of a composite function and inverse functions. Stationary points.

Fermat's theorem. Theorems of Rolle and Lagrange. Monotony and sign of the first derivative. Linear approximation.

Second derivatives, concavity, inflections. Plotting graphs of functions. Theorems of Cauchy and De l'Hopital. Applicative problems and optimizations. Taylor polynomial. Formula of the rest of Lagrange for n=2.

Introduction to indefinite and definite integrals. The problem of calculating the area of a flat region. The theorem of the integral average. The fundamental theorem of integral calculus. Integration by parts and substitution.

Introduction to the use of computer software for plotting functions.

James Stewart, Calcolo. Funzioni di una variabile, Apogeo Education - Maggioli Editore.

Robert A. Adams Calcolo Differenziale I ed. CEA (Casa Editrice Ambrosiana)

Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli



Bibliografia

James Stewart, Calcolo. Funzioni di una variabile, Apogeo Education - Maggioli Editore.

Robert A. Adams Calcolo Differenziale I ed. CEA (Casa Editrice Ambrosiana)

Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli

Courant, Robbins "Che Cos' è La Matematica?" Ed. Boringhieri