# MATHEMATICS 1

- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
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Basic knowledge of high school algebra, trigonometry and analytic geometry.

The exam is divided into three parts:

-a written test consisting in three to five exercises covering the main topics studied in the course, which will test the ability of students in applying the computational techniques learned in class;

-a second written test covering the theoretical aspects of the course, and consisting in giving statements and proofs of a few theorems seen in class, which will test the understanding of the underlying theory and the ability to express themselves in correct mathematical language;

- an oral part, which follows immediately the second written test, consisting in the discussion of the two written parts, where the ability of express themselves in correct mathematical language, and to independently recognize the validity of mathematical reasonings will be assessed. Each part will be evaluated with a grade in the range 0 to 30, and the final grade will be the average, if greater than or equal to 18, of the grades of the three parts. Access to the second part of the exam is conditioned to having obtained at least 14/30 in the first part.

The course aims at introducing students to fundamental methods and techniques of Mathematics, and in particular to the differential and integral calculus of one real variable and to sequences and series. A further goal is to train students in applying analytic techniques to other sciences.

Students will know the fundamental of differential and integral calculus of real functions of one real variable. In particular they will be able to study the qualitative graph of elementary function, solve elementary integration problems and to determine the convergence of sequences and series; they will be able to state and prove the main theorems of Analysis. Finally, students will be able to independently recognize the validity of mathematical reasoning and to express themselves in correct mathematical language.

Number sets: natural numbers and induction, summation symbol and sum of the first n integers. Factorials and binomial coefficients. Binomial formula. Rational numbers, irrationality of square root of 2. Ordering and sup property, axiomatic property of real numbers. Decimal expansions.

Generalities on functions of a real variable: domain, image, injectivity, surjectivity, invertibility, monotonicity and boundedness.

Real valued sequences and epsilon-delta definition of their limit. Converging, diverging and irregular. Properties of limits: uniqueness, monotonicity, sign theorem sandwich theorem. Arithmetic of limits and indeterminate forms. Limits and subsequences. Limit of monotonic sequences and the Neper number. Ratio test for sequences. Fundamental limits. Infinite and infinitesimal sequences. The scale of infinte sequences. The Landau symbols.

Series: converging, diverging and irregular. Necessary condition for convergence. Geometric series, Mengoli series and telescopic series. Comparison and asymptotic comparison for series eventually of constant sign. Harmonic series. Ratio and root test; examples. Signed series: absolute convergence and convergence. Leibniz test.

Elementary functions and their graphs. Transformations, symmetries and graphs.

Limits of functions: epsilon-delta definition and sequential limits. Properties and computation of limits. Landau symbols, infinitesimals and infinites.

Continuity: epsilon-delta definition and arithmetic properties. Continuity of the composition of continuous functions. Continuity and sequential continuity. Classification of discontinuities. Continuity of piecewise defined functions and continuous extensions. Global properties of continuous functions: Zeros, intermediate values and Weiestrass theorems. Invertible continuous functions.

Introduction to derivatives. Differentiability and continuity. Differentiation rules: sum, product and quotient rules. Chain rule. Differentiability of inverse function.

Local maximum and minimum points, Fermatâ€™s Theorem. Lagrangeâ€™s Theorem and monotonicity criterion. Characterization of constant functions on an interval. De Lâ€™Hospitalâ€™s Theorem. Higher order derivatives. Taylorâ€™s formula (with Peano and Lagrange remainder). Conditions of the second derivative to determine if a stationary point is an extremum point. Convexity. Study of the graph of a function.

Introduction to the Riemann integral: Riemann sums and geometric interpretation. Integrability of continuous function. Properties of the Riemann integral. Primitives and integration rules: tables of elementaty integrals, by substitution, by parts, integration of rational functions. Integral mean theorem and fundamental theorem of calculus. Generalized integrals and integral functions.

Number sets: natural numbers and induction, summation symbol and sum of the first n integers. Factorials and binomial coefficients. Binomial formula. Rational numbers, irrationality of square root of 2. Ordering and sup property, axiomatic property of real numbers. Decimal expansions.

Generalities on functions of a real variable: domain, image, injectivity, surjectivity, invertibility, monotonicity and boundedness.

Real valued sequences and epsilon-delta definition of their limit. Converging, diverging and irregular. Properties of limits: uniqueness, monotonicity, sign theorem sandwich theorem. Arithmetic of limits and indeterminate forms. Limits and subsequences. Limit of monotonic sequences and the Neper number. Ratio test for sequences. Fundamental limits. Infinite and infinitesimal sequences. The scale of infinte sequences. The Landau symbols.

Series: converging, diverging and irregular. Necessary condition for convergence. Geometric series, Mengoli series and telescopic series. Comparison and asymptotic comparison for series eventually of constant sign. Harmonic series. Ratio and root test; examples. Signed series: absolute convergence and convergence. Leibniz test.

Elementary functions and their graphs. Transformations, symmetries and graphs.

Limits of functions: epsilon-delta definition and sequential limits. Properties and computation of limits. Landau symbols, infinitesimals and infinites.

Continuity: epsilon-delta definition and arithmetic properties. Continuity of the composition of continuous functions. Continuity and sequential continuity. Classification of discontinuities. Continuity of piecewise defined functions and continuous extensions. Global properties of continuous functions: Zeros, intermediate values and Weiestrass theorems. Invertible continuous functions.

Introduction to derivatives. Differentiability and continuity. Differentiation rules: sum, product and quotient rules. Chain rule. Differentiability of inverse function.

Local maximum and minimum points, Fermatâ€™s Theorem. Lagrangeâ€™s Theorem and monotonicity criterion. Characterization of constant functions on an interval. De Lâ€™Hospitalâ€™s Theorem. Higher order derivatives. Taylorâ€™s formula (with Peano and Lagrange remainder). Conditions of the second derivative to determine if a stationary point is an extremum point. Convexity. Study of the graph of a function.

Introduction to the Riemann integral: Riemann sums and geometric interpretation. Integrability of continuous function. Properties of the Riemann integral. Primitives and integration rules: tables of elementaty integrals, by substitution, by parts, integration of rational functions. Integral mean theorem and fundamental theorem of calculus. Generalized integrals and integral functions.

Lecture notes available at:

http://www.matapp.unimib.it/~demichele/libro.pdf

M. Bramanti, C. Pagani, S. Salsa, Analisi matematica 1, Zanichelli

S. Salsa, A. Squellati, Esercizi di analisi matematica: 1 , Zanichelli

Frontal lectures: 24 hours

Exercise sessions: 36 hours

Office hours: by appointment (email the instructor)