Dr. Ciolli has some exercises on the topics of this course on his webpage.

Past exams by Prof. Carpi.

Recorded lecture videos can be watched on the Microsoft Teams. Access the team of the course, go to the channel "Lectures" and you find the video file below the session.

1. The exam consists of a written test (with Moodle) and a compulsory oral test, both online.
2. To access the oral, a grade of at least 18/30 must be obtained on the written test. The exam is passed if the final mark is at least 18/30.
3. The written test and the oral test must take place in the same call.
4. Under penalty of exclusion, during written tests the use of electronic devices and applications except those required (MS Teams and possibly a web browser to access to the Moodle quiz) is not allowed. It is not possible to use books and notes.
5. It is necessary to have a webcam and a microphone to have test.

A simulation of the written test is available on the site.

• Winter session 1st Written test 27/01/21, 9:00 (online), problems and solutions. 1st Oral assessment 27/1/21, 12:00, (online).
• Winter session 2nd Written test 16/02/21, 9:00 (online), problems and solutions. 2nd Oral assessment 16/2/21, 12:30, (online).
• Summer session 1st Written test 18/06/21, 9:00 (online), problems and solutions. 1st Oral assessment 27/1/21, 13:00, (online/in person).
• Summer session 2nd Written test 13/07/21 , 9:00 (online), problems and solutions. 2nd Oral assessment 14/07/20, 9:00 (online/in person).
• Autumn session 1st Written test 02/09/21, 9:00 problems and solutions, more details. 1st Oral assessment 31/08/20 (after written).
• Autumn session 2nd Written test 17/09/21, 14:30, problems and solutions. 2nd Oral assessment 17/09/20 (after written).

• 09/21 slides. Introduction of the course, mathematical symbols, axioms and proofs, interger, rational numbers and their basic properties.
• 09/23 slides. Order relations on rational numbers, naive set theory, ordered pairs and graphis, examples.
• 09/25 slides. Examples of irrational numbers, a proof of irrationality of √2, the axioms of real numbers, upper and lower bounds, sup and inf, a graphical representation of R, the Archimedean property, √2 is a real number.
• 09/28 slides. Intervals, operations on sets of real numbers (sum and product), some properties of upper and lower bounds, square root of real numbers, decimal representation of real numbers.
• 09/29 slides. Exercises on real numbers, sets, graphs, irrationality of certain roots, upper and lower bounds and the Archimedean property.
• 09/30 slides. Mathematical induction, the Peano axioms, the well-ordering principle, the summation and the product notations, useful formulas, the binomial theorem.
• 10/01 slides. Functions, domain and range, examples, injectivity and surgectivity, inverse functions, sum, product, division and composition of functions, absolute value and the triangle inequality.
• 10/02 slides. Exercises on intervals, inf and sup, intervals defined by polynomials, the summation symbol, the binomial theorem, the domain and inverse of some functions, operations on graphs.
• 10/05 slides. Convergence and divergence of sequences, examples, subsequences, bounded and monotone sequences, limits of sum, product and quotient, the limit of powers, the geometric series.
• 10/07 slides. Decimal representations of real numbers, repeaing recimal representations give rational numbers, limit of functions, continuity of functions, examples, continuity of sum, product and quotient, continuity of polynomial and rational functions.
• 10/08 slides. Continuity and sequences, the intermediate value theorem, continuity of composed and inverse functions, roots and power functions.
• 10/09 slides. Exercises on limit of sequences, decimal reprenstations, limit of functions, the intermediate value theorem, n-th roots.
• 10/12 slides. Pinching theorem, Cauchy sequences, convergence and Cauchy property, various limits including exponentials, definition of the exponential functions for real numbers and its graph.
• 10/14 slides. Some properties of exponential functions, Napier's number, logarithm, some properties of logarithm.
• 10/15 slides. The left and right limits, limits at infinity, the change of variables in limit, some notable limits of log and exp, hyperbolic functions.
• 10/16 slides. Exercises on exponential, logarithm and various limits including exp and log.
• 10/19 slides. Review of trigonometric functions. The general angle, certain values of sin and cos, their relations, the sum-product formula, the product-sum formula, some limit.
• 10/21 slides. Open and closed sets, examples, the Bolzano-Weierstrass theorem, the boundedness of continuous functions defined on bounded closed sets, examples.
• 10/22 slides. Maximum and minimum of functions, the Weierstrass theorem, examples, uniform continuity, the Heine-Cantor theorem, examples, preliminaries for derivatives.
• 10/23 slides. Exercises on trigonometric functions, open and closed sets, boundedness and uniform continuity of functions.
• 10/26 slides. Definition of derivative, examples (constant, polynomials, 1/x, log, exp, sin, cos), left and right derivatives, differentiability implies continuity.
• 10/28 slides. More rules of derivative: linearirty, the Leibniz rule, quotients, the chain rule, inverse functions, and examples.
• 10/29 slides. Slope of a line, tangent line to a graph and the derivative, local minima and maxima, stationary points and examples, implicit derivative and examples.
• 10/30 slides. Exercises on derivatives, linearity, the Leibniz rule, the chain rule and inverse function, the implicit differentiation.
• 11/02 slides. Roll's theorem, Lagranges' and Cauchy's mean value theorems, the sign of the derivative and the increasing/decreasing of the function, maxima and minima.
• 11/04 slides. Higher derivatives, maxima and minima by the second derivative, convexity and concavity, asymptotes, examples.
• 11/05 slides. Operations on functions and graphs, symmetry of graphs, curve sketching, examples and some applications.
• 11/06 slides. Exercises on increasing/decreasing of functions, minima and maxima, asymptotes, graph sketching and some applications.
• 11/09 slides. Indefinite limites, Bernoulli-de l'Hôpital rule in various cases, examples.
• 11/11 slides. Landau's o and O symbols, the first and second Taylor formula and examples.
• 11/12 slides. The n-th order Taylor formula, examples and some applications to limits.
• 11/13 slides. Exercises on Bernoulli-de l'Hôpital rule, Taylor formula and their applications to limits.
• 11/16 slides. Definition of definite integral. Lower and upper sum, lower and upper integral, examples.
• 11/18 slides. Integrability of continuous functions on closed bounded intervals, fundamental theorems of calculus, examples.
• 11/19 slides. Primitive, integral calculus of elementary functions and examples, the mean value theoreme for integral, some applications of integral.
• 11/20 slides. Exercises on upper and lower sums, primivite and definite integral, reviews of Taylor formula.
• 11/23 slides. Summary of derivatives and indefinite integrals, integration by parts, integration by substitution, examples.
• 11/25 slides. Integral of rational functions, partial fractions, integration by change of variables, examples.
• 11/26 slides. Change of variables in definite integral, integral of even and odd functions, logarithmic differentiation, Taylor formula with reminder, convergence of Taylor series for some functions, imporoper integrals and examples.
• 11/27 slides. Exercises on integration by parts, substutition, change of variables and integral of rational functions and improper integrals.
• 11/30 slides. Definition and properties of improper integral, examples. Definition of area as integral.
• 12/02 slides. Definition of area and length as integral, example, equivalence of the definition of length as the sum of length of segments.
• 12/03 slides. Convergence and divergence of series examples, telescopic series, geometric series and functions as series (exponential, sin and cos).
• 12/04 slides. Exercises on improper integral, convergence of improper integral, calculus of area and length.
• 12/09 slides. Convergence criteria for non-negative series: root test, ratio test, integral test, condensation principle and examples.
• 12/10 slides. Convergence criteria for general series: absolute convergence, Leibniz criteria, Dirichlet and Abel's test, examples.
• 12/11 slides. Exercises on computations and convergence/divergence of infinite series.
• 12/14 slides. Differential equations, general remarks, equations independent of x, linear homogeneous first-order equations, examples.
• 12/16 slides. General linear homogeneous and inhomogeneous first-order equations consequences of linearity, existence and uniqueness of solution, examples.
• 12/17 slides. Second-order linear differential equations with constant coefficients, some inhomogeneous cases, examples and physical applications.
• 12/18 slides. Exercises on first-order linear differential equations, second-order linear differential equations with constant coefficients and some physical examples.
• 12/21 slides. Implicit solutions of differential equations, separable equations and examples, integral curves and vector fields, examples by Wolfram Alpha.
• 12/22 slides. Existence and uniqueness of solutions to first-order differential equations (without proof), Euler method, numerical solutions with Python.
• 01/07 slides. Complex numbers, definition and basic properties, graphical representation, inverse and roots, algebraic equations.
• 01/08 slides. Triangle inequality of complex numbers, sequence and series of complex numbers, absolute convergence, complex extensions of exp, sin, cos.
• 01/11 slides. Exercises on differential equations and complex numbers.
• 01/12 slides. Exercises on limits, Taylor formula, Landau's symbol and convergence and divergence of series.
• 01/13 slides. Exercises on graph of functions, maxima and minima, monotonicity and derivatives.
• 01/14 slides. Exercises on various techniques of definite integrals and convergence and divergence of improper integrals.
• 01/15 slides. Exercises on various differential equations and complex numbers.