Basic Mathematics: a math precourse

The lecture notes, exercises and past exams:2020, 2021.

Recorded lecture videos from the year 2020/21 can be watched on the Microsoft Teams. Use the code for the course of the year 2020/21 (you can find it in a post in the year 2022/23), go to the channel "Lectures" and you find the video file below the session.

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

1. The exam consists of a written test (2h30m, with Moodle) and a compulsory oral test. The oral test takes place on the same day or on the next day(s) if there are many students (the schedule will be communicated on Teams and Delphi).
2. To access the oral, a grade of at least 18/30 must be obtained on the written test including the bonus if one has completed the assignments (in that case, 16 in the written).
3. The final mark is determined based on the written exam mark, with the adjustment according to the performance in the oral test. The oral test verifies the understanding of the basic material of the course, especially those in the written exam. This includes questions on the rough papers. The exam is passed if the student can explain well enough the rough calculations and the final mark including the bonus is at least 18/30.
4. The written test and the oral test must take place in the same call.
5. 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, notes and calculators, or to communicate with anyone other than the invigilators.
6. It is necessary to bring an identification document and the university booklet when attending both the written and oral tests.

• Winter session 1st Written test 01/02/23, 9:00 (Room 4), problems and solutions. 1st Oral assessment 01/02/23, 12:30 (Room 4), or 02/02/23 (Room C10).
• Winter session 2nd Written test 16/02/23, 9:00 (Room 4), problems and solutions. 2nd Oral assessment 16/02/23, 12:30 (Room 4), or 17/02/23 (Room C10).
• Summer session 1st Written test 03/07/23, 9:00 (Room 2), problems and solutions, more details. 1st Oral assessment 03/07/23, 12:30 (Room 2), or later.
• Summer session 2nd Written test 19/07/23, 9:00 (Room 4) problems and solutions. 2nd Oral assessment 19/07/23, 12:30 (Room 4) or later.
• Autumn session 1st Written test 08/09/23, 9:00 (Room 4) problems and solutions, more details. 1st Oral assessment 08/09/23 12:30, (Room 4) or later.
• Autumn session 2nd Written test 22/09/23, 9:00 (Room 4), problems and solutions. 2nd Oral assessment 22/09/23 12:30, (Room 4) or later.

• 09/28 Introduction of the course, mathematical symbols, axioms and proofs, interger, rational numbers and their basic properties.
• 09/29 Order relations on rational numbers, naive set theory, union, intersection, difference of sets, examples.
• 09/30 (11:30) Sets and logic (and, or, negation), sets of sets, ordered pairs and graphs, examples.
• 09/30 (14:00) Exercises on rational numbers, sets, graphs and logic.
• 10/03 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.
• 10/05 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.
• 10/06 Mathematical induction, the Peano axioms, the well-ordering principle, the summation and the product notations, factorial, powers.
• 10/07 Useful formulas, binomial theorem, exercises on decimal representations, sum and product.
• 10/10 Functions, domain and image, examples, injectivity and surgectivity, inverse functions, sum, product, division and composition of functions, absolute value and the triangle inequality.
• 10/12 Convergence and divergence of sequences, examples, subsequences, bounded and monotone sequences, limits of sum, product and division.
• 10/13 Limit of functions, continuity of functions, examples, continuity of sum, product and quotient, continuity of polynomial and rational functions, limits of powers.
• 10/14 Decimal representations of real numbers. Exercises on geometric series, limit of sequences, repeating decimal reprenstations and rational numbers, limits of functions.
• 10/17 Intermediate value theorem, continuity of composed functions and inverse functions, examples, roots.
• 10/19 Squeezing theorem, Cauchy sequences, convergence and Cauchy property, various limits including exponentials, definition of the exponential functions for real numbers and its graph.
• 10/20 Some properties of exponential functions, Napier's number, logarithm, some properties of logarithm.
• 10/21 Exercises on exponential, logarithm and various limits.
• 10/24 The left and right limits, limits at infinity, the change of variables in limit, some notable limits of log and exp.
• 10/26 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. Hyperbolic functions.
• 10/27 Open and closed sets, examples, the Bolzano-Weierstrass theorem, the boundedness of continuous functions defined on bounded closed sets, examples.
• 10/28 Exercises on some limit involving exp and log, trigonometric functions, hyperbolic functions, open and closed sets.
• 10/31 Maximum and minimum of functions, the Weierstrass theorem, examples, uniform continuity, the Heine-Cantor theorem, examples, preliminaries for derivatives.
• 11/02 Definition of derivative, examples (constant, polynomials, 1/x, log, exp, sin, cos), left and right derivatives, differentiability implies continuity.
• 11/03 More rules of derivative: linearity, the Leibniz rule, quotients, the chain rule, inverse functions, and examples.
• 11/04 Exercises on derivatives, linearity, the Leibniz rule, quotients, the chain rule and inverse function.
• 11/07 Slope of a line, tangent line to a graph and the derivative, local minima and maxima, stationary points and examples, implicit derivative and examples.
• 11/09 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.
• 11/10 Higher derivatives, maxima and minima by the second derivative, convexity and concavity, asymptotes, examples.
• 11/11 Exercises on increasing/decreasing of functions, minima and maxima, asymptotes and some applications.
• 11/14 Operations on functions and graphs, symmetry of graphs, curve sketching, examples.
• 11/16 Bernoulli-de l'Hôpital rule in various cases, examples.
• 11/17 Landau's o and O symbols, the first and second Taylor formula and examples.
• 11/18 Exercises on graph sketching, Bernoulli-de l'Hôpital rule, Taylor formula.
• 11/21 The n-th order Taylor formula, examples and some applications to limits.
• 11/23 Definition of definite integral. Lower and upper sum, lower and upper integral, examples.
• 11/24 Integrability of continuous functions on closed bounded intervals, fundamental theorems of calculus, examples.
• 11/25 Exercises on Taylor formula and their applications to limits and some integrals and fundamental theorem of calculus.
• 11/28 Primitive, integral calculus of elementary functions and examples, the mean value theoreme for integral, some applications of integral.
• 11/30 Summary of derivatives and indefinite integrals, integration by parts, integration by substitution, examples.
• 12/01 Integral of rational functions, partial fractions, integration by change of variables, examples.
• 12/02 Exercises on integration by parts, substutition, change of variables and integral of rational functions.
• 12/05 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.
• 12/07 Definition and properties of improper integral, criteria for convergence and divergence, examples.
• 12/07 (14:00) Definition of area and length as integral, example, equivalence of the definition of length as the sum of length of segments.
• 12/09 Exercises on improper integral, convergence of improper integral.
• 12/12 Convergence and divergence of series, examples, telescopic series, geometric series and functions as series (exponential, sin and cos).
• 12/14 Convergence criteria for non-negative series: root test, ratio test, integral test, condensation principle and examples.
• 12/15 Convergence criteria for general series: absolute convergence, Leibniz criteria, Dirichlet and Abel's test, examples.
• 12/16 Exercises on computations and convergence/divergence of infinite series.
• 12/19 Differential equations, general remarks, equations independent of y, linear homogeneous first-order equations, examples.
• 12/21 Linear differential equations, second-order linear differential equations with constant coefficients, examples and physical applications.
• 12/21 (14:00) Implicit solutions of differential equations, separable equations and examples.
• 12/22 Exercises on first-order linear differential equations, second-order linear differential equations with constant coefficients.
• 01/09 Integral curves and vector fields, Existence and uniqueness of solutions to first-order differential equations (without proof), Euler method.
• 01/11 Complex numbers, definition and basic properties, graphical representation, inverse and roots, fundamental theorem of algebra (without proof).
• 01/12 Triangle inequality of complex numbers, sequence and series of complex numbers, absolute convergence, complex extensions of exp, sin, cos.
• 01/13 Exercises on limits, Taylor formula, Landau's symbol.
• 01/16 Exercises on convergence and divergence of series and complex numbers.
• 01/18 Exercises on graph of functions, maxima and minima, monotonicity and derivatives.
• 01/19 Exercises on various techniques of definite integrals.
• 01/20 Exercises on convergence and divergence of improper integrals and various differential equations.