Basic Mathematics: a math precourse

The lecture notes, exercises and past exams: 2020, 2021, 2022, 2023.

During the course, there are assignments. If you do them all, you will get a bonus in the exam. You need to have an account of Microsoft 365. Ask the student office for it.

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 2023/24), 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. Assignment 1: sets and real numbers, deadline 2024 Oct. 23.
2. Assignment 2: continuity and limits, deadline 2024 Nov. 13.
3. Assignment 3: derivative, deadline 2024 Dec. 4
4. Assignment 4: integral, deadline 2023 Dec. 22

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. The "basic" part of the written test must result more than 80%. To access the oral, a grade for the main written part of at least 15/25 must be obtained, including the bonus if one has completed the assignments (in that case, 13 in the written).
3. The final mark is determined based on the written exam mark (max 25) and the oral test (max 5) with possible corrections of careless mistakes in the written exam (max 3 points), and the assessment of the understanding of the material of the written exam (possibly negative marks). 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, one tab for Moodle) 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 21/01/25, 9:00 (Room 2), problems and solutions. 1st Oral assessment 22/01/24, 12:30 (Room 1), or 23/01/24 (Room 9).
• Winter session 2nd Written test 05/02/24, 9:00 (Room 4), 2nd Oral assessment 05/02/24, 12:30 (Room B7), or 06/02/24 (Room B10).

• 09/23 Introduction of the course, reviews of high school mathematics (equations, inequalities, identities, roots, trigonometric functions, exp, log, graphs).
• 09/25 statements, logic, mathematical symbols, axioms and proofs, interger, rational numbers and their basic properties.
• 09/26 Order relations on rational numbers, naive set theory, union, intersection, difference of sets, examples.
• 09/27 Sets and logic (and, or, negation), sets of sets, ordered pairs and graphs, examples.
• 10/02 Examples of irrational numbers, a proof of irrationality of √2, the axioms of real numbers, upper and lower bounds, sup and inf, intervals, a graphical representation of R, the Archimedean property.
• 10/03 Existence of square root of 2, mathematical induction, the Peano axioms, some properties of upper and lower bounds, the well-ordering principle.
• 10/04 Exercises on rational and real numbers, sets, graphs and logic.
• 10/07 Decimal representation of real numbers, the summation and the product notations, factorial, powers, useful formulas, binomial theorem.
• 10/09 Functions, domain and range, examples, injectivity and surgectivity, inverse functions, sum, product, division and composition of functions, absolute value and the triangle inequality.
• 10/10 Convergence and divergence of sequences, examples, subsequences, bounded and monotone sequences, limits of sum, product and division of sequences.
• 10/11 Limits of powers. Exercises on sums of sequences, functions limit of sequences, limits of functions and inverse functions.
• 10/14 Decimal representations of real numbers. Limit of functions, continuity of functions, examples, continuity of sum, product and quotient, continuity of polynomial and rational functions, sequences and continuity of functions.
• 10/16 The intermediate value theorem, continuity of composed functions and inverse functions, examples, root functions.
• 10/17 Squeezing theorem, Cauchy sequences, convergence and Cauchy property, various limits including exponentials, definition of the exponential functions for real numbers.
• 10/18 Exercises on continuity of functions, limits of functions and sequences, squeezing, roots of positive numbers.
• 10/21 Some properties of exponential functions, Napier's number, logarithm, some properties of logarithm.
• 10/23 The left and right limits, limits at infinity, the change of variables in limit, some notable limits of log and exp.
• 10/24 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.
• 10/25 Exercises on exponential, logarithm, various limits, and trigonometric functions.
• 10/28 Open and closed sets, examples, the Bolzano-Weierstrass theorem, the boundedness of continuous functions defined on bounded closed sets, examples.
• 10/30 Maximum and minimum of functions, the Weierstrass theorem, examples, uniform continuity, the Heine-Cantor theorem, examples, preliminaries for derivatives.
• 10/31 Definition of derivative, examples (constant, polynomials, 1/x, log, exp, sin, cos), left and right derivatives, differentiability implies continuity.
• 11/04 More rules of derivative: linearity, the Leibniz rule, quotients, the chain rule, inverse functions, and examples.
• 11/06 Slope of a line, tangent line to a graph and the derivative, local minima and maxima, stationary points and examples, implicit derivative and examples.
• 11/06 (13:45) Exercises on open and closed sets and their union and intersection, minimum and maximum of continuous functions, uniform continuity and derivative.
• 11/07 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.
• 11/08 Exercises on derivatives, linearity, the Leibniz rule, quotients, the chain rule and inverse function, logarithmic derivative, stationary points.
• 11/11 Higher derivatives, maxima and minima by the second derivative, convexity and concavity, asymptotes, examples.
• 11/13 Operations on functions and graphs, symmetry of graphs, curve sketching, examples.
• 11/14 Bernoulli-de l'Hôpital rule in various cases, examples. Landau's o and O symbols.
• 11/18 The n-th order Taylor formula, examples and some applications to limits.
• 11/20 Definition of definite integral. Lower and upper sum, lower and upper integral, examples.
• 11/20 (14:00) Exercises on increasing/decreasing of functions, asymptotes, graphs of functions.
• 11/21 Integrability of continuous functions on closed bounded intervals, fundamental theorem of calculus.
• 11/22 Fundamental theorem of calculus (continued). Exercises Bernoulli-de l'Hôpital rule, Taylor formula and definite integrals.
• 11/25 Primitive, integral calculus of elementary functions and examples, the mean value theoreme for integral.
• 11/27 Summary of derivatives and indefinite integrals, integration by parts, integration by substitution, examples.
• 11/27 (14:00) Integral of rational functions, partial fractions, integration by change of variables, examples.
• 11/28 Change of variables in definite integral, integral of even functions, Taylor formula with reminder, convergence of Taylor series for some functions and examples.
• 11/29 Exercises on integration by parts, substutition, change of variables and integral of rational functions.
• (Butterley)
• 12/02 Definition and properties of improper integral, criteria for convergence and divergence, examples.
• 12/04 Definition of area and length as integral, example, equivalence of the definition of length as the sum of length of segments.
• 12/05 Convergence and divergence of series, examples, telescopic series, geometric series and functions as series (exponential, sin and cos).
• 12/09 Convergence criteria for non-negative series: root test, ratio test, integral test, condensation principle and examples.
• 12/11 Convergence criteria for general series: absolute convergence, Leibniz criterion, uniform convergence of sequences and series of functions, examples.
• 12/11 (14:00) Exercises on improper integral, convergence of improper integral.
• 12/12 Differential equations, general remarks, equations independent of y, linear homogeneous first-order equations, examples.
• 12/13 Exercises on computations and convergence/divergence of infinite series.
• 12/16 Linear differential equations, second-order linear differential equations with constant coefficients, examples and physical applications.
• 12/18 Implicit solutions of differential equations, separable equations and examples.
• 12/19 Integral curves and vector fields, Existence and uniqueness of solutions to first-order differential equations (without proof), Euler method, solution by power series.
• 12/20 Exercises on first-order linear differential equations, second-order linear differential equations with constant coefficients and separable equations.
• 01/08 Complex numbers, definition and basic properties, graphical representation, inverse and roots, fundamental theorem of algebra (without proof).
• 01/09 Triangle inequality of complex numbers, sequence and series of complex numbers, absolute convergence, complex extensions of exp, sin, cos.
• 01/10 More details and examples related to complex numbers.
• (Tanimoto)
• 01/13 Exercises on convergence and divergence of series and complex numbers.
• 01/15 (14:00) Exercises on limits, Taylor formula, Landau's symbol.
• 01/15 Exercises on graph of functions, maxima and minima, monotonicity and derivatives.
• 01/16 Exercises on various techniques of definite integrals.
• 01/17 Exercises on convergence and divergence of improper integrals and various differential equations.