AIMS AND CONTENT

LEARNING OUTCOMES

The aim of this teaching is to introduce to the rigorous treatment of analysis while developing the methods of differential and integral calculus in the context of real functions of one real variable, with the purpose of acquiring logical rigor, attaining a good command of calculus, and knowing the main proof techniques.

AIMS AND LEARNING OUTCOMES

The expected learning outcomes are that the student knows how to handle the basic tools of analysis and calculus. It is expected that the student has understood proofs  and knows how to set up and write demonstrations of simple statements.

TEACHING METHODS

The teaching includes coordinated theory and exercise lectures. Tutoring is provided parallel to the course to enable the student to monitor his or her preparation in progress. Exercise tests will be uploaded onto the aulaweb at the end of each topic covered.

SYLLABUS/CONTENT

1. Real numbers.  The natural numbers and the integers. Rational numbers and their geometric representation. The axiom of completeness and its consequences. 

2. Functions. Relations, functions, domain, codomain, image and graph of functions. Composition of functions. Invertible functions. Operations on real functions. Monotone functions. Polynomials and rational functions. Other elementary functions. Trigonometric functions. The exponential function in the rational body.

3. Limits. Metric and topological properties of R. Definition of continuity. Operations with continuous functions. Limits and their properties. Operations on limits. Theorems of comparison. Limits of monotone functions. Limits of compound functions and changes of variables. Successions and their limits. Subsuccessions. Bolzano-Weierstrass theorem. Cauchy successions. Use of successions in the study of limits. Limits of successions defined by recurrence. Neper's e-number.

4. Global properties of continuous functions. Weierstrass theorem. Theorem of zeros. Theorem of intermediate values. Continuity and monotony. Continuity of the inverse. Uniform continuity. Heine Cantor's theorem. The exponential function in the real body.

5. Differential calculus, The derivative: definition and first properties. Differentiability. Algebraic properties of the differential. Derivation of compound functions and the inverse function. Derivatives of elementary functions. Derivatives of higher order. Rolle's, Lagrange's and Cauchy's theorems and their consequences. De l'Hopital theorem. Local comparison between functions. Infinities and infinitesimals. Taylor's formula. Study of the monotonic and convexity properties of a function through the signs of the derivatives. Convex functions. Newton's method. Iterative methods for solving equations.

6. The indefinite integral. Rules of integration. Integration of some elementary functions. Integration by parts and by substitution. Integration of rational functions.

8. Geometric, telescopic series. Convergence.Numerical series with nonnegative terms: comparison, root and ratio criteria; condensation, order and integral criteria. Series with alternate signs. Leibniz's theorem.

9. (FOR CdL FISICA - 8 hours) Differential equations. Differential equations with separable variables. First-order linear equations. Second-order linear equations with constant coefficients.

9. (FOR CdL MATEMATICA and SMID - 22 hours) Differential calculus of functions of several variables.

TEACHERS AND EXAM BOARD

LESSONS

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

ASSESSMENT METHODS

Written tests. 

1. Two intermediate written tests will be given during the course of the year. If a student obtains an average mark greater than or equal to 18/30 and if he or she obtains at least 15/30 in both, the average of the two marks counts as the written test and takes its place.

2. A written test with a mark of 17/30 or higher gives access to the oral test.

3. If a student hands in a written test, any written tests handed in previously are deemed to have been cancelled.

The written test will include several exercises on the topics of the program to assess the student's ability to critically use the tools learnt during the course.

Oral tests. During the oral test, the committee questions the entire syllabus. In particular, knowledge of the definitions of the main concepts, and of the statements an