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Calculus GR. A-B: PROF. ANNA TORRIERO; GR. C-DE: PROF.ROSA ALBANESE ; GR. DI-LA: PROF. MONICA BIANCHI; GR. LE-O: PROF. ENRICO MIGLIERINA; GR. P-SA: PROF. ALVISE MERINI ; GR. SB-Z: PROF. SALVATORE VASSALLO COURSE AIMS The course has two objectives: to present several fundamental mathematical tools for dealing with economic-financial problems, and to help students to acquire a precise and essential language. The course will emphasize how to develop a view toward critically re-examining mathematical concepts which students will find in their academic pursuits, and how to stimulate the capacity to use mathematical methods, tools and models in a wide array of applications. The course will cover basic topics in linear algebra, differential and integral calculus, and optimization; taken together, these concepts will prove an effective tool for analysing economic and business phenomena. COURSE CONTENT BASIC KNOWLEDGE (pre-course class) Natural, integer, rational and real numbers. Fundamentals of logic and basic set theory. Elementary algebra, arithmetic, analytical geometry, trigonometry. REAL FUNCTIONS OF ONE VARIABLE − Introductory concepts: Domain. Maximum, minimum, upper and lower bounds. Bounded functions, monotonic functions, composition of functions, inverse function. Convex functions. − Limits and continuity: Limits and related theorems. Operations on limits and indecision forms. Continuity of functions and related theorems. Asymptotes. − Differential calculus: Incremental ratio and derivative. Differentiable functions. Rules of differentiation. Derivative of composite and inverse functions. Differential calculus fundamental theorems. Global and local maxima and minima, points of inflexion. Necessary and/or sufficient conditions for the existence of maxima and minima. Concavity, convexity. − Integral Calculus: The Riemann (definite) integral. The Mean value theorem. The Torricelli-Barrow theorem. The indefinite integral. Some techniques of integration. ELEMENTS OF LINEAR ALGEBRA Vectors and matrices and corresponding operations. Determinants. Inverse matrix. Matrix rank. Systems of linear equations. Rouchè-Capelli theorem, Cramer’s rule. REAL FUNCTION OF TWO REAL VARIABLES The euclidean space R2. Domain. Level sets. Partial derivatives. Global and local maxima and minima. Saddle points. Unconstrained optimization: first and second order conditions. Constrained optimization via the level set approach and the Lagrange multiplier method. READING LIST L. SCAGLIANTI-A.TORRIERO, Manuale di Matematica, III edition, Cedam, 2009. M. SCOVENNA-R.GRASSI, Esercizi di Matematica, III edition, Cedam, 2010. M.BIANCHI-L. SCAGLIANTI, Precorso di Matematica, II edition, Cedam, 2010. F. BREGA-G. MESSINEO, Esercizi di Matematica generale, Voll. I (2006) e II (2008), Giappicchelli. Online instructional material is available on Blackboard. TEACHING METHOD Lectures (course and pre-course classes), assignments. ASSESSMENT METHOD Grading will be based on a written exam in which students will be required to answer theoretical and practical questions. The oral exam is optional for students having achieved a grade of 18/30 or higher on the written test, whereas it has to be taken by students with a grade on the written test of 15/30, 16/30 or 17/30 and also in other cases as specified in Blackboard. The oral exam concerns all the programme and some simple proofs can be asked in addition to the proofs of the following theorems: Uniqueness of limits, Sign preservation, Continuity of a differentiable function, Fermat’s theorem, Rolle’s Theorem, Mean value theorem, Mean value theorem for integral calculus, Fundamental theorem of integral calculus. For all students it is possible to take two partial tests (one during the class period and the other at the end) as an alternative to the written exam. More detailed information on the partial tests will be available on Blackboard. NOTES Further information can be found on the lecturer's webpage http://www2.unicatt.it/unicattolica/docenti/index.html or on the Faculty notice board. at Basic knowledge will be included in the exam, and thus attendance at the pre-course classes is highly recommended. More detailed information on the pre-course will be available on Blackboard. An on line pre-course Teorema is also accessible to the address http://teorema.cilea.it.