Dipartimento di Scienze della Vita e dell'Ambiente - Guida degli insegnamenti (Syllabus)

Basic elements of Calculus and Analytic Geometry

There will be lectures and exercises for a total of 32 two-hour lessons each (8CFU).

Knowledge:

The course aims to introduce students to some theoretical, methodological and applicative elements of differential and integral calculus for real functions of one real variable. It aims to provide students with the elements necessary for the understanding of analytical models in use for describing the scientific phenomena they will encounter in their later studies.

Ability to apply the knowledge:

The course aims to develop the ability to perform studies of functions, derivation, integration and solve simple differential equations

Soft skills:

Classroom and individual resolution of many problems and exercises will improve learning ability and independence of judgment. The study of deductive logical topics and the correct use of logical mathematical language develops communication skills.

Sets, Relations and Functions. Composition, invertibility. Natural, Integer, Rational and Real numbers. The Induction principle. Supremum, infimum, maximum, minimum. Modulus and powers. Exponential, logaritmic and angular functions. Limit of real sequences and its properties. Indeterminate forms. Monotone sequences. The Neper's number and related limits. Asymptotic comparison. Limits of real function of real variable. Properties. Indeterminate forms. Monotone functions. Asymptotic comparison. Continuity; The Weierstrass's and the Intermediate Values Theorems. Derivative and Derivative Formulas. Successive Derivative. The Fermat's, Rolle's, Lagrange's and Cauchy's Theorems. Derivative and monotonicity. Convexity. Primitives. The De L'Hospital's Theorems. Asymptotes and the study of the graphs of functions. Definite Integral and its properties. Fundamental Theorem and Formula of the Integral Calculus. Indefinite Integral and integration methods: sum decomposition, by parts and substitution. General Integral for first order linear ordinary differential equations. The Cauchy Problem. The Bernoulli's equations. The Malthus and Verhulst models for the population dynamics.

Methods for assessing learning outcomes:The exam consists of two tests. The first test (written) consists of a multiple choice test consisting of ten questions. Each correct answer corresponds to a score of 3 points. The wrong answers or no dates are worth 0 points. The test will be a satisfactory result if the total score will be at least 18. The first trial allows access to the second trial (which must be made within the time of the next written test, the result of the first test is invalidated if not) the second test consists of 4 theoretical questions each rated up to a maximum score of 8 points. The second test will be a satisfactory result if the total score will be at least 18.

Criteria for assessing learning outcomes: In the written test the student must demonstrate the ability to solve simple exercises. In the second test the student must demonstrate that he has learned the theoretical themes proposed in the lessons. Particular attention will be given to evaluate the student's ability to justify rigorously his assertions and in the proper use of logical mathematical language.

Criteria for measuring learning outcomes:The final grade is attributed out of thirty. The exam is passed when the rating is greater than or equal to 18. It is possible the award of full marks with honors (30 e lode).

Criteria for conferring final mark:The final grade is assigned equal to the average of marks obtained in the two tests.

F.G Alessio, P. Montecchiari, Note di Analisi Matematica uno, Esculapio editore

P. Marcellini - C. Sbordone, Elementi di Calcolo, Liguori editore

P. Marcellini - C. Sbordone, Esercitazioni di matematica vol. 1 (parte I e II), Liguori editore

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**Università Politecnica delle Marche**

P.zza Roma 22, 60121 Ancona

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