Search    Print

Analisi Matematica 1 (ELE)
Mathematics 1
Flaviano Battelli

Seat Ingegneria
A.A. 2015/2016
Credits 9
Hours 72
Period I
Language ENG

Prerequisites
Trigonometry, elements of analytical geometry in the plane and the space

Learning outcomes
To know and understand the theory of functions of one real variable (limits, continuity, differentiability, optimization, integration) and of the sequences and series in R and their applications to solve concrete problems. Students should be able to apply these notions to solve real world problems.

Program
Elements of set theory. Natural, relative and rational numbers. The field of real numbers. Completeness, l.u.b. and g.l.b. of non empty subsets. Limit points. Open and closed sets. Functions from R to R. 1:1 and onto functions, bijections. The inverse function. The absolute value. Limit of a real valued function. Neighborhoods and limits. Uniqueness of limits. Comparison theorem. Right and left limits. Permanence of sign. Algebra of limits. Few noticeable limits. Monotone functions and their limits. Landau symbols. Algebra of little-o and big-O. The principle of substitution of little-o. Limits of sequences in R. Divergent sequences. Uniqueness of limits. Compairison principle. Operations with limits. Few noticeable limits. Monotone sequences and their limits. The number e. Series in R. Convergence, divergence and indeterminacy. The Mengoli, the geometric and the telescopic series. Series with terms of constant sign. The armonic and generalized armonic series. Convergence criteria. Absolute convergence. Leibnitz theorem. Continuous functions from R to R and their algebra. Zeri, Weierstrass and intermediate values theorems and their consequences. Discontinuous functions. Continuity of the composition map. Continuity of monotone functions and of the inverse function. Derivative of a function from R to R. Differentiable functions. Differentiability and continuity. Differentiability rules, the chain rule. Right and left derivative. The derivative of the inverse function. Max and min. Fermat, Rolle and Lagrange Theorems and their consequences. De l'Hopital rule. Monotony test. Recognising critical points. Taylor and Mac Laurin approximation of functions with remainder in Peano form. Recognizing critical points with Taylor formula. Taylor formula and limits. Convex and concave functions. Drawing the graph of a function. Riemann integral for real functions of a real variable. Linearity and monotonicity of the integral. Riemann Criterium. Integrability of continuous and monotonic functions. Mean value and weighted mean value theorems. Fundamental theorem of calculus. Integration rules. Integration of rational functions and some irrational functions. Taylor formula and the remainder term in integral form and consequences (Cauchy, Lagrange and Schlomilch remainders). Indefinite integral. Improper integrals. Comparaison criterium. Integrability of some elementary functions. Absolute integrability. Improper integrals and series. The integral criterium. Analitic function in the real field ans power series. Linear differential equations.

Development of the examination
LEARNING EVALUATION METHODS
Written proof followed by oral proof. To be admitted to the oral proof the student has to receive a mark of at least 15/30 in the written proof. The student must pass the oral proof in the same session of the written proof. Should the oral proof give a negative result the student must repeat also the written proof.

LEARNING EVALUATION CRITERIA
The student must prove to have understood the topics of the lectures and to know how to use them to solve concrete problems.

LEARNING MEASUREMENT CRITERIA
At each proof a mark is given between 0/30 and 30/30

FINAL MARK ALLOCATION CRITERIA
The student will pass the exam if his mark at the written proof is not less than 15/30 and the final mark is at least 18/30. The final mark is obtained adding the 2/5 of the mark obtained in the written proof to the 3/5 of the mark obtained in the oral proof. Full marks and honors is given to the student who deserves full marks and has shown to be particularly brilliant during the oral proof

Recommended reading
Bertsch, Dal Passo, Giacomelli, Analisi Matematica, Mc Graw Hill. Other reference texts: 1 - Marcellini, Sbordone; Analisi Matematica 1; Liguori 2 - Fusco, Marcellini, Sbordone; Analisi Matematica 2; Liguori 3 - Giusti, Analisi Matematica 1; Bollati Bor

Courses

• Ingegneria Elettronica (Corso di Laurea Triennale (DM 270/04))