Facoltà di Ingegneria - Guida degli insegnamenti (Syllabus)

Program

Analisi Matematica 1 (ELE)
Mathematics 1
Flaviano Battelli

Seat Ingegneria
A.A. 2016/2017
Credits 9
Hours 72
Period I
Language ENG

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

Learning outcomes
KNOWLEDGE AND UNDERSTANDING:
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, Taylor, and Fourier, and their applications to solve concrete problems. Students should be able to apply these notions to solve real world problems.
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
The student must develop the ability of solving problems by applying theorems, tools and methods.
TRANSVERSAL SKILLS:
The student will be able to critically evaluate different practical solutions in order to identify the most appropriate objectives to be pursued.

Program
Elements of set theory. Natural, relative, rational and real numbers. Completeness, l.u.b. and g.l.b.. Limit points. Open and closed sets. Functions from R to R. The inverse function. The absolute value. Limits of a real valued function. Uniqueness of limits. Right and left limits. Permanence of sign. Comparison principle. 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. Monotone sequences and their limits. The number e. Series in R. Convergence, divergence and indeterminacy. The geometric and the telescopic series. Convergence criteria for series with terms of constant sign. The armonic and generalized armonic series. Absolute convergence. Leibnitz theorem. Continuous functions from R to R. Zeri, Weierstrass and intermediate values theorems. 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. Right and left derivative. The derivative of the inverse function. Max and min. Fermat, Rolle and Lagrange Theorems. De l'Hopital rule. Monotony test. 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. Asymptotes. Drawing the graph of a function. Riemann integral and its properties. 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 in integral, Cauchy, Lagrange and Schlomilch form. Indefinite integral. Improper integrals. Integrability criteria. Integrability of some elementary functions. Absolute integrability. Improper integrals and series. The integral criterium. Analitic function in the real field ans power series.Fourier series.

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 develop the ability of solving problems by applying theorems tools and methods taught in lectures

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