Bachelor's Degree in Electrical Engineering and Information Technology

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Calculus I

COURSE LANGUAGE: English

YEAR OF THE DEGREE PROGRAMME (I, II, III): I

SEMESTER (I, II, ANNUAL): I

CFU: 12

REQUIRED PRELIMINARY COURSES (IF MENTIONED IN THE COURSE STRUCTURE “REGOLAMENTO”) 
None.

PREREQUISITES (IF APPLICABLE) 
The mathematical content of secondary school programs

LEARNING GOALS 
To provide the fundamental concepts, in view of the applications, related to infinitesimal, differential and integral calculus for the real functions of a real variable; make the students acquire adequate logical formalization skills and conscious operational skills.

EXPECTED LEARNING OUTCOMES (DUBLIN DESCRIPTORS) 

Knowledge and understanding 
The students will have to show knowledge of the notions (definitions, statements, proofs if provided by the program) related to infinitesimal, differential and integral calculus for real functions of a real variable and the calculation tools developed, and to be able to understand related topics by elaborating the notions acquired.

Applying knowledge and understanding 
The students must demonstrate that he knows how to apply what he has learned in solving verification exercises developed by the teacher, related to topics such as: domains of functions, limits of sequences and functions, numerical series, graph of a function, integral calculus.

COURSE CONTENT/SYLLABUS 
Numerical sets - Naturals, integers, rationals. The axioms of real numbers. Infimum, supremum, maximum, minimum of a set. Archimedes' principle. Density of Q in R; roots; powers with real exponent.
Principle of induction. Bernoulli's inequality. Binomial formula. Elementary functions. Sequences - Limit of a sequence; first properties of limits: uniqueness of the limit, comparison theorem, sign permanence. Operations with limits and indeterminate forms. Monotone sequences: regularity theorem; the number “e”. Ratio criterion. N-th Root criterion. Arithmetic mean and geometric mean.
Ratio-root criterion. Cauchy convergence criterion. Subsequences. Bolzano-Weierstrass theorem. Numerical series - Definitions and first properties; operations with series. Geometric series, harmonic series and generalized harmonic series. Cauchy criterion for series. Series with non-negative terms: root, ratio, comparison, asymptotic comparison criteria. Euler-Mascheroni constant. Series with alternating signs: Leibniz criterion; estimate of the reminder. Absolutely converging series and their properties. Functions - Topology of the real line: cluster points, closed, open, compact sets. Limits of functions and their properties. Equivalent definition of limit. Operations with limits and indeterminate forms. Monotone functions: regularity theorems; continuous functions; Lipschitz functions; inverse functions; 3 composite functions. Maxima and minima: Weierstrass theorem. Zero value theorem, theorem of intermediate values. Uniformly continuous functions, Cantor's theorem. Differential calculus - Definition of derivative and its geometric meaning. Rules of derivation; derivatives of elementary functions. Relative extremes: necessary condition of the first order. Rolle and Lagrange theorems; characterization of monotone functions in intervals. Relative extremes: sufficient conditions of the first order. Theorem of extension of the derivative. First theorem of de L’Hôpital; second theorem of de L’Hôpital; calculation of limits that occur in an indeterminate form. Infinitesimal and infinite: principles of cancellation. Taylor's formula with remainder in the form of Peano. Taylor's formula with remainder in Lagrange form. Outline of Taylor series. Relative maxima and minima: necessary conditions and sufficient conditions of the second order. Geometric meaning of the second derivative. Convexity and concavity in an interval; characterization of convex functions in intervals; inflected; asymptotes; graphs of functions. Integral calculus - Outline of measure according to Peano-Jordan. Riemann integral of a bounded function in a compact interval. Area of the trapezoid. Integrability of monotone functions in compact intervals. Integrability of continuous functions in compact intervals. Properties of the definite integral. Integral mean theorem. Fundamental theorem of integral calculus. Primitives and indefinite integration. Indefinite integration rules: sum decomposition, integration by parts, integration by substitution, integration of rational functions. Generalization of the concept of integral: summability. Summability criteria.

READINGS/BIBLIOGRAPHY 
Check the teacher's website.

TEACHING METHODS 
The lessons will be face to face, and about one third of the lessons will be of exercises.

EXAMINATION/EVALUATION CRITERIA 

Exam type:

  • Written and oral.

In case of a written exam, questions refer to:

  • Multiple choice answers.
  • Open answers.
  • Numerical exercises.

Evaluation criteria:
The grade is formulated by the Examination Commission based on the outcome of written test, in particular on the basis of the consistency and accuracy of the exercises performed and the adequacy of the answers provided by the student to the theory questions. The final grade is also suitably motivated to the student.