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B024394 - MATHEMATICAL ANALYSIS I
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Further information
Type of Assessment
Course program
Academic Year 2015-16
Coorte 2015 - 3-years First Cycle Degree (DM 270/04) in Electronics and Telecommunications Engineering
Course year
First year - First Semester
Belonging Department
Information Engineering (DINFO)
Modulo di sola Frequenza of
Scientific Area
MAT/05 - MATHEMATICAL ANALYSIS
Credits
6
Teaching Hours
54
Teaching Term
21/09/2015 ⇒ 18/12/2015
Attendance required
No
Type of Evaluation
Giudizio Finale
Course Content
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Course program
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Lectureship
Teaching Language
Italian
Course Content
Numbers; sequences and series; functions of one real variable; limits and continuity; differential calculus in one variable and approximation; Riemann integral.
Suggested readings (Search our library's catalogue)
1) M. Bramanti, C.D. Pagani, S. Salsa,"Analisi Matematica 1", Zanichelli (2008)
2) S. Salsa, A. Squellati, "Esercizi di Analisi Matematica" Vol. 1, Zanichelli (2011)
3) G. Anichini, G. Conti, "Analisi Matematica 1", Pearson (2010).
4) Carlo Sbordone, Paolo Marcellini, "Esercitazioni di matematica Volume I", Parte prima e parte seconda
2) S. Salsa, A. Squellati, "Esercizi di Analisi Matematica" Vol. 1, Zanichelli (2011)
3) G. Anichini, G. Conti, "Analisi Matematica 1", Pearson (2010).
4) Carlo Sbordone, Paolo Marcellini, "Esercitazioni di matematica Volume I", Parte prima e parte seconda
Learning Objectives
Acquisition of mathematical tools needed for the description and understanding of physical phenomena. Strengthening and development of the attitude to both the analytical and logical deductive reasoning to identify the essential data in the analysis and synthesis of the presentation of possible problems.
Prerequisites
Elementary logic and algebra; literal calculus. Euclidean geometry in 2 and 3 dimenions. Lenght, area, volume of elementary hapes. Analytic geometry: polar coordinates and graphcs of elementary functions. Trigonometry.
Teaching Methods
The course includes the carrying out of lectures and classroom exercises.
Further information
Exams
See http://sol.unifi.it/docprenot/docprenot
More information and exercises can be found in the didactical section of the home page of
teacher:
https://e-l.unifi.it/enrol/index.php?id=448
See http://sol.unifi.it/docprenot/docprenot
More information and exercises can be found in the didactical section of the home page of
teacher:
https://e-l.unifi.it/enrol/index.php?id=448
Type of Assessment
The exam consists of a written part and an theoretical one. Partial tests are planned during the course, passing these tests is equivalent to passing the written exam.
Course program
THE NUMBERS
Sets. Set operations: union, intersection, difference, complement, cartesian product, inclusion. sum. Summation formulas and properties. Sum of numbers in a geometric progression. Natural numbers. Principle of induction. Factorial of n and binomial coefficients. Newton binomial. Rational numbers. Real numbers. Absolute value. Upper and lower bounds. Supremum and infimum, maximum and minimum of a set.
REAL FUNCTIONS
Domain, image, graph. Bounded, injective, surjective, monotone functions. Composition of functions. Inverse function. Exponential, logarithmic functions. Trigonometric functions. Hyperbolic functions. Inverse trigonometric and inverse hyperbolic functions.
SEQUENCES
Formal definition. Bounded sequences. Limit of a sequence. Uniqueness of the limit. Convergent, divergent, oscillating sequences. Operation with limits. Indeterminate forms. Squeeze theorem. Permanence of sign. Monotone sequences. The number e. Use of the asymptotic. Fundamental limits.
LIMIT OF FUNCTIONS. CONTINUOUS FUNCTIONS.
Intervals in R. Definition of limit. Relations with limits of sequences. Right-handed and left-handed limit. Operations. Uniqueness, permanence of sign, squeeze theorems. Horizontal, oblique, vertical asymptotes. Fundamental limits. Continuous functions. Continuity of the sum, product, ratio and composition of functions. Weierstrass theorem. Bolzano and intermediate value theorem. Continuity of monoton functions. Inverse of continuous functions.
DIFFERENTIAL CALCULUS OF ONE VARIABLE FUNCTIONS
Derivative and geometrical interpretation. Derivatives of elementary functions. Operation with derivatives. Differentiability and continuity. Derivatives of inverse functions. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. First derivative sign and monotonicity. Higher order derivatives. Convex and concave functions. Inflection points. Graphic representation of a function. L'Hospital’s rule. A sufficient condition of differentiability.
Linear approximation. Little o notation. Taylor's theorem. Peano and Lagrange forms of the remainder.
INTEGRAL CALCULUS OF ONE VARIABLE FUNCTIONS
Integral of bounded functions. Geometric interpretation. Class of integrable functions. Non-integrable functions. General properties: linearity, additivity, comparison. The integral mean value theorem. Primitives. The fundamental theorem of calculus. Methods of integration: substitution, decomposition by parts, by recurrence. Generalized integrals. Comparison and asymptotic comparison tests. Absolute convergence. Integral functions.
NUMERICAL SERIES
Convergent, divergent, oscillating series. Geometrical series. Necessary condition of convergence.
Comparison and asymptotic comparison. Root and ratio tests. Integral test. Generalized harmonic series. Absolute convergence. Leibniz criterion.
Sets. Set operations: union, intersection, difference, complement, cartesian product, inclusion. sum. Summation formulas and properties. Sum of numbers in a geometric progression. Natural numbers. Principle of induction. Factorial of n and binomial coefficients. Newton binomial. Rational numbers. Real numbers. Absolute value. Upper and lower bounds. Supremum and infimum, maximum and minimum of a set.
REAL FUNCTIONS
Domain, image, graph. Bounded, injective, surjective, monotone functions. Composition of functions. Inverse function. Exponential, logarithmic functions. Trigonometric functions. Hyperbolic functions. Inverse trigonometric and inverse hyperbolic functions.
SEQUENCES
Formal definition. Bounded sequences. Limit of a sequence. Uniqueness of the limit. Convergent, divergent, oscillating sequences. Operation with limits. Indeterminate forms. Squeeze theorem. Permanence of sign. Monotone sequences. The number e. Use of the asymptotic. Fundamental limits.
LIMIT OF FUNCTIONS. CONTINUOUS FUNCTIONS.
Intervals in R. Definition of limit. Relations with limits of sequences. Right-handed and left-handed limit. Operations. Uniqueness, permanence of sign, squeeze theorems. Horizontal, oblique, vertical asymptotes. Fundamental limits. Continuous functions. Continuity of the sum, product, ratio and composition of functions. Weierstrass theorem. Bolzano and intermediate value theorem. Continuity of monoton functions. Inverse of continuous functions.
DIFFERENTIAL CALCULUS OF ONE VARIABLE FUNCTIONS
Derivative and geometrical interpretation. Derivatives of elementary functions. Operation with derivatives. Differentiability and continuity. Derivatives of inverse functions. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. First derivative sign and monotonicity. Higher order derivatives. Convex and concave functions. Inflection points. Graphic representation of a function. L'Hospital’s rule. A sufficient condition of differentiability.
Linear approximation. Little o notation. Taylor's theorem. Peano and Lagrange forms of the remainder.
INTEGRAL CALCULUS OF ONE VARIABLE FUNCTIONS
Integral of bounded functions. Geometric interpretation. Class of integrable functions. Non-integrable functions. General properties: linearity, additivity, comparison. The integral mean value theorem. Primitives. The fundamental theorem of calculus. Methods of integration: substitution, decomposition by parts, by recurrence. Generalized integrals. Comparison and asymptotic comparison tests. Absolute convergence. Integral functions.
NUMERICAL SERIES
Convergent, divergent, oscillating series. Geometrical series. Necessary condition of convergence.
Comparison and asymptotic comparison. Root and ratio tests. Integral test. Generalized harmonic series. Absolute convergence. Leibniz criterion.