MATHEMATICAL ANALYSIS I.A
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- Versione italiana
- Academic year
- 2017/2018
- Teacher
- DAMIANO FOSCHI
- Credits
- 6
- Didactic period
- Primo semestre (primi anni)
- SSD
- MAT/05
Training objectives
- Aim of the course is to provide the basic tools of mathematical analysis, especially for what concerns the differential calculus for functions of one real variable and its applications to the resolution of problems based on mathematical models.
At the end of the unit, students will know the theoretical contents and the methods proper of mathematical analysis calculus.
They should be able to properly apply the concepts learnt to the resolution of different kinds of problems and to identify the most appropriate approach to do so.
Students will have to master the mathematical language and the logical-deductive approach, demonstrating the ability to illustrate their problem solving strategies in a logical, effective, pertinent and synthetic way. Prerequisites
- Buona conoscenza di:
elementi di base della teoria degli insiemi
proprietà elementari dei numeri Naturali, Interi e Razionali
risoluzione di equazioni e disequazioni di primo e secondo grado
algebra dei polinomi: fattorizzazione di un polinomio date le sue radici, calcolo della divisione tra polinomi
teorema di Pitagora e teoremi di Euclide per i triangoli rettangoli
trigonometria: definizione di seno, coseno, tangente; formule di addizione per seno e coseno
proprietà elementari delle funzioni esponenziali e logaritmiche
geometria analitica nel piano (rette, parabole, circonferenze, ellissi, iperboli). Course programme
- 1) Natural numbers, induction principle.
2) Algebraic properties of numerical fields, rational numbers.
3) Handling of summations. Arithmetic Progressions and Geometric Progressions. Telescopic sums.
4) Elementary combinatorics. Newton's binomial formula.
5) Decimal Alignments, Real Numbers.
6) Ordering of the real straight line, supremum and infimum, completeness properties.
7) Bernoulli's inequality. Relationship between arithmetic and geometric mean.
8) Nepero's number.
9) Definition of powers with integer, rational, and real exponents; roots, logarithms.
10) Functions: Domain, Graphs, Image, Counter-Image, Injectivity, Suriectivity, Bijectivity. Restrictions and extensions.
11) Composition of functions. Invertibility. Inverse function.
12) Limits. Incremental ratios, monotonicity and convexity. Symmetries, periodicity.
13) Elementary functions: powers with integer exponent, radicals, powers with real exponent.
14) Elementary functions: exponentials and logarithms. Hyperbolic functions and their inverses.
15) Trigonometric functions and their inverse. Trigonometric formulas.
16) Absolute value, positive and negative part; step, sign, whole part, mantissa.
17) Graphing manipulation: compositions with translations, homotheties, and elementary symmetries.
18) Euclidean structure of R^n: norm, scalar product, triangular inequality.
19) Metric and topological structure of R^n: distance, balls, open and closed sets, boundary. Extended Real line. Intervals.
20) Accumulation points, isolated points. Locally and definitively valid properties.
21) Limits: topological and metric definition. Definition of continuity.
22) Uniqueness of limit. Limits of restrictions. Right, left, by above, by below limits.
23) Comparison and sign permanence properties. Limits of monotone functions. Upper and lower limits.
24) Asymptotic relationships.
25) Infinities and infinitesimals. Algebraic properties of limits. Indefinite forms.
26) Limits of Rational Functions.
27) Limits and continuity of composite functions. Method of substitution in the calculus of limits.
28) Continuity and remarkable limits of elementary functions.
29) Classification of discontinuity points.
30) Compact sets. Bolzano-Weierstrass theorem. Compact sets of R^n. Weierstrass theorem.
31) Continuous function defined on a interval: the zero theorem, intermediate values theorem.
32) First order approximation, tangent line, derivative.
33) Differentiation implies continuity. Derivatives of sums, products, quotients, polynomials.
34) Derivative of elementary functions.
35) Derivative of composite functions.
36) Derivative of inverse functions.
37) Stationary Points. Fermat's theorem. Rolle's theorem.
38) Lagrange's theorem. First derivative, monotonicity intervals, maximum or minimum local points.
39) Optimization Problems.
40) Convexity and monotonicity of the first derivative. Second derivative, convexity intervals, flex points.
41) Cauchy's theorem. Rule of De L'Hopital.
42) Higher order approximation.
43) Taylor's Polynomial.
44) Computation of limits using Taylor's approximation.
45) Estimates for the remainder in Taylor's approximation. Approximation problems.
46) Vertical, horizontal, oblique asymptotes. Angular points, cusps.
47) Study of the graph of a function.
48) Summary exercises. Didactic methods
- The course provides 48 hours of classroom lectures with presentation of theoretical aspects, applications and exercises on the blackboard.
There will also be periodic tutoring sessions (two hours per week) with exercises and review of the topics. Learning assessment procedures
- The course learning assessment is done through a written test and an oral interview.
- In the written test the student is required to solve problems and exercises related to the course topics. The expected time for the written test is approximately 3 hours. It is not permitted to consult texts, use PC, tablet or smartphone. However, the student may still consult a personal hand written sheet (A4) in which he or she can write anything, that sheet must be delivered together with the test solutions. The student can use a pocket scientific calculator as long as it does not have graphic capabilities and can not be programmed. At the end of the written test, a score of at most 30 points is awarded. To pass the test and gain access to the oral interview you must get a score of at least 15 points.
- In the oral interview the student will be required to present some aspects of the course topics, illustrating some definitions, examples, properties, formulas, theorems, proofs, or applications. More than the mnemonic knowledge of the topics, we want to evaluate the logical understanding of concepts, the accuracy and rigor of the mathematical language used to describe them, and the ability to grasp the relationship between abstract aspects and concrete applications. The time for the oral test is about 30 minutes. If the outcome of the interview is not considered sufficient, the student will be able to retry it at a later date without necessarily having to repeat the written test.
The final vote is proposed at the end of the oral interview and will take into account all the elements that allow the teacher to evaluate the student's preparation: active participation during lectures, correctness and completeness in the written test, quality of exposure in the oral interview.
Passing the exam certifies to have acquired the knowledge and skills specified in the learning objectives of the course. Reference texts
- M.Bertsch, R.Dal Passo, L.Giacomelli - "Analisi Matematica" -McGraw-Hill (ISBN978-88-386-6234-8).
M.Bramanti, C.D.Pagani,S.Salsa-"Analisi matematica 1" con elementi di geometria e algebra lineare - Zanichelli (ISBN 978-88-08-25421-4)
Exercises: Marco Bramanti Esercitazioni di Analisi Matematica I Società Editrice ESCULAPIO ISBN 978-88-7488-444-5
