ANALISI MATEMATICA 1 PARTE B

Knowledge and understanding

The course provides the basic knowledge of infinitesimal calculus for functions of one variable and contributes to developing inductive and deductive reasoning skills. Students will learn the fundamental mathematical concepts and develop the ability to compute and manipulate the most common objects of Mathematical Analysis, including differential and integral calculus.

Applying knowledge and understanding

Students will not simply learn individual concepts but will be able to apply the acquired knowledge to the mathematical modelling of classical problems arising from other scientific fields.

Making judgements

Students will be encouraged to deepen their knowledge independently and to work on exercises related to the topics covered. Seminars are planned to illustrate the topics studied, together with tutorials in which students can critically discuss with one another to find the correct solutions to the exercises.

Communication skills

Attending lectures and reading the recommended textbooks will help students become familiar with mathematical language. Through tutorials and seminars, they will learn to communicate the acquired knowledge rigorously and clearly, both orally and in writing. By the end of the course, students will have learned that mathematical language is a useful tool for clear scientific communication.

Learning skills

Students will be guided in refining their study method. In particular, through guided tutorials they will be able to independently approach new topics, recognising the prerequisites required to understand them.

The course is structured into 9 ECTS credits, of which 7 credits (49 hours) are dedicated to lectures and 2 credits (24 hours) to practice sessions.

The concepts and methods covered by the course will be presented through lectures (Direct Teaching). For each topic, the instructor will carry out an adequate number of tutorials. To develop judgement autonomy and communication skills, and to make participation in lectures more active and productive, some hours will be devoted to guided tutorials (Interactive Teaching), in which various exercises will be proposed. Students may work individually or in groups and discuss with one another.

If the course is delivered in blended or remote mode, appropriate adjustments may be made to the above, in order to ensure consistency with the syllabus.

1. Further Topics on Series. Main convergence tests; representation in base (b); complex-valued series; complex exponential; brief introduction to series of functions and power series.

2. Differential Calculus. Definition of the derivative and its geometric interpretation. Differentiability and continuity. Derivatives of elementary functions. Algebra of derivatives. Derivatives of composite and inverse functions. Higher-order derivatives. Fermat's, Rolle's, Cauchy's and Lagrange's theorems. Characterization of monotonicity for differentiable functions on an interval. Functions with zero derivative on an interval. Darboux's theorem. de l'Hôpital's theorems. Taylor's formula. Sufficient conditions for local extrema. Convex functions on an interval and their characterization in terms of derivatives. Necessary condition for inflection points. Applications: curve sketching; recursively defined sequences.

3. Integral Calculus. Riemann integral: definition, properties and geometric interpretation. Example of a non-integrable function. Integrability of continuous functions, monotone functions, and piecewise continuous functions. Properties: linearity with respect to the integrand, monotonicity, and additivity over intervals. Mean value theorems. Antiderivatives and indefinite integrals. Example of a function without an antiderivative. Fundamental Theorem of Calculus, counterexamples and variants. Torricelli's theorem. Integration by parts and substitution. Integration of rational functions and of certain classes of irrational and transcendental functions. Generalized and improper integrals. Summability, absolute summability, and sufficient conditions. Improper integrals and numerical series. Applications: Stirling's formula and Taylor's formula with integral remainder.

4. Differential Equations. Definition of a differential equation and of a solution. Cauchy problem. Sufficient condition for uniqueness*. First-order differential equations: separable equations, linear equations, Manfredi's equations, and Bernoulli equations. Higher-order linear differential equations with constant coefficients. Method of variation of constants (Lagrange). Application to harmonic motion.

1. J. R. Hass, G. B. Thomas, M. D. Weir, Thomas Calculus, Pearson 
   
2. M. Spivak, Calculus, Cambridge University Press
3. T. Tao , Analysis 1, Springer Nature
4. W. Rudin, Principles of Mathematical Analysis, Mc Graw Hill 
   
5. B. Demidovich, Problems in mathematical analysis, Mir Publishers
   

The examination consists of a written test, upon passing which students are admitted to a compulsory oral examination to be taken within the examination session. The written test is structured as follows:

1. Multiple-choice test consisting of five questions, lasting 15 minutes. The correct answers are provided at the end of the test. In order to proceed to the second part of the written examination, students must have answered at least three questions correctly.

2. Solution of four theoretical and practical exercises concerning the topics covered during the course. The duration of the second part is 120 minutes.

The final grade for the written examination will be calculated as the grade obtained in Part 2 (from 6 to 8 points per exercise, depending on its difficulty), plus one point for each question answered correctly in Part 1. The written examination is passed with an overall score of at least 18 points.

Learning assessment may also be carried out on-line, should the conditions require it. To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l'integrazione Attiva e Partecipata – Servizi per le Disabilità e/o i DSA) referring teacher within their department (https://www.cinap.unict.it/content/referenti).

• Determine the behaviour of a series, possibly depending on a parameter
• Determine the qualitative behaviour of the graph of a function
• Use Taylor expansion to compute a limit, possibly depending on a parameter
• Compute the limit of a recursively defined sequence
• Compute a definite/indefinite/improper integral
• Determine the summability of a function, possibly depending on a parameter
• Solve a Cauchy problem