ANALISI MATEMATICA 1 PARTE B

Students should be capable of studying qualitative and quantitative properties of single-variable functions and applying them to both theoretical and practical problems. They should be able to graph a function, justifying it based on the theoretical knowledge acquired. Additionally, they should know how to solve relatively simple differential equations. The course provides the necessary skills to effectively tackle the subsequent course in Mathematical Analysis II.

1.General educational objectives of the course in terms of expected learning outcomes:

1. 2.Knowledge and understanding: The course aims to provide a theoretical foundation and some applications related to Differential and Integral Calculus for functions of a real variable.

2. Applying knowledge and understanding: Students will acquire the necessary skills to study simple models.

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4. 3.Autonomy in judgment: Through concrete examples and exercises, students will be able to independently develop their own solutions to some simple problems.

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6. 4.Communication skills: Students will gain further communication skills and expressive appropriateness in using theoretical language in the context of Mathematical Analysis.

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8. 5.Learning skills: The course aims to provide students with the necessary theoretical and practical methodologies to independently address and solve problems that may arise during their design activities.

Blackboard lectures with proofs.

Attending every class is Strongly recommended.

1. Definition of derivative and its geometric meaning. Differentiability and continuity. Derivatives of elementary functions. Algebra of derivatives. Derivatives of composite functions and inverse functions. Higher order derivatives. Relative maxima and minima. Fermat's theorem. Rolle's, Cauchy's, and Lagrange's theorems. Monotonicity characterization for differentiable functions on an interval. Functions with zero derivative on an interval. L'Hôpital's theorems. Taylor's formula. Convex functions on an interval. Derivatives of functions with complex values. Recursive sequences. Numerical solution of equations: Newton's method and the method of chords.

2.Riemann Integral. Definition, properties, and geometric significance. Example of a non-integrable function. Integrability of continuous functions. Integrability of monotonic functions. Integrability of generally continuous and bounded functions. Properties: linearity with respect to the integrand, positivity, monotonicity, additivity with respect to the integration interval. Integrability of the absolute value of an integrable function and its estimation. Mean value theorems. Primitives. Example of a function with no primitives. Integral function of a continuous function and the Fundamental Theorem of Calculus. Torricelli's Theorem. Indefinite integral. Integration by parts and substitution. Integration of rational functions. Integration by rationalization of certain classes of irrational and transcendental functions. Generalized and improper integrals. Criteria for summability and absolute summability. Improper integrals and series of numbers. Integrals of functions with complex values.

3. Definition of a differential equation and definition of a solution. Cauchy problem. Sufficient condition for uniqueness*. Separable variable differential equations. First-order linear differential equations. Homogeneous differential equations (of Manfredi). Bernoulli's differential equations. Higher-order linear differential equations with constant coefficients. Differential equations with complex coefficients. Lagrange's method of variation of parameters. Application to harmonic motion.

1. M.Giaquinta - G. Modica Mathematical Analysis: Functions of one variable Birkhäuser (2013)
2. W. Rudin - Principles of Mathematical Analysis 3 ed Mc Graw Hill

Solving a list of exercises and eventually an oral discussion.