Industrial pharmacy REPLICA LATINA

Numbers Review of sets and set operations, fundamental numerical sets and their properties, powers and roots, powers with real exponents, proportions and percentages, scientific notation, order of magnitude, absolute value (modulus). Review of solving quadratic equations and inequalities (both integer and fractional). Exponentials and logarithms, logarithmic and exponential equations and inequalities, absolute value equations and inequalities. Irrational equations and inequalities. Trigonometric equations and inequalities. Functions of One Variable Concept of a function. Real functions of a real variable: general principles. Properties of functions: injective, surjective, bijective functions; monotonic functions; bounded functions; even and odd functions; periodic functions. Inverse and composite functions. Domain of a function, intersections with the axes, sign of a function. Algebraic functions: polynomial and rational functions; irrational functions. Transcendental functions: exponential and logarithmic functions; trigonometric functions. Piecewise-defined functions. Limits and Continuity Intuitive approach to the concept of limits. Limits of functions, asymptotes. Fundamental properties of limits and continuity. General limit theorems (with proofs). Notable limits (including lim ⁡ 𝑥 → 0 sin ⁡ 𝑥 𝑥 = 1 ). Indeterminate forms, infinitesimals, and infinite quantities. Definition of continuity, examples of continuous functions. Continuity of inverse and composite functions. Theorems on continuous functions (with graphical interpretation). Types of discontinuity. Numerical Sequences Definition of sequence. Limit of a sequence and related theorems. Differential Calculus Introduction to differential calculus. Definition and properties of the derivative. Fundamental theorems on derivability and continuity (with proofs). Geometric and trigonometric interpretation of the derivative. Fundamental derivatives, algebra of derivatives. Derivatives of inverse and composite functions. Tangent lines, angular points, cusps, vertical tangents. Higher-order derivatives. Rolle’s theorem, Cauchy’s theorem, Lagrange’s mean value theorem (with consequences). De L’Hôpital’s theorem. Relative and absolute extrema, Fermat’s theorem. Concavity and convexity, inflection points. Graph analysis of functions. Integral Calculus Introduction to integral calculus. Primitive function and indefinite integrals. Properties of integrals, integration techniques. Fundamental theorems of integral calculus (Torricelli-Barrow theorem). Definite integrals. Differential Equations General differential models. First-order differential equations. Cauchy problem. Linear differential equations. Vectors Definition of 𝑛 -dimensional vectors and components. Vector operations. Linear systems, compatibility, and Rouché-Capelli theorem. Matrices and Transformations Definition and properties of matrices. Matrix operations, determinants, matrix rank. Applications to linear systems. Probability Calculus Random events and probability, combinatorial analysis. Conditional probability, independent and dependent events. Bayes' theorem. Random Variables Discrete and continuous random variables, probability distributions. Mean value, variance, standard deviation. Binomial and Poisson distributions. Normal (Gaussian) distribution. Elements of Computer Science Information representation. Computational thinking, algorithms, and data structures. Problem-solving approaches. Programming languages.

Basic mathematics common to all five-year upper secondary school courses

Recommended Books: - D. Benedetto, M. Degli Esposti, C. Maffei - Matematica per le scienze della vita - CEA - S. Barbero, S. J. N. Mosconi, A. Portaluri – Matematica per le scienze con elementi di probabilità e statistica - Pearson - C. Bisi, R. Fioresi - Metodi matematici per le scienze applicate - CEA - P. Marcellini, C. Sbordone - Analisi Matematica Uno - Liguori Editore - P. D’Ancona, M. Manetti - Istituzioni di Matematiche – 2018 (disponibile in rete Note: The student is free to study the course topics using other texts.

Attendance at lectures is not mandatory, but it is strongly recommended.

The evaluation of the course consists of a written exam, followed by a possible oral exam. The Computer Science portion of the syllabus will be assessed in the oral exam. There will be five regular exam sessions and two extraordinary exam sessions throughout the academic year. The written exam, lasting 2 hours, consists of six questions (problems, exercises, proofs of formulas and/or theorems). The oral exam, to be taken in the same session as the written test, includes questions related to the written exam and the topics covered during the course. The oral exam is mandatory for students who score between 15/30 and 18/30 on the written exam, while it is optional for those scoring 18/30 or higher. Students wishing to improve their written exam score or those who obtained the highest possible score (30/30) and aspire to receive distinction (cum laude) must take the oral exam. It should be noted that in such cases, the final grade will reflect the performance in both tests and may therefore be lower than the written exam score. If the oral exam is not passed, the student must retake both the written and oral components. The purpose of the exam is to certify students' ability to apply acquired knowledge and mathematical tools to problem-solving. The oral exam further evaluates the ability to construct appropriate reasoning to support the chosen solution approach and the use of correct technical language. Evaluation Criteria The elements considered for grading include: Knowledge of the subject matter Use of appropriate language Demonstrated reasoning ability during the exam Autonomous study skills using the recommended texts A sufficient understanding of the covered topics and basic proficiency in problem-solving is required to pass the exam with the minimum grade. To achieve 30/30 with distinction, students must demonstrate an excellent grasp of the subject matter, an ability to select the appropriate mathematical tools, and proficiency in using them effectively. They must also construct clear, correct, and complete arguments while employing mathematical language appropriately. Exam Rules and Accommodations During the written exam, students are not allowed to leave the examination room. Use of electronic devices capable of accessing files, sending photos, or connecting externally via wireless or phone networks is strictly prohibited—except for non-programmable scientific calculators. Students with DSA certification, disabilities, or other special educational needs are encouraged to contact the instructor at the beginning of the course to arrange appropriate teaching and exam accommodations that respect the learning objectives while addressing individual learning styles and compensatory tools.

Lessons are typically lecture-based, alternating between theory and exercises, some of which actively involve students, with the possible use of innovative teaching tools.