Course program
Preliminary:
Numerical sets (natural, relative, rational and real numbers). Maximum, minimum, upper and lower extremes. ranges, boundaries, distance: open and closed sets. Cartesian equation of a plane line, perpendicularity and parallelism between lines.
Functions of one real variable:
Set of definition, image, graph, limited functions, symmetrical, monotonous, periodic, composite and inverse functions. Elementary functions: Potencies, Exponentials, logarithms, trigonometric functions. Limits: Definition and main properties (comparison theorem, algebraic operations etc.), indeterminate forms and notable limits. Continuous functions, algebraic operations with continuous functions and composition of continuous functions. Consequences of continuity in closed and limited intervals: permanence of the sign, existence of the zeros, intermediate values and existence of maxima and minima. Derivative: definition and geometrical meaning. Tangent line. Derivatives of elementary functions, derivation rules, derivative of composite functions and inverse functions; Application of derivatives to the calculation of limits: De Hopital theorem. Application of derivatives to the study of functions: Fermat's theorem, Lagrange's theorem and consequences. Maximum and minimum, secondary derivative, convexity and concavity. Asymptotes. Study of the graph of a function.
Integral calculus for functions of one real variable:
Definition and properties of definite and undefined integral, fundamental theorem of integral calculus, integration methods; Calculation of areas of flat figures.
Prerequisites
No prerequisites
Books
Every book of Mathematical Analysis of university level. Among these: P. Marcellini, C. Sbordone "Istituzioni di matematica", Liguori Editore or "Mathematica 1 " by G. Crasta and A. Malusa Ed. Pitagora or any other text of theory of calculus;
Every exercise book of Mathematical Analysis of university level. Among these: "Esercitazioni di matematica" by P. Marcellini, C. Sbordone (Volume 1) or any other book of exercises.
Exam mode
The goal of the exam is to assess the level of preparation through one written and one oral exam (if requested by the student). The written exam has a duration of about 2h and can be replaced by two written intermediate written exams of 2h each, one at the half and another at the end of the lesson cycle.
The written part are exercises on the main arguments of the course while the oral examination is substantially devoted to verify the comprehension of the proof of the arguments explained in the lessons.
To pass the exam a mark not less than 18/30 is required. To get this, the student must show a sufficient knowledge of the simpler techinques and the most important arguments touched during the course.
To get full marks, 30/30 e lode, the student must show an excellent knowledge of all topics covered during the course (and hence it is mandatory the oral examination).
