Academic Year 2015/2016 - 1° Year

MAT/05 - 8 CFU - 2° Semester

Subsets of real numbers and their properties. Maximum and minimum, supremum and infimum of a numerical set. Topology in the set of real numbers. accumulation/limit points. Functions of one real variable; elementary functions and their properties. Graph of a function. trigonometric functions and their "inverse functions". Definition of limit of a function. Unicity of limit, theorem of sign permanence, the Squeeze Theorem. Algebra of limits, indeterminate forms. Definition of continuous function. Theorem of Weierstrass. Theorem of existence of zeros. Intermediate value theorem. Limit of Neper. Asymptotic approximations. Differential calculus: definition of derivative and its geometrical meaning. Differentiable functions. Rules of derivation: derivative of a product, of a reciprocal, of a ratio. Derivative of a composite function: the chain rule. Derivative of elementary functions. Fermat's, Rolle's, Lagrange's and Cauchy's theorems; their applications to the study of the monotony and extremes of a function. Corollaries of the Lagrange's theorem. De l'Hopital's rule. Integral calculus. Primitive/antiderivative of a function. Indefinite integrals. Techniques of integration. Definition of definite integral. Fundamental theorem ofcalculus. Sequences of real numbers. regular sequences. Monotone sequences. Limit of a sequence of real numbers. Number of Neper. Systems of linear equations. Matrix algebra. Matrix operations, product of matrices. Determinant of a square matrix, rank of a matrix. Solution of a linear system, matrix representation of a linear system. Cramer's theorem, Rouchè-Capelli-Kronecker theorem. Vector calculus. Two-dimensional Euclidean spaces. Applied vectors of ordinary space. Vector algebra, sum and external product. Scalar product and vector product. Angle between two vectors. Plane cartesian geometry: Cartesian coordinates, distance between two points, the midpoint of a segment. Cartesian and parametric representations of the line. Parallelism and orthogonality conditions. Angle between two lines. Formula of the distance between a point and a line. Circle in euclidean plane and its equation. Some mathematical models for Biology.