Università Roma Tre

quantifiers. Numbers: natural, integers, rational and real. Axioms of real numbers.
Cartesian coordinates in the plane. Points and vectors. Distance: formal definition. Absolute value. Density of Q in R.
Linear algebra: vector sum, scalar product. Matrices. Matrix operations of sum and product, determinant, rank of a matrix.
Matrix representation of linear transformations. Geometric meaning of the determinant.
Rotation matrices and omotethy. Parametric equation of the line. Orthogonality conditions.
Introduction to real functions. Graphs.Q
Working with graphics, absolute value of a graph. Exponential, logarithm of a function for which you know the plot.
Accumulation points. Limits. Operations with limits. Comparison theorem. Continuous functions. Theorems on continuous functions.
Asymptotes. Derivatives: definition, geometric meaning. Operations: sum, product, quotient, scalar product. Main rules of derivation. Equation of the tangent line at a point to the graph.
Derivative of a composite function and inverse functions. Stationary points.
Fermat's theorem. Theorems of Rolle and Lagrange. Monotony and sign of the first derivative.
Second derivatives, concavity, inflections. Plotting graphs of functions. Theorems of Cauchy and De l'Hopital. Word problems.
Taylor polynomial. Formula of the rest of Lagrange. Hyperbolic functions, conic sections as geometric loci. Classification of conic sections.
Introduction to the problem of calculating the area of a flat region. The fundamental theorem of calculus, definite integrals.
The theorem of the average. Integration by parts and substitution. Integration of rational functions.
Definition of parametric curve. From parametric to cartesian equations and viceversa.
Examples: circumference cycloid, conical. Vector and unit vector tangent vector and the unit vector normal. Length of a curve. Curvature

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