1. Linear Algebra
Numerical space with n dimensions. Vector space of n dimensions. Basis of a vector space. Fundamental theorem of the dimension. Ordinary vector spaces. Some generalizations. Elements of geometry in R. Plans. Straight. Parallelism. Hyperplanes.
Scalar product. Perpendicularity between lines, lines and planes. Angles between two vectors.
Matrices. Vector space structure. Operations between matrices.
Definition of determinant. Laplace's theorem. Properties of determinants. Rank of a matrix. Rank and linear dependence.
Inverse of a matrix. Inverse theorem. Reverse by distribution. Inverse by means of elementary transformations.
Linear systems. Normal systems. Non-normal systems. Homogeneous systems.
2. Elementary functions, limits and continuous functions
Nth power and nth root. exponential function and logarithm. Sine and cosine functions and their inverses. Tangent and cotangent functions and their inverses. Elementary functions.
Absolute value of a real number. Limits of sequences. Uniqueness theorems of the limit, of the compound function and of the restriction. Special cases of R, expanded R, R.
Sequences of real numbers. Regularity criteria. Fundamental theorems on the limits of sequences.
Limits of numerical and vector functions. Fundamental theorems on the limits of numerical functions. Elementary functions and their limits.
Continuous functions Definitions and first examples. Calculation of limits by substitution. Continuous function theorems. Complements on connected sets.
Infinitesimal and infinite. Definition and first properties. Theorems on infinitesimals and infinites. Applications of theory and exercises.
3. Derivation
Incremental and derivative relationship. Examples. Differentiable functions. Left derivative, right derivative. Left and right derivative theorem. Tangent to a curve. Differential theorem.
Rules of derivation. Linearity theorem. Derivative of the product and of the quotient. Derivative of the compound function. Derivative of the inverse function. Derivatives of elementary functions. Derivatives of higher order than the first. Differentials of higher order than the first.
Crescence, decrease, maximums and minimums. Maximum and minimum relative and absolute. Necessary and / or sufficient conditions for increasing, decreasing, maximum and minimum.
Rolle's theorem. Lagrange's theorem and its consequences. Cauchy's theorem. Hospital theorems. Taylor's formula with the remainder of Peano. Taylor's formula with Lagrange's remainder. Use of Taylor's formula for the theorems relating to increase, decrease, maximum and minimum and for the approximate calculation of the values assumed by a function.
Asymptotes. Convex and concave functions. Flex. Theorems for the research of the points of concavity and convexity and of inflections. Study of the graph of a function. Calculation of limits with the help of Hospital's theorems and Taylor's formula. Exercises and complements on the Taylor formula.
4. Integration
Integrals of one-variable functions. Geometric definition and interpretation. Lower and upper integral. Some properties of integrals. Comparison and mean theorem. The integrable functions. Absolute value theorem. Applications of the integral. The fundamental theorem of integral calculus. Theorems on primitives.
Definition of indefinite integral and immediate integrals. Transformation theorem of the integrand function. Integration theorem by parts. Substitution theorem. Integrations of fractional rational functions. Methods for calculating definite integrals: exact and approximate methods. Taylor's formula with integral remainder. Form of Cauchy and Lagrange. Applications of Taylor's formula for the computation of integrals. Exercises on integrals.
5. Differential equations
Linear differential equations of the first and second order. Existence and uniqueness theorems for solutions. Integration of exact differential equations, with separable variables or attributable to them. Integrating factor.
