Mathematics I (2 º Sem 2019/2020)
Program
1. LINEAR ALGEBRA
1.1. Vectors
Vectors in Euclidean spaces. Vector algebra. Linear combination. Linear dependence and linear independence.
1.2. Matrices
Basic definitions and matrix algebra. The rank of a matrix and the Gauss method. Inverse matrix.
1.3. Determinants
Definition and expansion by minors. Properties.
1.4. Systems of linear equations
Definition. Matrix form and the Gauss method for classifying and solving systems of linear equations. The homogeneous system and the Cramer?s system.
2. CALCULUS
2.1. The real number system and infinite series.
Mathematical induction. Axioms of the real number system. Topology. Infinite series and power series.
2.2. One-variable functions
Basic definitions. Trigonometric functions and inverse trigonometric functions.
2.3. Limits and continuity
The notion of limit. Continuous functions and continuous extension. Theorems about continuous functions.
2.4. Differentiation
The notion of derivative. Theorems about differentiable functions. Taylor polynomials and Taylor?s formula. Optimisation.
2.5. Antidifferentiation and integration
Definitions and basic results. Methods of antidifferentiation. The definite integral. Improper integrals. Areas of plane regions. The fundamental theorem of calculus.
1.1. Vectors
Vectors in Euclidean spaces. Vector algebra. Linear combination. Linear dependence and linear independence.
1.2. Matrices
Basic definitions and matrix algebra. The rank of a matrix and the Gauss method. Inverse matrix.
1.3. Determinants
Definition and expansion by minors. Properties.
1.4. Systems of linear equations
Definition. Matrix form and the Gauss method for classifying and solving systems of linear equations. The homogeneous system and the Cramer?s system.
2. CALCULUS
2.1. The real number system and infinite series.
Mathematical induction. Axioms of the real number system. Topology. Infinite series and power series.
2.2. One-variable functions
Basic definitions. Trigonometric functions and inverse trigonometric functions.
2.3. Limits and continuity
The notion of limit. Continuous functions and continuous extension. Theorems about continuous functions.
2.4. Differentiation
The notion of derivative. Theorems about differentiable functions. Taylor polynomials and Taylor?s formula. Optimisation.
2.5. Antidifferentiation and integration
Definitions and basic results. Methods of antidifferentiation. The definite integral. Improper integrals. Areas of plane regions. The fundamental theorem of calculus.