English class
Final grade
Final Exam 100% (2 hours, no calculator, no consultation)
Old exams
Program
Part I - Linear algebra and linear systems of equations
-
Vectors
- Definition, geometrical representation, elementary operations, inner product, norm, distance, Cauchy-Schwarz inequality.
- Linear combination and linear independence of vectors
- Matrices
- Definition, properties, sum and scalar product, transpose
- Product of matrices
- Gauss elimination method, rank
- Gauss method as product of matrices
- Systems of
k linear equations with
n variables
- No solutions
- Unique solution
- Infinite solutions
- Systems of
n linear equations with
n variables
- Determinant of a matrix
- Properties of determinants
- Inverse matrix
- Cramer's rule
Part II-Mathematical analysis: real-valued functions
- Functions and series
- Definition of a function, sequences, odd and even functions
- Series, arithmetica and geometrical series, convergence
- Real-valued functions, limits, continuity, linear and non-linear functions
- Composition of functions
- Polynomial and exponential functions
- Derivatives
- Definition, geometrical interpretation
- Rate of change, monotonicity, elasticity
- Chain rule, derivative of the inverse function
- Implicit differentiation
- L'Hôpital/Cauchy's rule
- Polynomial approximation of functions
- Linear and quadratic approximations
- Taylor series
- Intermediate value theorem
- Mean value theorem
- Optimization
- Stationary and extreme points
- Second derivative classification
- Convexity and concavity
- Integral
- Definition
- Anti-derivatives: definition, properties
- Computation of areas
- Fundamental theorem of calculus
- Further integration methods
- By parts
- By substitution