# Mathematics 1

## Syllabus

1. Systems of linear equations and vectors. System of linear equations; Gauss elimination; vectors; operations with vectors; vector space; dimension; basis.

2. Matrices. Definition; operation with matrices; rank of a matrix; Fundamental theorem for linear systems; Leontief Input-Output model.

3. Determinants and special matrices. Determinants; properties of determinants; transpose of a matrix; identity matrix; inverse matrix; finding the inverse matrix.

4. Matrix equations, eigenvalues and eigenvectors. Matrix equations; solving systems of linear equations; eigenvalues and eigenvestors.

5. Functions. Definition; demand and supply functions; profit, revenue and cost functions; properties of functions; operations on functions; Polynommials; Horner's scheme; rational functions; trigonometric functions; exponential and logarithmic functions.

6. Limits. Continuity; limit; properties of limits; application of limits; asymptotes.

7. Derivatives. Definition; basic formulae; rules for computing the derivatives; higher order derivatives; tangent line to a curve; maxima and minima; limits and derivatives; L'Hospital's rule.

8. Application of derivatives. Marginal cost, marginal revenue and marginal profit; elasticity of demand; profit maximization in a competitive market; profit maximization in a monopolistic market; taxation in a competitive market.

9. Integrals. Indefinite integral; basic formulae; rules for computing indefinite integrals; definite integral; rules for computing definite integrals; improper integral; numerical integration methods; Fundamental theorem of calculus.

10. Application of integrals. Consumer's surplus; producer's surplus; continuous income streams.

11. Taylor's polynomials. Taylor's polynommial and its application.

12. Repetition.

13. Repetition.

Keywords: Linear algebra; systems of linear equations; matrices; Leontief Input-Output model; determinants; Gauss elimination; vector space; eigenvalues and eigenvectors; mathematical analysis; functions in one variable; limits; derivatives; elasticity of demand; profit maximization; taxation in a competitive market; indefinite, definite and improper integrals; consumer's and producer's surplus; continuous income streams; Taylor's polynomials.