This online course program establishes the minimum requirements for the knowledge and skills of the student and determines the content and types of training sessions and reporting. The program is designed for teachers leading the discipline, teaching assistants and students of economics or humanities. The program is developed according to the working curricula of different universities and institutes.
Course syllabus
Higher Mathematics for Students in Economics and the Humanities
Elements of linear algebra
- The determinants of the second and third orders, and their properties. Algebraic complements and minors. Calculation of the determinant by the expansion of the row (column). The rule of triangle, the calculation of the third-order determinant.
- The determinants of the second and third orders and their properties. The algebraic .
- Matrices. Operations with them. Inverse matrix.
- Rank of a matrix. Elementary transformations of matrices.
- Systems of two and three linear equations. Cramer’s rule, the matrix way of solving a system of linear equations.
- A system of n linear equations. Gauss method for solving systems. Lesson 1.
Elements of analytical geometry.
- Vectors. Linear operations on vectors. The basis. Decomposition of a vector into a basis.
- Orthonormalized basis. Rectangular Cartesian coordinate system. Point coordinates, vector coordinates in the Rectangular Cartesian coordinate system.
- Scalar product of vectors and its properties. Vector length and angle between vectors in coordinate form. Orthogonality condition of two vectors.
- The directional cosines of a vector. A unit vector.
- Vector product of two vectors. The condition of collinearity. Vector product in coordinate form. Area of a parallelogram and a triangle.
- Mixed product of vectors. Geometrical sense. Condition of coplanarity of vectors.
- The equation of a line in the plane. Various equations of a line in the plane. Angle between lines. Distance from a point to a line. Mutual arrangement of lines.
- Equation of the plane. Angle between planes. Mutual arrangement of planes. Distance from point to plane.
- Various equations of lines in space. Mutual arrangement of lines. A line in space and a plane. Their mutual arrangement. Angle between a line and a plane. Angle between straight lines.
Introduction to Mathematical Analysis
Function. The area of its definition. The vicinity of a point. Limit of a function at a point. One-sided limits.
Infinitesimal functions. Properties of infinitesimals. Comparison of infinitesimals. Infinitely large functions.
Theorems about limits. Uncertainties. Revealing them.
The first and second remarkable limits.
Continuity of functions. Properties of functions that are continuous on a segment: boundedness, existence of greatest and least values, existence of intermediate values.
Differential calculus of functions of one and several variables.
- The problem of the tangent. The derivative of a function. The derivative of a complex function. The table of derivatives. The differential.
- Theorems on differentiable functions: Fermat, Rolle, Langrange.
- Application of derivatives to the study of functions: signs of monotonicity of a function, necessary and sufficient conditions of extremum; inflection point, condition of convexity of the graph of a function. Asymptotes.
- A general scheme for the study of a function of one variable and the construction of its graph.
- The concept of a function of several variables. The graph of a function of two variables. Limit, continuity of a function of two variables, properties of a function of two variables that is continuous in a bounded closed region.
- Partial derivatives. Directional derivative. Gradient.
- Extremum of a function of two variables: necessary and sufficient conditions. The maximum and minimum values of a function of two variables in a closed bounded area.
Elements of integral calculus.
- The original and the indefinite integral. Properties of an indefinite integral. Table.
- Methods of integration: table integration, substitution of variables, by parts. Integration of trigonometric functions.
- The problem of calculating the area of a curvilinear trapezoid. A definite integral. Substitution of a variable in a definite integral. The concept of the non-integral integral over an infinite interval. Calculation of areas of plane figures using the definite integral.
Elements of differential equations.
- The concept of a first-order differential equation. The Cauchy problem. First-order equations with separating variables, linear, homogeneous.
- Second order differential equations. Second order linear differential equations, homogeneous and inhomogeneous. A characteristic equation. General solution of a homogeneous equation depending on the type of roots of the characteristic equation.
- A general solution of a linear inhomogeneous equation with constant coefficients. Finding partial solutions to such an equation by the type of the right-hand side using the method of indefinite coefficients.