Mathematics 0600-S1-CM-M

Semester I, Lecture:

Quantifiers:
- introducing two types of recording
- simple examples and consistent application during the lecture

Sequences of real numbers:
- definition of sequence and basic properties (monotonicity, limitation)
- definition of sequence convergence and the concept of a boundary
- the concept of sequence divergence and the definition of sequence divergent to infinity
- basic theorems on sequence limits
- number e

Function limits:
- definition of the limit of the function at a point according to Heine
- one-sided borders
- boundaries in infinity
- limits not appropriate at the point
- calculus of finite limits

Continuity of functions:
- definition of continuity in point and examples
- continuity of a function on a set of arguments
- basic properties of continuous functions
- continuity of elementary functions, rational functions, composite functions

Function derivative:
- differential quotient
- definition of the derivative of a function at a point
- geometric interpretation of the derivative and the relationship to the tangent to the graph
- a function "derivative of function"
- a derivative of elementary functions
- derivatives of sum, product and quotient of functions
- a derivative of a composite function
- higher-order derivatives
- examples of the use of derivatives in physics and chemistry

Application of derivatives of functions:
- Taylor series, with an introduction introducing the concept of series, its convergence and simple examples
- de l'Hospital rule
- diagnostics of the properties of functions with the use of derivatives, such as monotonicity, extremes of functions, inflection points, concavity
- a study of the function variability

Indefinite integral:
- the concept of an antiderivative function
- elementary integrals
- fundamental properties of integrals
- integration by parts
- integration by substitution

Definite integral:
- the concept of a whole sum
- relationship with indefinite integral - Newton-Leibnitz theorem
- application for calculating surface areas
- improper integrals

Elements of linear algebra:
- Cartesian product
- the concept of linear space
- vectors and matrices and operations on them
- permutations and the sign of the permutation
- determinants and their basic properties
- scalar and vector products in R^3
- inverse matrix
- matrix eigenvalues and vectors

Systems of linear equations:
- matrix representation of a system of linear equations
- Cramer's rule

Multivariable Functions:
- partial derivatives
- local extremes of functions of several variables
- a study of local extremes of functions of two variables

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Tutorials:

Sets:
- actions on sets (sum, cross-section, difference)
- Cartesian product
- graphical interpretation of the Cartesian product in a two-dimensional Cartesian system

Basic information about the functions:
- domain and range
- creating a graph of a function
- properties such as monotonicity, limitation, evenness, oddness, periodicity, one-to-one
- function composition, inverse function
- illustration based on simple examples
- discussion of the above properties for all functions introduced in further parts of the course

Quadratic function:
- fundamental concepts and properties, product and canonical form
- quadratic equations and inequalities

The absolute value:
- definition
- equations and inequalities

Polynomials:
- definition and graphs
- operations on polynomials, with particular attention to the long division
- Bezout theorem
- zero places and algebraic equation
- finding particular solutions for polynomials with rational coefficients
- product form of a polynomial
- equations and inequalities with polynomials

Rational functions:
- definition and properties
- equations and inequalities with rational functions

Trigonometric functions:
- definitions, essential properties and graphs
- reduction formulas
- basic trigonometric identities
- the relationship of reduction formulas with identities for the sum of angles
- relations between trigonometric identities
- basic trigonometric equations and inequalities

Exponential and logarithmic functions:
- definitions, essential properties and graphs
- exponential and logarithmic equations and inequalities
- particular attention to the importance of monotonicity of both functions when solving inequalities
- power functions

Sequences of real numbers:
- calculating the limits of sequences of real numbers with the use of basic theorems (about the limit of the product or quotient of sequences, about three sequences)
- recognition of sequences converging to the number e or its powers

Function limits:
- calculation of function limits at a point, including left and right limits
- using the calculus of finite limits
- computing improper limits at a point and finding vertical asymptotes
- calculating function limits at infinity and finding horizontal and vertical asymptotes
- solving problems with functions converging to the number e and with the function sin(x)/x

Continuity of functions:
- testing the continuity of a function at a point and on an interval
- continuity of elementary functions
- using theorems about the continuity of functions built of continuous functions, including composition of such functions

Function derivative:
- derivatives of elementary functions
- calculation of derivatives of sum, product, quotient and composition of elementary functions
- higher-order derivatives

Application of derivatives of functions:
- finding Taylor series for simple functions
- application of de l'Hospital rule to find function limits for different cases of indeterminate expressions
- testing the properties of functions with the use of derivatives (monotonicity, extremes, inflection points, concavity)
- a study of the course of function variability and the construction of the graph

Indefinite integral:
- integrals of elementary functions
- usage of basic properties of integrals
- integration by parts, including multiple
- integration by substitution

Definite integral:
- calculation of definite integrals using the Newton-Leibnitz theorem
- calculation of surface areas of flat areas
- calculation of surface areas and volumes of revolving solids
- calculation of improper integrals of various types

Elements of linear algebra:
- calculating the sum and product of matrices
- illustration of the concepts: transposed, unit, symmetric, orthogonal, and triangular matrix
- calculation of determinants of 2x2 and 3x3 matrices from the rules of Sarrus
- the Laplace expansion for computingh matrix determinants of higher degrees
- determining the order of the matrix
- calculation of scalar and vector products in R ^ 3, finding the length of a vector and finding orthogonality
- determination of the inverse matrix
- finding matrix eigenvalues and eigenvectors

Systems of linear equations:
- saving systems of linear equations in matrix form
- finding solutions to systems of linear equations from Cramer's rule

Multivariable Functions:
- first and second-order partial derivatives
- local extremes of functions of multiple variables
- a study of local extremes of functions of two variables