HIGHER MATHEMATICS
THE FIRST SEMECTER
LECTURE 1 (10.10.2023)
Real numbers. Real numbers as points on a number scale
The absolute value of a real number
Variables and constants
The range of a variable
Ordered variables. Increasing and decreasing variables. Bounded variables
Function
Ways of representing functions
Basic elementary functions. Elementary functions
LESSON 1 (11.10.2023)
Basic elementary functions. Elementary functions
LECTURE 2 (16.10.2023)
The graphs of rational functions
LESSON 2 (17.10.2023)
Algebraic functions and their graphs
LESSON 3 (18.10.2023)
The inverse trigonometric functions and their graphs
LECTURE 3 (24.10.2023)
The curves set parametrically
LESSON 4 (25.10.2023)
Polar coordinate system and equations of the curves in this system
LECTURE 4 (30.10.2023)
The functions set implicitly
LESSON 5 (31.10.2023)
The limit of a variable. An infinitely large variable
LESSON 6 (01.11.2023)
The limit of a function
LECTURE 5 (07.11.2023)
A function that approaches infinity. Bounded functions
Infinitesimals and their basic properties
Basic theorems on limits
The limit of the function sin x/x as x → ∞
LESSON 7 (08.10.2023)
The limit of a function
LECTURE 6 (13.11.2023)
The number e
Natural logarithms
Continuity of functions
Certain properties of continuous functions
Comparing infinitesimals
LESSON 8 (14.11.2023)
The second remarkable limit
LESSON 9. Control work 'Limits' (15.11.2023)
LECTURE 7 (21.11.2023)
Velocity of motion
The definition of a derivative
Geometric meaning of the derivative
Differentiability of functions
The derivative of the function y = xn, n a positive integer
Derivatives of the functions y = sin x, y = cos x
Derivatives of a constant, the product of a constant by a function, a sum, a product, and a quotient
The derivative of a logarithmic function
The derivative of a composite function
Derivatives of the functions y = tan x, y = cot x, y = ln |x|
LESSON 10 (22.11.2023)
Derivatives of the functions
LESSON 11 (24.11.2023)
Basic differentiation formulas
LECTURE 8 (27.11.2023)
An implicit function and its differentiation
Derivatives of a power function for an arbitrary real exponent, of a general exponential function, and of a composite exponential function
An inverse function and its differentiation
Inverse trigonometric functions and their differentiation
Basic differentiation formulas
LESSON 12 (28.11.2023)
Basic differentiation formulas
Logarithmic differentiation formulas
LESSON 13 (29.11.2023)
Properties of derivatives
Logarithmic differentiation formulas
LECTURE 9 (5.12.2023)
Parametric representation of a function
The equations of some curves in parametric form
The derivative of a function represented parametrically
Hyperbolic functions
The differential
The geometric meaning of the differential
LESSON 14 (6.12.2023)
Differentiation of Implicit Functions
LECTURE 10 (11.12.2023)
Derivatives of different orders
Differentials of different orders
Derivatives (of various orders) of implicit functions and of functions represented parametrically
The mechanical meaning of the second derivative
The equations of a tangent and of a normal. The lengths of a subtangent and a subnormal
The geometric meaning of the derivative of the radius vector with respect to the polar angle
LESSON 15 (12.12.2023)
Differentials. Derivatives of different orders
LESSON 16 (13.12.2023)
Derivatives of different orders
LESSON 17 (19.12.2023)
Equations of a tangent and a normal. Lengths of a subtangent and a subnormal
LESSON 18 (20.12.2023)
Preparation to control work
LECTURE 11 (25.12.2023)
SOME THEOREMS ON DIFFERENTIABLE FUNCTIONS
A theorem on the roots of a derivative (Rolle’s theorem)
The mean-value theorem (Lagrange’s theorem)
The generalized mean-value theorem (Cauchy’s theorem)
The limit of a ratio of two infinitesimals (evaluating indeterminate forms of the type 0/0)
The limit of a ratio of two infinitely large quantities
Taylor’s formula
Expansion of the functions ex, sin x, and cos x in a Taylor series
LESSON 19. Control work 'Derivatives' (26.12.2023)
LESSON 20 (09.01.2024)
Rolle’s Theorem, Lagrange’s Theorem
LESSON 21 (16.01.2024)
Calculation the limits
LECTURE 12 (17.01.2024)
INVESTIGATING THE BEHAVIOUR OF FUNCTIONS
Statement of the problem
Increase and decrease of a function
Maxima and minima of functions
Testing a differentiable function for maximum and minimum with a first derivative
Testing a function for maximum and minimum with a second derivative
Maximum and minimum of a function on an interval
Applying the theory of maxima and minima of functions to the solution of problems
Testing a function for maximum and minimum by means of Taylor’s formula
Convexity and concavity of a curve. Points of inflection
Asymptotes
General plan for investigating functions and constructing graphs
Investigating curves represented parametrically
LESSON 22 (17.01.2024)
Taylor’s formula
LESSON 23. Control work 'Some theorems on differentiable functions' (19.01.2024)
LECTURE 13 (22.01.2024)
THE CURVATURE OF A CURVE
Arc length and its derivative.
Curvature.
Calculation of curvature.
Calculating the curvature of a curve represented parametrically.
Calculating the curvature of a curve given by an equation in polar coordinates.
The radius and circle of curvature. The center of curvature. Evolute and involute.
The properties of an evolute.
Approximating the real roots of an equation.
LESSON 24 (23.01.2024)
The extrema of the functions.
The maximum and minimum values of the function on the indicated
intervals.
LESSON 25 (02.02.2024)
The points of inflection and the intervals of convexity and concavity.
The asymptotes.
LECTURE 14 (05.02.2024)
COMPLEX NUMBERS. POLYNOMIALS
Complex numbers. Basic definitions.
Basic operations on complex numbers.
Powers and roots of complex numbers.
Exponential function with complex exponent and its properties.
Euler’s formula. The exponential form of a complex number.
Factoring a polynomial.
The multiple roots of a polynomial.
Factoring a polynomial in the case of complex roots.
Interpolation. Lagrange’s interpolation formula.
Newton’s interpolation formula.
Numerical differentiation.
On the best approximation of functions by polynomials. Chebyshev’s theorem.
LESSON 26 (05.02.2024)
Investigating the functions and plotting their graphs.
LESSON 27. Control work 'Investigation the behavior of functions' (06.02.2024)
LESSON 28 (08.02.2024)
The curvature.
The radius of curvature.
The center of curvature.
LESSON 29 (08.02.2024)
The curvature.
The radius of curvature.
The center of curvature.
The evolute.
LESSON 30. Control work 'The curvature of the curve' (13.02.2024)
LESSON 31 (14.02.2024)
Complex numbers.
Polynomials.
LESSON 32. Control work 'Complex numbers, polynomials' (14.02.2024)