Lecture I. Sequences
Statement of a number sequence, different types of sequences, arithmetic operations on sequences, finite and infinite limits of sequences, theorems on limits of sequences, limits of important sequences, examples of calculations of limits.
Lecture II. Series
Statement of a number series, a sequence of partial sums and the sum of a series. Examples of convergent and divergent series. Necessary condition for convergence. Convergence of geometric series. Convergence tests. Convergence of Dirichlet's series. Series with arbitrary and alternating terms.
Lecture III. Limits and continuity.
Definition of the limit of a function at a point ( by Heine and by Cauchy). One-sided limits. Infinite limits. Theorems on limits. Continuity of a function. Operations on continuous functions. Theorems on continuous functions.
Lecture IV. Derivatives
Statement of a difference quotient and its graphical interpretation. Finite derivatives. Derivatives of elementary functions. Tangent to a function at a point, geometrical interpretation of a derivative. One-sided derivatives, infinite derivatives.
Lecture V. Theorems on differentiable functions
Rolle's theorem. The mean value theorem and graphical interpretation. Monotonicity, relative extrema and the minimal and maximal value of a function.
Lecture VI. Applications of derivatives I
L'Hospital's rule. Higher order derivatives. Investigation of functions: asymptotes, convexity and concavity, inflection points, graphs of functions.
Lecture VII. Applications of derivatives II
Approximation of functions using Taylor polynomial. Taylor and MacLaurin function series. Newton's method of solving nonlinear equations.
Lecture VIII. Antiderivative, indefinite integral
Antiderivatives and indefinite integrals. Linearity of indefinite integrals. Integration by parts and substitution rule. Integration of rational, trigonometric and some irrational functions.
Lecture IX. Definite integral
Integral sum and Riemann integral. Geometrical interpretation of a definite integral.Integration by parts and substitution rule. Properties of definite integrals. Fundamental theorems of the calculus - continuity and differentiability of a function of upper limit of integration.
Lecture X. Applications of definite integrals, numerical integration
Areas of plane figures, an arc length, volume and surface area of a
solid of revolution. Applications of definite integrals in physics.
Methods of approximate computation of definite integrals.
Lecture XI. Ordinary differential equations I
General and actual solutions of of an ordinary differential equation. Separable differential equations. Homogeneous equations. Linear homogeneous first order differential equations.
Lecture XII. Ordinary differential equations II
Methods of solution of nonhomogeneous linear differential equations of the first order. Linear differential equations of the second order with constant coefficients. Examples of applications of differential equations.
Lecture XIII. Multivariable functions
Sets in an n-dimensional cartesian space. Real n-variable functions. The limit and continuity of a 2- and 3-variable function. Partial derivatives.
Lecture XIV. Directional derivative
Derivative of a composite function. Directional derivative. Gradient. Relative extrema of 2-variable functions.
Lecture XV. Integrating functions of several variables
Double integral, geometrical interpretation of double integral. Double integral over a normal region. Triple integral. Triple integral over a parallelepiped and over a normal region. Geometrical interpretation of the triple integral.
Remark
Mastering of parts of the lecture marked with the symbol (i) is not necessary - it is an extension of the basic Analysis course. However they may be very helpful in deeper understanding of the text and thus in solving the attached problems.
The aim of this course is to review the basic notions and problems of mathematical analysis that are relevant for a well educated engineer.
Discrete Mathematics and Linear Algebra.