Introduction to Mathematical Analysis

Published

18 October 2026

Passport

  • Where and when. In MIPT’s official curricula for the 2025 intake, the course is in semester I of the FPMI programs 01.03.02 β€œApplied Mathematics and Informatics” (profile β€œMathematics”) and 09.03.01 β€œInformatics and Computer Engineering” (profile β€œApplied Mathematics, Computer Science and Engineering”).
  • Workload. For 01.03.02: 270 hours, including 30 hours for exam preparation and the exam itself, 6 credits; 120 contact hours: 60 lecture hours and 60 seminar hours. For 09.03.01: the same 270 hours and 6 credits, but 150 contact hours: 60 lecture hours and 90 seminar hours.
  • Department. Department of Higher Mathematics.
  • Lecturers in the published 2025/26 programs. For the FPMI stream, A. L. Lukashov’s program is published; a separate program for FPMI and VSHPI is by A. I. Tyulenev. This does not mean that these particular lecturers will teach the course next year: the assignment of streams changes.
  • Final assessment. Exam in semester I.

Contents of the published program

Below is the complete list of topics in the exam program of A. L. Lukashov’s stream for the 2025/26 academic year, presented in a form convenient for the website. Items marked with an asterisk in the original document are examined without proofs if the student does not wish to prove them.

  1. Real numbers: existence and uniqueness of the supremum and infimum of a bounded set; countability of \(\mathbb Q\) and uncountability of \(\mathbb R\).
  2. Cantor’s nested intervals theorem.
  3. Limit of a numerical sequence and its uniqueness; operations with limits and inequalities; limit of a monotone bounded sequence; the number \(e\); infinitesimal and infinitely large sequences.
  4. Subsequences and subsequential limits; the Bolzano–Weierstrass theorem; the Cauchy criterion for the existence of a finite limit; upper and lower limits of a sequence.
  5. Limit of a function of one variable: the Cauchy and Heine definitions and their equivalence; properties of the limit, the Cauchy criterion, limit of a composition; one-sided limits of a monotone function.
  6. Continuity at a point and one-sided continuity; operations with continuous functions and composition; passing to the limit under the sign of a continuous function; classification of discontinuities and discontinuities of monotone functions.
  7. Continuous functions on a closed interval: boundedness, attainment of the supremum and infimum, intermediate values, uniform continuity and the inverse function.
  8. Continuity of elementary functions; definition and properties of the exponential function; the remarkable limits.
  9. Derivative and differential of a function of one variable, one-sided derivatives, geometric meaning; rules of differentiation, derivative of a composition and of an inverse function, elementary and parametrically defined functions; invariance of the form of the first differential.
  10. Higher-order derivatives and differentials; the Leibniz formula; non-invariance of the form of the second differential; FaΓ  di Bruno’s formula\(^*\).
  11. Fermat’s theorem; the theorems of Rolle, Lagrange and Cauchy; Taylor’s formula with remainder in the Peano and Lagrange forms, the basic expansions; L’HΓ΄pital’s rule for the indeterminate forms \(0/0\) and \(\infty/\infty\); Darboux’s theorem on intermediate values of the derivative.
  12. Studying functions with the derivative: necessary and sufficient conditions for monotonicity and a local extremum, convexity, inflection points and asymptotes.
  13. Metric, normed linear and Euclidean spaces; the Cauchy–Bunyakovsky–Schwarz and Minkowski inequalities; open and closed sets, the topology of a metric space.
  14. Limits of sequences in \(\mathbb R^n\); the Bolzano–Weierstrass theorems and the Cauchy criterion in \(\mathbb R^n\); compactness in metric spaces and in \(\mathbb R^n\); the fundamental theorem of algebra\(^*\).
  15. Smooth plane and space curves: tangent line, admissible change of parameter, length and the natural parameter; functions of bounded variation; curvature, the moving trihedron, torsion and the Frenet formulas\(^*\); evolute and involute\(^*\).

Assessment

A general grading scheme for the course has been published (the document itself does not name a stream), 120 points: colloquium 0–7, semester work 0–7, test on the third assignment 0–4, three assignments 0–2 each, theory check 0–3, seminar work 0–3, written exam paper 0–30, oral exam 0–60. Productive work in the practicum gives up to 3 points, but the semester total is capped at 30.

Scale: 112–120 β€” 10, 103–111 β€” 9, 94–102 β€” 8, 85–93 β€” 7, 76–84 β€” 6, 67–75 β€” 5, 54–66 β€” 4, 41–53 β€” 3, 27–40 β€” 2, 0–26 β€” 1. For a grade above 2, the oral answer is converted to \(6N\), where \(N\) is the preliminary grade on the ten-point scale; for \(N=1,2\) the final grade equals \(N\).

Sources and currency

Contents and lecturers: FPMI exam program, 2025/26 and the FPMI and VSHPI version; assessment: general grading scheme. Workload and semester: curriculum 01.03.02, 2025 intake and curriculum 09.03.01, 2025 intake. The official documents take precedence over this reference page.