Irena Penev

Mathematical Analysis 1 -- NMAI054 (summer 2026)

Lecture:
  • Monday 14:00-15:30, N1

Tutorials:

  • Thursday 10:40-12:10, N7
  • Thursday 12:20-13:50, N2
  • Thursday 14:00-15:30, N6

Office hours: Right after lecture on Mondays (in the IMPAKT corridor); on Fridays 14:00-15:00 in my office (S323); or by appointment.

Contact (by e-mail): ipenev [at] iuuk [dot] mff [dot] cuni [dot] cz



Course matrials:
Lecture Notes (will be updated as the semester progresses)



Course content
Real numbers and their relation to rationals, complex numbers.
Sequences of real numbers: Basic properties of limit, accumulation points, liminf and limsup. (Bolzano-Weierstrass theorem, limits of monotone sequences, etc.)
Informative series of real numbers.
Basic properties of functions (monotonicity, convexity, ...), definition by a series, basic approximations.
Function limits: methods of calculation.
Continuity of functions: extreme value theorem, intermediate value theorem.
Derivatives of functions: methods of calculation, usage - l'Hospital's rule, mean-value theorem, graphing a function. Taylor's polynomial.
Introduction to integral calculus: Newton integral (and methods of calculation), Riemann integral, applications (areas, volumes, lengths, estimates of sums).

Course requirements and evaluation
Students enrolled in the lecture are required to also enroll in one of the tutorials. Tutorial credit ("zápočet") is a prerequisite for the exam.

The final exam will be written. Exam problems will be similar to examples/exercises seen in lecture/tutorial/quizzes/practice HW problems. In rare cases, a student may be invited to take an oral exam (in addition to the written exam).

To obtain tutorial credit, students obtain (cumulatively) at least 50% on weekly/biweekly quizzes (the lowest quiz score will be dropped). There will be no graded HW, but I will ocasionally assign practice problem (not to be turned in); similar problems may be on the exam.

All quizzes, as well as the exam, will be closed book. Students will not be allowed to leave the room during the quizzes/exam (for example, to go to the restroom); exceptions are possible only for medical reasons, and with proper documentation from a Czech/Slovak doctor or from the university. The exam will be 90-120 minutes long (to be determined).




Lectures
Lecture slides will be updated as the semester proceeds. Slides are closely based on the Lecture Notes.

Lecture 1: Number systems: rational and real numbers. An introduction to limits of sequences (slides)

Lecture 2: Properties of limits. The Monotone Sequence Theorem and the Squeeze Theorem (slides)

Lecture 3: The Bolzano-Weierstrass Theorem. Accumulation points. Divergence to infinity (slides)

Lecture 4: The limit superior and limit inferior. Series (part I) (slides)

Lecture 5: Series (part II) (slides)

Lecture 6: Limits of functions (part I) (slides)

Lecture 7: Limits of functions (part II) (slides)

Lecture 8: Limits of functions (part III) (slides)

Lecture 9: Differentiation (slides)

Lecture 10: The Chain Rule. The Inverse Function Theorem. Local extrema (slides)

Lecture 11: The Mean Value Theorems. L'Hôpital's Rule (slides)

Lecture 12: Local Extrema: the First and Second Derivative Test. Convexity and concavity (slides)

Tutorial 9: Sketching graphs of functions (slides)

Lecture 13: The indefinite integral (slides)

Lecture 14: The definite (Riemann) integral and the Fundamental Theorem of Calculus (slides)

Lecture 15: Improper integrals. Applications of integration (slides)


Tutorials
Tutorial 1 (solution of Exercise 1(t))

Tutorial 2

Tutorial 3

Tutorial 4

Tutorial 5

Tutorial 6 (some solutions)

Tutorial 7 (Solution of Exercise 5)

Tutorial 8 (Solution of Exercise 2)

Tutorial 9

Tutorial 10

Tutorial 11 (Solution of Exercise 3)

Tutorial 12


Practice problems (not to be turned in)
Practice problems 1 (a subset of these problems will be on the quiz on Thursday, February 26)

Practice problems 2 (a subset of these problems will be on the quiz on Thursday, March 5)

Practice problems 3 (a subset of these problems will be on the quiz on Thursday, March 26) [edited on Sunday, March 22, at 11 a.m.]

Practice problems 4 (a subset of these problems will be on the quiz on Thursday, April 9)

Practice problems 5 (there will be a quiz on Thursday, April 30; click on the link for the details)




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