Mathematics 1 in WS 2022

Syllabus

• Sets, statements and logic
• Introduction to linear algebra - basic properties of vector spaces, operations with vectors, linear independency, dimensions, bases
• Matrices - Gaussian elimination, rank of matrices, determinants, system of linear equations, the Cramer rule
• Sequences - basic properties (monotonicity, boundedness), limits
• Functions - domain and range, basic properties (monotonicity, periodicity, even and odd functions etc), inverse function, elementary functions
• Limits - finite and infinite limits of functions, one-sided limits, properties and calculations, definition of number e
• Derivatives - definitions, properties and calculations, geometrical and physical meaning, second derivative
• Application of derivatives - the l'Hospital rule, extremes, convexity and concavity
• Analysis of arbitrary function - domain, asymptotes, local extremes, inflection points, graph
• Integration - Definition of the Riemann and Newton integral, their connection, integral of elementary functions, rules for computations, substitution and integration by parts, integration of rational functions, applications in probability

Recommended literature

• Neustupa, J.: Mathematics I, CTU Publishing House, Prague, 1996
• Neustupa, J. and Kracmar, S.: Problems in Mathematics I, CTU Publishing House, Prague, 1999

• Neustupa, J. and Kracmar, S.: Selected problems from the textbook Problems in Mathematics I

• Examples of exam problems

• Keisler, H. J.: Elementary Calculus: An Infinitesimal Approach, 2nd edition, Prindle, Weber & Schmidt, 1986.
• Calculus Volume I., Volume II., Volume III., provided by https://cnx.org/
• College algebra, provided by https://cnx.org/

Lectures of Dr. Neustupa

• 1. (Intro, vector spaces)
• 2. (linear dependency, base, matrices)
• 3. (rank of matrix, determinant)
• 4. (Frobenius' theorem)
• 5. (Sequences)
• 6. (Intro to functions)
• 7. (Derivation)
• 8. (Second derivative of function)
• 9. (Local extrema)
• 10. (Indefinite integral)
• 11. (Integration methods)
• 12. (Definite integral)

Exercises of Dr. Valasek

• my primary comucation channel is email - Jan.Valasek at fs.cvut dot cz
• only 2 absences are allowed without additional HWs
• active work including homeworks is required
• collect 8 of 10 points from small tests (problems for additional points is HERE)
• prepare yourself for exam! calculate!


• We 28.9. is holiday
• 5.10. we start with 5min test on linear independency of vectors
• 12.10. small test on rank of matrix
• 19.10. small test on inverse matrix
• 26.10. small test on Frobenius' theorem
• 2.11. small test on sequences
• 9.11. small test on domain of function
• 16.11. small test on derivation of function
• 23.11. small test on tangent of function
• 30.11. small test on local extrema
• 7.12. no test
• 14.12. small test on integration per partes
• 11.1. voluntary tutorial