Mathematics 110491-D
Ćwiczenia (CW) Semestr zimowy 2024/25

Limit of a sequence, Euler number e. Algebraic properties of proper and improper limits. Indeterminate forms. Limit of a function. Squeeze theorem.

2. General properties of a function - injection, surjection, bijection, superposition of functions. Inverse functions - natural logarithm and cyclometric functions. Images and preimages. Continuity of a function, Darboux property. Asymptotes. Infimum and supremum of a subset of real numbers, Weierstrass theorem.

First derivative of a function, Leibniz notation. Geometric interpretation, tangent line and linear approximation. Elasticity of a function.

Monotonicity, local and global extrema, infimum and supremum of a function on a subset. De l'Hospital rule.

Second derivative of a function. Concavity and convexity, inflection points, sketching graphs of functions.

Vector space R^n: vectors, linear combination, linear independence. Basic geomentry in R^n: lines, hyperplanes, half-spaces. Parametric equations of a line and a plane.

Matrices, operations on matrices. Rank of a matrix. Non-singular matrix and inverse matrix. Matrix equations.

Elementary row operations on matrices. Calculating rank of a matrix and inverse matrices by elementary row operations. Determinant of a square matrix and its applications - calculating rank and inverse matrix by adjugate matrix.

Systems of linear equations. Kronecker-Capelli theorem. General solution. Solving systems of linear equations by elementary row operations. Cramer's rule. Geomeric interpretations of systems of linear solutions and their general solution in the vector space R^3.

Multivariable functions - domain, contour lines, local, global and constrained extrema. Application of contour map to optimization. Directions of increase and decrease of the function.

Partial derivatives and directional derivatives, gradient. Partial elasticities. Derivatives of the second order. Partial elasticities. First and second order conditions for the local extrema.

Lagrange multipliers. Constrained extrema - necessary and sufficient condition. Maximal and minimal value of a function on a closed and bounded set.

Indefinite integral, primitive function (anti-derivative). Integrating by parts and by substitution.

Definite integral and its geometric interpretation. Improper integral.

Complementary classes/repetition.