Practise

  1. Functions: Brief review; limits of functions: explanation of two-sided and one-sided limits, determining limits from graphs, operations with infinities and indeterminate forms, limits of polynomial and rational functions, especially at points where the function is not defined.
  2. Limits of Functions: Limits of exponential and logarithmic functions, limits of the form and one-sided limits, limits of composite functions, continuity of functions.
  3. Derivatives: The concept of the slope of a linear function, formal definition of the derivative at a point and its geometric interpretation, estimating derivatives from graphs, differentiation rules and formulas, routine differentiation of functions.
  4. Applications of Derivatives: L’Hôpital’s Rule, local and absolute extrema, finding local extrema over the entire domain, finding absolute extrema of a continuous function on a closed interval.
  5. Higher-Order Derivatives: Definition and significance for the behavior of functions (identifying intervals of convexity/concavity and inflection points, both graphically and analytically), Taylor polynomial (definition, approximation of functions and function values, construction of Taylor polynomials, equation of the tangent line).
  6. Integration: Notation, the concept of an antiderivative, definition of the indefinite integral, basic integration formulas, evaluation of simple integrals using formulas and algebraic manipulations, absence of general formulas for the integral of a product or a composite function, integration by parts.
  7. Innovation week (without scheduled lessons).
  8. Midterm Exam: Covers the material from the first five weeks, i.e., everything up to and including higher-order derivatives, but excluding integration.
  9. Integration: Substitution method, including integrals of the form , definition and evaluation of the definite integral.
  10. Integration: Computing the area under a curve and between two curves, definition and evaluation of improper integrals.
  11. Functions of Two Variables: Graphs and domains, local and absolute extrema, saddle points, partial derivatives (calculation, geometric interpretation, estimation from graphs), finding local extrema over the entire domain.
  12. Constrained Optimization: Explanation of the concept, graphical identification, finding local constrained extrema on open curves extending to infinity (e.g., lines, parabolas), finding absolute constrained extrema on closed curves (e.g., circles, ellipses).
  13. Reserve/Review Session for the Final Exam.