Matematika A1 és Analízis 1 ütemterv

2006. őszi félév

(Az alábbi időbeosztás csak hozzávetőlegesnek tekinthető!)

1. hét. Basics: sets, logic, functions, relations, and cardinality. Operations with sets, Boolean algebra, operations with logical statements, truth tables, one-to-one functions, onto functions, inverse, equivalence, complete ordering, finite sets, countable infinite sets and uncountable infinite sets.

2. hét. Real numbers. Natural numbers, mathematical induction, integers, rationals, ordered field, reals, least upper bound (sup).

3. hét. Complex numbers. Algebraic and trigonometric forms, operations, nth root, polynomials, factoring.

4. hét. Vectors in Space. Planar vectors, spatial vectors, coordinates, basic operations, dot product, cross product, lines and planes.

5. hét. Limits. Limits of functions, limit combination theorem, sandwich theorem, one-sided limits, limits at infinity and infinite limits.

6. hét. Sequences. Limit of a sequence, monotonic sequences, subsequences, Cauchy property.

7. hét. Continuous functions. Algebraic operations, composition, Max-Min Theorem, Intermediate Value Theorem.

8. hét. Derivatives. Definition, tangent lines, differentiation rules, linear approximation, chain rule, implicit functions.

9. hét. Applications of derivatives. Related rates, maxima and minima, mean value theorems, increase and decrease, convexity and concavity, asymptotes, curve sketching, optimization, Newton’s method.

10. hét. Integration. Antiderivatives, upper and lower sums, upper integral, lower integral, Riemann sums, integration rules, the Fundamental Theorem of Calculus, numerical integration.

11. hét. Elementary transcendental functions. Logarithmic and exponential functions, indeterminate forms, L’Hospital rule, inverse trigonometric functions, hyperbolic functions.

12. hét. Techniques of integration. Integration by parts, integration by substitution, integration of rational functions.

13. hét. Applications of integrals. Areas, volumes of solids of revolution, length of curves, areas of surfaces of revolution.

14. hét. Improper integrals. Integration on unbounded intervals, integration of unbounded functions, p-integrals, comparison test, absolute and conditional convergence.

Szabados Tamás