Analysis over R
Mathematical induction. Basic properties of real numbers. Limits of a sequence. Series of real numbers. Absolute and conditional convergence. Main tests of series convergence. Notions of topology in the real line. Continuous functions; operations. Intermediate value theorem. Weierstrass theorem on extrema of continuous functions. Uniform continuity. Derivative. Chain rule. Relationship between derivative and growth. Mean-value theorem. Convex functions. Integrable functions. Fundamental theorem of calculus. Change of variables. Integration by parts. Theorem of the mean. Taylor formula.
References:
LIMA, E. L. – Análise Real, Vol. 1, Rio de Janeiro, IMPA. Coleção Matemática Universitária, 1999.
LIMA, E. L. – Curso de Análise, Vol.1, Rio de Janeiro, IMPA, Projeto Euclides, 1989.
LANG, S. – Analysis I, Reading, Mass., Addison-Wesley, 1968.
RUDIN, W. – Principles of Mathematical Analysis. 2nd ed., New York, McGraw-Hill, 1964.
LIMA, E. L. – Análise Real, Vol. 1, Rio de Janeiro, IMPA. Coleção Matemática Universitária, 1999.
LIMA, E. L. – Curso de Análise, Vol.1, Rio de Janeiro, IMPA, Projeto Euclides, 1989.
LANG, S. – Analysis I, Reading, Mass., Addison-Wesley, 1968.
RUDIN, W. – Principles of Mathematical Analysis. 2nd ed., New York, McGraw-Hill, 1964.
* Standard program. The teacher has the autonomy to make any changes.
