Calculus Ι (7,5 ECTS)

Course Code: 

Axiomatic foundation of the system of real numbers. Axioms of domain and order, axiom of the least upper bound and the Archimedean property. Monotonic and bounded functions, continuity of a function, Bolzano's theorem, Mean-value theorem, extreme value theorem and uniform continuity. Elements of set theory. Derivative of a function, calculus of derivatives and derivatives of higher order, theorems of Rolle, Mean-Value and L’Hospital, local extremes. Riemann' s integral, properties of the integral (additivity, triangular inequality, linearity), continuity and differentiability of the integral function, integrability of continuous functions, Mean-value theorem for integrals, indefinite integral of a function, Fundamental theorem of Infinitesimal Calculus. Techniques of integration (change of variable, integration by parts, etc.), logarithm and the exponential function, generalized integrals, examples and applications. Subsets of R, points of accumulation, sequences of real numbers, monotonic sequences, subsequences and the Cauchy criterion of convergence, Bolzano-Weierstrass theorem, theorems of sequence convergence. Series of real numbers, series with positive terms, criteria of convergence and absolute convergence of series. Taylor’s theorem and Taylor series.

Recommended reading

  • Αθανασιαδης Χ.Ε, Γιαννακουλιας Ε., Γιωτοπουλος Α. (2010). Γενικά Μαθηματικά, Απειροστικός Λογισμός, Τόμος 1, Εκδόσεις Συμμετρία.
  • Spivak, M. (1991). Διαφορικός και Ολοκληρωτικός Λογισμός, 2η έκδοση, Πανεπιστημιακές Εκδόσεις Κρήτης.
  • Finney R.L., Weir M.D., and Giordano F.R. (2004). Aπειροστικός Λογισμός, τόμος Ι, Πανεπιστημιακές Εκδόσεις Κρήτης.
  • Apostol, T. M. (1967). Calculus, Vol.1, 2nd edition, Wiley.