ΠΕΡΙΓΡΑΜΜΑ ΜΑΘΗΜΑΤΟΣ

The course is designed to provide the basic tools of advanced mathematics, including mainly elements of differential and integral calculus of functions of one variable. In particular, it focuses on the detailed presentation of mathematical concepts, theorems and propositions but also on problem-solving techniques related to them. For this purpose, extensive use is made of examples that find use in practical applications from the field of engineering.
As a background course, it offers the engineer the mathematical knowledge and the way of thinking in order to develop his / her ability to express mathematically and to face methodological practical problems.
Consistent and successful course attendance has as expected learning outcomes for the student:
to achieve the gradual theoretical logical subtraction from the real numbers, in the sense of the variable, in the definition of a function, in the sense of the differential of a function,
to connect and be able to study the representations of a function (analytical form, graphical representation, verbal description),
to understand theoretically and in practice the basic theorems of differential calculus,
to understand the concept of the integral of a function and relate it to practical applications,
to learn all the necessary techniques related to the differentiation and integration of functions,
to identify and distinguish problem-solving methods related to the differentiation and integration of functions,
to make him/her capable to apply the above methods to engineering problems,
to analyze and interpret the obtained results,
to be able to attend, without significant learning gaps, more specialized courses of the department.

Research, analysis and synthesis of data and information, using corresponding technologies, Adaptation to new situations
Independent work, Teamwork – distribution of responsibilities, Intellectual competences, Societal competence.

• Foundation of the real number system. Field and order axioms, the least upper bound axiom and the Archimedean principle.
• Monotone and bounded real-valued functions, continuation of a real-valued function, Bolzano theorem, and intermediate value theorem, extreme value theorem, uniform continuity.
• Elements of set theory, the system of real numbers.
• Function derivative, derivative calculus and higher order derivatives, Rolle, Mean Value, and L’Hospital theorems, local extrema.
• The Riemann integral, integral properties (sum-difference rule, triangular inequality, linearity), differentiability and continuity, integral at points of discontinuity of the integrable function, integrability of continuous functions, mean value theorem, indefinite integral, fundamental theorem of integral calculus.
• Integration techniques (variable change, integration by parts, etc.), logarithm and exponential function, generalized integrals, examples and applications.
• Subsets of R, accumulation points, sequences of real numbers, monotonic sequences, subsequences and Cauchy’s convergence criterion, Bolzano-Weierstrass theorem, convergence theorems for sequences.
• Series of real numbers, series with positive terms, convergence and absolute convergence tests of series. Taylor’s theorem and Taylor series.

Συγγράμματα

Calculus, Fourth Edition, by Michael Spivak
Thomas’ Calculus, 14th edition, by Joel Hass, Christopher Heil, Maurice Weir
Calculus, Second Edition, by William Briggs, Lyle Cochran, Bernard Gillett