Calculus

The course introduces students to the fundamental concepts and methods of Mathematical Calculus, essential for their theoretical and practical background in Computer Science. Topics include sequences, series, power series, improper integrals, applications of definite integrals, simple differential equations, the basic theory of multivariable functions, and multiple integrals. A basic introduction to complex numbers is also provided.

Applications of these mathematical concepts to Computer Science are illustrated through examples in data and algorithm analysis, cost function optimization, and numerical simulation. The course also introduces suitable software tools (e.g. MATLAB/Octave) and web-based applications for solving practical computational problems.

Upon successful completion of the course the student will be able to:

• perform complex numbers operations and find any root, real or not, of a polynomial
• study real sequences and calculates limits
• apply convergence tests and calculate sums of series
• calculate the radius of convergence of power series and Taylor series of a function
• apply convergence tests and calculate improper integrals
• calculate lengths, areas and volumes using integrals
• solve simple differential equations
• apply basic theory of functions of several variables
• calculate multiple integrals

Complex numbers – Sequences - Series – Power Series – Taylor Series – Improper Integrals – Beta and Gamma functions – Calculation of lengths, areas, and volumes – Separable Differential Equations – First Order Differential Equations – Bernoulli Differential Equations – Functions of Several Variables - Limits and Continuity – Partial Derivatives – Directional Derivatives – Maxima, Minima, and Saddle Points – Absolute Maxima and Minima – Constrained Maxima and Minima – Double Integrals – Polar Coordinates - Triple Integrals

• Adams, R. A., Essex, Ch., Calculus and Vector Analysis, P.Ch. Paschalidis Publications, 2023, ISBN: 978-9925-35-151-0.
• Finney, R. L., Weir, M. D., Giordano, F. R., Calculus, Vol. I, ITE-Crete University Press, 2009, ISBN: 978-960-524-183-4.
• Finney, R. L., Weir, M. D., Giordano, F. R., Calculus, Vol. II, ITE-Crete University Press, 2009, ISBN: 978-960-524-184-1.
• Panagiotopoulos, A. Ch., Sapounakis, A., Calculus, Stamoulis Publications, 1990.

• Lectures
• Tutorial exercises
• Student participation in tutorials

• Support of the educational process via the e-learning platform Opencourses
• Use of ICT in teaching and education (specialized software, electronic lecture notes)
• Use of ICT for communication with students

Written final examination with problem-solving questions