Courses Catalogue

Syllabus of the course: General Mathematics I (Exercises)

In this web page we provide the syllabus of the course General Mathematics I (Exercises), offered by the Department of Physics.
The list of the courses offered during the current accademic year is available here.
The list of all courses offered by the Department of Physics is available here.

InstructorG. Athanasiu
ProgramMonday, 15:00-17:00, Amphitheater ST
Web page
Goal of the courseThe course is addressed to first year students. It covers and expands material presented in the last years of high school, including functions, basic calculus, limits, derivatives and integrals. It also focuses on direct applications of the material covered to a number of simple physics problems.

During the spring semester to course is offered in the form of selfstudy with only one two-hour lecture per week for the students that did not pass it during the fall semester. Emphasis is given in problem solving.
SyllabusIntroductory material: Straight lines, functions and plots, exponential fubctions, inverse functions and logarithms, trigonometric functions and their inverses, parametric equations (0.5 weeks)

Limits and Continuity: Rates of change and limits, computing limits and side limits, infinite limits, continuity, tangent lines (1.5 weeks).

Derivatives: The derivative as a function, the derivative as a rate of change, derivative of a fraction and a negative power, derivatives of trigonometric functions, derivation chain rule, derivative of an implicit function (1 week).

Applications of derivatives: Extrema of functions, mean value theorem, rules for plotting functions, construction of optimization models, differentials, Newton's algorithm, introduction of taylor's formula (2 weeks).

Integration: Indefinite integrals, integration rules, integration by substitution, intergral estimates from finite sums, Riemann sums and definite integrals, mean value theorem, fundamental theorem of integration, evaluation of definite integrals by substitution, numerical integration, main types of integration, integration by parts, partial fractions, trigonometric substitutions, computational algebra systems, Monte Carlo integration, generalized integrals (3 weeks).

Applications of integrals:Calculation of volumes by slicing and rotation around an axis, by using cylindrical shells, calculation of lengths of curves in the plane, springs, pumps, elevators, fluid forces, center of mass and torques. (2 weeks).

Non-algebraic functions: Logarithms, exponentials, derivatives of inverse trigonometric functions, integrals of hyperbolic functions.(1 week)

Infinite Series: Limits of sequences, subsequences, bounded sequences, Picard's method, infinite series, non-negative series, alternating series, absolute convergence, conditiuonal convergence, power series, Taylor and McLaurin series, applications of power series. (2 weeks).
Bibliography«THOMAS, Calculus» – J. Haas, Ch. Heil, M.D. Weir., (14th Edition 2018) «higher mathematics», M.R. Spiegel.
«Differential and Integral Calculus» - M. Spivak.

University of Crete - Department of Physics  - Voutes University Campus - GR-70013 Heraklion, Greece
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