Calculus

Subject description

1. Real numbers, complex numbers, sequences, limits and convergent sequences, series.
2. Functions: basic properties, graph. Continuity and limits, properties of continuous functions, bisection, secant method, functions of several variables.
3. Derivatives: definition and geometric interpretation of derivative, rules for differentiation, partial derivatives, differential, linear aproximation, l’Hostpial’s rule, gradient. Applications: critical points and local extrema, global extrema, solving optimization problems, Taylor polynomial and Taylor series.
4. Integral: indefinite integral, definite integral and areas, numerical integration (trapezoid and Simpson's rule), fundamental theorem of calculus (connection between indefinite and definite integrals), examples of nonelementary functions.
5. Differential equations: growth models, solutions, separable equations, linear first degree differential equations, examples.

Objectives and competences

The goal of this course is to provide a broad understanding of the basic concepts of mathematical analysis, such as convergence, derivative and integral, and demonstrate how they can be applied to solve problems in computer science and science as a whole.

General competences:

• Ability of critical thinking.
• Developing skills in critical, analytical and synthetic thinking.
• Understanding and using mathematical concepts and mathematical thinking
• Understanding the concept of abstraction

Subject specific competencies

• Basic skills in computer and information science, which includes basic theoretical skills, practical knowledge and skills essential for the field of computer and information science;
• Basic skills in computer and information science, allowing the continuation of studies in the second study cycle.
• Understanding the concepts of convergence, continuity, derivatives and integrals
• Ability to use basic mathematical concepts like sequences, series, functions, derivatives and integrals in solving problems from computer science and other relevant fields.

Teaching and learning methods

Lectures, lab exercises with oral presentations, homework problems. Special attention will be given to continuing work with homework problems and group work.

Expected study results

Knowledge and understanding:

After completion of the course the student will

• master the basic concepts and principles of calculus and understand the connection between their symbolic, graphic and numeric representations
• master simple proofs with mathematical induction and basic manipulations with real and complex numbers
• understand the concepts of sequence and convergence
• understand the concept of functional dependence and continuity
• understand the concept of derivative, be able to compute relatively simple derivatives and use them in function analysis and in simple optimization problems
• understand the concept of integral, master basic principles for computing integrals, and understand the connection between indefinite and definite integrals
• know basic principles and examples of applying these concepts to computer algorithms and to real world problems

Basic sources and literature

1. G. Tomšič, B. Orel, N. Mramor: Matematika I, Matematika II; Ljubljana, Založba FE in FRI.
2. J. Stewart: Calculus: early transcendentals (8th edition), Cengage Learning, 2016, poglavja 1-8 in 14.