COURSE GOALS: Course goals are acquiring knowledge of the basic mathematical notions, gaining operational knowledge of techniques of differentiation and understanding the related theoretical concepts.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.2. recognize and follow the logic of arguments, evaluate the adequacy of arguments and construct well supported arguments;
2.3. use mathematical methods to solve standard physics problems
4. COMMUNICATION SKILLS
4.2. present complex ideas clearly and concisely;
5. LEARNING SKILLS
5.1. search for and use professional literature as well as any other sources of relevant information.
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon successful completion of the course Mathematical Analysis 1, a student will be able to:
* give correct definitions and interpretation of fundamental concepts of mathematical analysis (basic properties of a function of one real variable, limit of a function, continuity, derivative of a function)
* list elementary functions and their properties and use them in practical computations
* calculate the derivatives of elementary functions
* apply the rules of differentiation (derivative of the sum, the product, the composition of functions, the inverse function, etc.)
* investigate the behaviour of a function and draw its graph using derivatives, limits and continuity
COURSE DESCRIPTION:
Mathematical Analysis 1 (hours of lectures + hours of exercises)
* basic mathematical concepts and symbols; number sets; elements of mathematical logic (3 + 2)
* the concept of a function and general properties of functions (domain and range, graph, injective, surjective and bijective functions, composition of functions, inverse function) (3 + 4)
* the set of real numbers, open and closed intervals, neighbourhoods, absolute value and distance, ? - ? (epsilon - delta) symbols, supremum and infimum (3 + 4)
* a review of elementary functions of one real variable (6 + 6)
* sequences of real numbers (1 + 3)
* limit of a function, intuitive concept of a limit, operations with limits, limit at infinity, asymptotes (5 + 5)
* limit of the function (sin x) / x when x tends to 0; the precise definition of a limit (3 + 1)
* the number e as a limit, limits related to the exponential and logarithmic functions (2 + 3)
* continuous functions, properties of continuous functions, calculation of approximate zeroes of functions and approximate solutions of equations (3 + 2)
* the concept of a derivative of a function, interpretation of a derivative in geometry (slope of the tangent) and physics (velocity of a motion) (2 + 1)
* derivatives of elementary functions, the rules for derivatives of the sum, the product, the composition of functions and the inverse function, logarithmic derivative, derivatives of functions given in an implicit or parametric form, derivatives of higher order (7 + 9)
* the relation between properties of continuity and differentiability of a function, application of the differential to local linear approximation (2 + 1)
* analysis of a function (intervals of monotonicity, extreme values), mean value theorems, sketching the graph of a function (5 + 4)
REQUIREMENTS FOR STUDENTS:
During the course, the students are required to solve short written exams (tests consisting of 2-3 problems) on a weekly basis. There are two main written exams, namely midterm and term-end exams. The total of 100 points is divided 50 - 50 between tests and main exams. A minimum of 45 points out of 100 is required for admission to the final exam. Otherwise, a minimum of 20 points is required for admission to further written exams which encompass the full contents of the course.
GRADING AND ASSESSING THE WORK OF STUDENTS:
Having achieved at least 45 points in written exams, a student may accept the final grade according to the following scale: 88-100 excellent; 74-87 very good; 60-73 good: 45-59 sufficient, without taking an oral examination. Additionally, an oral exam may be taken in order to improve the grade achieved in written exams, but only by one grade. At the oral exam, the emphasis is set on the understanding of theoretical concepts, on the ability to connect and apply various facts from the acquired knowledge and also on the skill of communication in mathematical terms.
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