SYLLABUS
COURSE TITLE
FACULTY/INSTITUTE
COURSE CODE
DEGREE PROGRAMME
CALCULUS I
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
FIELD OF STUDY MATHEMATICS
DEGREE LEVEL 1
FORMA
STUDIÓW/STUDY
MODE STATIONARY
MATHEMATICS
FIRST DEGREE
FULL-TIME
COURSE FORMAT
YEAR AND SEMESTER
NAME OF THE TEACHER
BASIC
1 YEAR, 1 SEMESTER
DR HAB. STANISLAWA KANAS
COURSE OBJECTIVES
THE MAIN AIM OF THE COURSE IS TO PROVIDE FOR THE STUDENTS A KNOWLEDGE OF THE THEORY
AND PRACTICE APPLICATIONS OF THE SET OF REAL NUMBERS AND ITS SUBSET, SEQUENCES,
NUMBER SERIES AND FUNCTIONS OF ONE REAL VARIABLE. THE COURSE PROVIDE THE SUFFICIENT
KNOWLEDGE TO UNDERSTAND MATHEMATICAL NOTIONS AS WELL AS TO USE THEM IN PRACTICE,
I.E. TO MASTER TECHNIQUES OF CALCULATIONS. THE STUDENT IS ASSUMED TO BE VERSED IN THE
STANDARD PRE-CALCULUS TOPICS OF FUNCTIONS, GRAPHING AND SOLVING EQUATIONS AND THE
EXPONENTIAL, LOGARITHMIC AND TRIGONOMETRIC FUNCTIONS.
PREREQUISITES
LEARNING OUTCOMES
Basic knowledge of mathematics on secondary school level,
Secondary-school certificate
KNOWLEDGE: BASIC PROPERTIES OF A SET OF REAL NUMBERS AND
ITS SUBSETS. SUPREMUM AND INFIMUM OF A SET. MATHEMATICAL
INDUCTION. THEORY OF THE SEQUENCES, NUMBER SERIES AND
FUNCTIONS OF ONE REAL VARIABLE. DERIVATIVES AND
APPLICATIONS. GRAPH SKETCHING.
APPLICATIONS OF THE
KNOWLEDGE IN THE SEVERAL TECHNICAL AND ECONOMICAL
PROBLEMS
SKILLS: DETERMINE SUPREMUM AND INFIMUM OF A REAL SUBSET.
DETERMINE THE LIMIT OF A SEQUENCE. PROVE THAT THE
SEQUENCE IS DIVERGENT. USE BOTH THE LIMIT DEFINITION AND
RULES OF DIFFERENTIATION TO DIFFERENTIATE FUNCTIONS.
SKETCH THE GRAPH OF A FUNCTION USING ASYMPTOTES, CRITICAL
POINTS, THE DERIVATIVE TEST FOR INCREASING/DECREASING
FUNCTIONS, AND CONCAVITY. APPLY DIFFERENTIATION TO SOLVE
MAX/MIN PROBLEMS. USE L'HOSPITAL RULE TO EVALUATE CERTAIN
INDEFINITE FORMS.
FINAL COURSE OUTPUT - SOCIAL COMPETENCES KNOWS THE
LIMITATIONS OF THEIR KNOWLEDGE AND UNDERSTAND THE NEED
FOR FURTHER EDUCATION. ABILITY TO FIND RELEVANT
INFORMATION AND THEIR APPLICATIONS. ABILITY TO FIND THEIR
PLACE IN THE GROUP
COURSE ORGANISATION –LEARNING FORMAT AND NUMBER OF HOURS
LECTURES,CLASSES - 120 HOURS
 Set of real numbers. Axioms. (4)
 Basic properties of subsets of real numbers. Concept of supremum and infimum of a
sets. Mathematical induction. (6)
 Sequences, boudedness, monotinicity, convergence and divergence of a sequences,
limit, indeterminate forms. (8)
 Number series, basic definitions, sum of series. Convergence and divergence tests,
absolute and conditional convergence tests. (10)
 Concepts of function, Limits and continuity. Continuity on a bounded segment.
Asymptotes. (8)
 Derivative, geometric interpretation. Differentiation rules, extremum problems. (10)
 Convexity and concavity, application to graphing. (8)
 L'Hôspital's Rule, applications. (6)
COURSE DESCRIPTION
A SET OF REAL NUMBERS. AXIOMS OF REAL NUMBERS. SEVERAL SUBSET OF REAL NUMBERS.
NOTION OF SUPREMUM AND INFIMUM OF A SET. PRINCIPLE OF INDUCTION. SEQUENCES, LIMIT OF
SEQUENCES. INDETERMINATE FORMS. NUMBER SERIES. CONVERGENCE ANS SUM OF THE SERIES.
ABSOLUTE AND CONDITIONAL CONVERGENCE. CONVERGENCE TESTS. FUNCTIONS OF ONE
VARIABLE. COMPOSITE FUNCTIONS, ONE-TO-ONE FUNCTIONS, INVERSE FUNCTIONS. A LIMIT OF A
FUNCTION. PROPERTIES AND COMPUTATIONS OF LIMITS. LIMIT, CONTINUITY, ASYMPTOTES, THE
DERIVATIVE OF A FUNCTION AND ITS INTERPRETATIONS. COMPUTATIONS OF DERIVATIVES FORMULAS AND RULES. THE MEAN VALUE THEOREM. GRAPH SKETCHING AND PROBLEMS OF
EXTREMA. THE LOGARITHMIC, EXPONENTIAL, INVERSE TRIGONOMETRIC AND HYPERBOLIC
FUNCTIONS. VARIOUS GEOMETRIC AND PHYSICAL APPLICATIONS. L'HOSPITAL RULE. HIGHER
DERIVATIVES OF FUNCTIONS. CONCAVITY OF FUNCTIONS. POWER SERIES. TAYLOR'S FORMULA. THE
DERIVATIVE OF A VECTOR FUNCTION.
METHODS OF INSTRUCTION
REQUIREMENTS AND ASSESSMENTS
ORAL COMUNICATION, SOLVING TASKS, INDIVIDUAL WORK
THE STUDENT KNOWS HOW TO USE THE MATHEMATICAL
INDUCTION, INVESTIGATE PROPERTIES OF A NUMBER SET.
FIND THE LIMIT OF A SEQUENCE ANS SHOW ITS
BOUNDEDNESS. USE A CONVERGENCE TESTS TO PROVE OR
DISPROVE THE CONVERGENCE (ABSOLUTE, CONDITIONAL)
OF NUMBER SERIES.
INVESTIGATE PROPERTIES OF
FUNCTION AND SKETCH ITS GRAPH. APPLY THE HOSPITAL
RULE. STUDENTS ARE ASSESSED REGULARLY SOLVING TASKS
WRITING.
GRADING SYSTEM
GRADING SCORE – 3.0 FOR 50 - 60%, 3.5 FOR 61 - 70 %,
4.0 FOR 71 – 80%, 4.5 FOR 81 – 90%, 5.0 FOR 91 – 100
%
TOTAL STUDENT WORKLOAD
NEEDED TO ACHIEVE EXPECTED
LEARNING OUTCOMES EXPRESSED
IN TIME AND ECTS CREDIT POINTS
LANGUAGE OF INSTRUCTION
INTERNSHIP
275 HOURS – 11 ECTS
ENGLISH
MATERIALS
PRIMARY OR REQUIRED BOOKS/READINGS:
1. Lecture Notes in Calculus, Raz Kupferman, The
Hebrew University, Jerusalem 2013.
2. Calculus, David Guichard, San Francisco,
California, USA 2011.
3. First Year Calculus For Students of Mathematics and
Related Disciplines, Michael M. Dougherty and
John Gieringe, USA
SUPPLEMENTAL OR OPTIONAL BOOKS/READINGS:
1. INTRODUCTION TO METHODS OF APPLIED
MATHEMATICS or Advanced Mathematical
Methods for Scientists and Engineers, SEAN MAUCH
2. Calculus in context, J. Callahan, K. Hoffmann, D.
Cox, Donald O. Shea, H. Pollatsek, L. Senechall, New
York University, USA 2008.