SYLLABUS
For the training course applicable from the academic year 2014/15
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, 2 SEMESTER
DR HAB. STANISLAWA KANAS
COURSE OBJECTIVES
The main aim is
 to provide students the fundamental knowledge concerning of the range of Calculus I, with
its applications in geometry and practice;
 vocational skills in mathematical reasoning;
 solving mathematical and practical problems;
 the use of different mathematical studies;
 develop further skills and knowledge in the future.
PREREQUISITES
Basic knowledge of mathematics on secondary school level,
Secondary-school certificate
KNOWLEDGE:
 Student has knowledge of the characteristic properties
LEARNING OUTCOMES



of the set of real numbers; its subsets and sequences
of real numbers; supremum and infimum and
mathematical induction;
defines the basic notions of sequences and series
numbers, real functions of one variable , including the
concept of boundary features, continuous function,
derivative, integral, and sequences and series of real
numbers;
knows and understands the basic theorems of
differential and integral calculus;
knows the basic examples and counterexamples to
illustrate concepts of the limit, the convergence of
series, continuity and differentiability of functions.
SKILLS:
 Formulate the correct definitions and theorems in the
field of account differential and integral calculus;


know how to prove theorems in the field of the
selected account and illustrate them giving some
examples;
calculate the limits of sequences and explores the
convergence of series, apply theorems and methods of
calculus I in the optimization problems; use integral
calculus of functions of one variables in the
applications of geometrics and physics.
FINAL COURSE OUTPUT - SOCIAL COMPETENCES
 Student knows the limitations of his own knowledge



and understands the need for further education;
formulate opinions on key issues in the field of
differential and integral calculus and their use in daily
life and in different areas of expertise;
searches for relevant information in the literature and
properly apply them;
find its place in a group.
COURSE ORGANISATION –LEARNING FORMAT AND NUMBER OF HOURS
LECTURES - 120 HOURS,
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)
 Indefinite integrals, introduction and basis properties (4).
 Techniques of integrations (integration by substitution, by parts, by partial fractions,
integration of trygonometric and exponential functions) (12).
 The Riemann integral, mean value theorem for integrals, fundamental theorem of
calculus (12).
 Numerical methods of integration, application of integrals to geometry and science
(12).
 Sequences and series of functions. Power series, Fourier series. Pointwise and
uniform convergence. The Stone-Weierstrass theorem, the radius of convergence.
Divergence of series. Transmission of properties. (12)
 Expansion in power and Fourier series. Applications (8)
COURSE DESCRIPTION
 Description the set of real numbers, finding a supremum and infimum. Formulations
and proving the axioms and properties. (4)
 Use of mathematical induction for several mathematical problems. (6)
 Finding the limit of sequences, checking boudedness, monotinicity, convergence and
divergence of a sequences. Transformation of indeterminate forms. (8)
 Describing of a convergence of number series, finding a sum of series. Convergence
and divergence tests, absolute and conditional convergence tests. (10)
 The specification of elementary functions and its domains. Computations of limit of
function, and checking its continuity and continuity on a bounded segment.
Calculations of the asymptotes of a functions. (8)
 Finding the derivative, describing a geometric interpretation. Use of differentiation
rules, finding an extremum. (10)
 Checking a convexity and concavity of functions, application to graphing. (8)
 Using L'Hôspital's Rule, applications in several mathematical problems. (6)
 Calculation of the indefinite integrals, proving its basis properties (4).
 Practising the techniques of integrations (integration by substitution, by parts, by
partial fractions, integration of trygonometric and exponential functions) (12).
 Calculating the Riemann integral, using mean value theorem for integrals,
fundamental theorem of calculus (12).
 Using numerical methods of integration, application of integrals to geometry and
science (12).
 Checking pointwise and uniform convergence of series. Application of the StoneWeierstrass theorem, finding the radius of convergence and the range of divergence
of series. Transmission of properties. (12)
 Expansion in power and Fourier series. Applications (8)
METHODS OF INSTRUCTION
REQUIREMENTS AND ASSESSMENTS
ORAL COMUNICATION, SOLVING TASKS, INDIVIDUAL WORK
Student defines the basic notions and formulate
fundamental theorems concerning sequences and
series of numbers, induction, functions of one variable,
optimization problems and integrals. Students know
how to prove fundamental properties and apply the
knowledge in practice tasks.
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
550 HOURS – 22 ECTS (11+11)
ENGLISH
INTERNSHIP
MATERIALS
PRIMARY OR REQUIRED BOOKS/READINGS:
1. Elementary Calculus, An Infitesimal Approach. Secon
Edition. H. Jerome Keisler, University of Wisconsin,
2012.
2. Lecture Notes in Calculus, Raz Kupferman, The
Hebrew University, Jerusalem 2013.
3. Calculus, David Guichard, San Francisco,
California, USA 2011.
4. 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.