Module also offered within study programmes:
General information:
Name:
Mathematics
Course of study:
2013/2014
Code:
RMS-1-101-s
Faculty of:
Mechanical Engineering and Robotics
Study level:
First-cycle studies
Specialty:
-
Field of study:
Mechatronics with English as instruction languagege
Semester:
1
Profile of education:
Academic (A)
Lecture language:
English
Form and type of study:
Full-time studies
Course homepage:
 
Responsible teacher:
dr Jarnicka Jolanta (jarnicka@wms.mat.agh.edu.pl)
Academic teachers:
dr Jarnicka Jolanta (jarnicka@wms.mat.agh.edu.pl)
Module summary

Description of learning outcomes for module
MLO code Student after module completion has the knowledge/ knows how to/is able to Connections with FLO Method of learning outcomes verification (form of completion)
Social competence
M_K001 Students can assess their level of understanding of the problem and its possible solution methods. They understand the need for continuous training MS1A_K01 Activity during classes,
Examination,
Test
Skills
M_U001 The student has ability to use the rules of strict, logical thinking in the analysis of physical and technical processes. MS1A_U07 Activity during classes,
Examination,
Test
M_U002 The student is able to use the acquired knowledge of mathematics to describe and analyze the basic physical and technical problems. MS1A_U07 Examination,
Test,
Execution of exercises
M_U003 The student can obtain information from literature, databases and other sources, can make the selection, interpretation, and draw conclusions. MS1A_U01 Activity during classes
M_U004 The student is able to work independently and in a team. MS1A_U02 Activity during classes
Knowledge
M_W001 The student knows the basic laws of logic and knows how to use them to draw the correct conclusions. MS1A_W01 Activity during classes,
Examination,
Test
M_W002 The student knows the basic concepts and theorems of mathematical analysis in the field of calculus (including differentiation, integration, and differential equations), as well as their applications. MS1A_W01 Activity during classes,
Examination,
Test
M_W003 The student knows the basic concepts and theorems of algebra and analytic geometry, as well as elements of applied mathematics MS1A_W01 Activity during classes,
Examination,
Test
FLO matrix in relation to forms of classes
MLO code Student after module completion has the knowledge/ knows how to/is able to Form of classes
Lecture
Audit. classes
Lab. classes
Project classes
Conv. seminar
Seminar classes
Pract. classes
Others
Zaj. terenowe
Zaj. warsztatowe
E-learning
Social competence
M_K001 Students can assess their level of understanding of the problem and its possible solution methods. They understand the need for continuous training + + - - - - - - - - -
Skills
M_U001 The student has ability to use the rules of strict, logical thinking in the analysis of physical and technical processes. + + - - - - - - - - -
M_U002 The student is able to use the acquired knowledge of mathematics to describe and analyze the basic physical and technical problems. + + - - - - - - - - -
M_U003 The student can obtain information from literature, databases and other sources, can make the selection, interpretation, and draw conclusions. + + - - - - - - - - -
M_U004 The student is able to work independently and in a team. + + - - - - - - - - -
Knowledge
M_W001 The student knows the basic laws of logic and knows how to use them to draw the correct conclusions. + + - - - - - - - - -
M_W002 The student knows the basic concepts and theorems of mathematical analysis in the field of calculus (including differentiation, integration, and differential equations), as well as their applications. + + - - - - - - - - -
M_W003 The student knows the basic concepts and theorems of algebra and analytic geometry, as well as elements of applied mathematics + + - - - - - - - - -
Module content
Lectures:
  1. Analytic geometry (in R^2 and R^3)

    Vectors, properties of vectors, length of a vector operations on vectors, scalar product, the angle between the vectors, Euclidean norm, vector product and scalar triple product, applications, equation of a line and plane in R^3, orthogonal projection, distance and angle between lines and planes.

  2. Differential equations

    Differential equations: first order ordinary differential eq., separable and linear equations (variation of parameters, predictor-corrector method), initial-value problems, second order linear equations with constant parameters.

  3. Series

    Infinite series as a limit of partial sums, necessary condition for the convergence of the series, geometric series, the sum of a series, tests for the convergence of a series (comparison test, ratio and root tests, asymptotic and limit asymptotic test, and "2^k’’ test), alternating series test, conditional and absolute convergence of a series, power series, the radius of
    convergence.

  4. Limits of functions and continuity

    Limit of a function at a point and its properties, one-sided limits,
    improper limits, asymptotes, continuity of a function at a point – definition and properties, continuous functions and their properties, continuity of elementary functions, intermediate value theorem, extreme value theorem.

  5. Derivatives and their applications

    Derivative at a point – definition, geometric interpretation, differentiability, basic formulas and techniques of differentiation (sum/difference rule, product rule, quotient rule, chain rule, logarithmic differentiation), mean-value theorems and their consequences, l’Hospital’s rule, monotonicity, minima and maxima, first derivative test, higher order derivatives, second derivatives test, global extreme values, closed interval method, concavity, investigation of functions,Taylor’s formula with remainder, Taylor expansions of real functions, Taylor series, Maclaurin series.

  6. Elements of combinatorics and probability Combinations, variations, permutations, set of elementary events, the probability-calculation methods, properties, the law of total probability, Bayes’ theorem, probability as a measure, random variable, continuous and discrete random variables, distributions of random variables, central limit theorem.
  7. Integrals and their applications

    Indefinite integrals: first fundamental theorem of calculus, antiderivative, basic formulas and techniques of integration, substitution rule, integration by parts, integration of rational functions-decomposition into partial fractions, integration of irrational, trigonometric functions.
    Definite integral: definition, the net change theorem, substitution rule for a definite integral, applications, including polar co-ordinates and parametric equations, improper integrals, integral test for the convergence of a series.

  8. Functions of two or more variables

    Function of two variables-definition and properties, quadratic form, limits and continuity, partial derivatives and differentiability, total differential, chain rules, tangent planes, directional derivatives and gradients, maxima and minima of functions of two variables, Lagrange multipliers, implicit functions, double integrals-definition, properties, and applications, line integrals, Green’s theorem.

  9. Elements of logic and number sets

    Basic logical connectives and quantifiers, important logical laws, sets, operations on sets, natural numbers, integers, rational numbers, real numbers, intervals, finite, infinite, bounded, and unbounded sets, principle of mathematical induction, binomial theorem.

  10. Functions

    Definition, graphs, properties of functions, inverse function, composite function, review of elementary functions and their properties: constant functions, power and root functions, polynomials, rational functions, exponential and logarithmic functions, trigonometric and inverse trigonometric functions, absolute value.

  11. Complex numbers

    Representation of a complex number: algebraic, polar, and exponential form, algebraic operations, conjugate, module, and argument of a complex number, the fundamental theorem of algebra, equations in the set of complex numbers, de Moivre’s theorem, nth root of a complex number.

  12. Matrices & systems of linear equations

    Definition of a matrix, operations on matrices, matrix equations, determinant of a matrix (definition, properties, and methods of calculation), inverse matrix, rank of a matrix, eigenvalues ​​and eigenvectors, systems of linear equations, Cramer’s rule, Gaussian elimination, Kronecker-Capelli theorem.

  13. Sequences

    Bounded sequence, monotone, arithmetic, and geometric sequence, limit of a sequence and its properties (arithmetic operations on limits, squeeze theorem, etc.), finite limits, improper limits, indeterminate forms, methods of finding limits of sequences, Euler’s number as a limit, Cauchy’s condition, subsequences.

Auditorium classes:

Program of the classes coincides with the program of the lectures.

Student workload (ECTS credits balance)
Student activity form Student workload
Summary student workload 453 h
Module ECTS credits 18 ECTS
Participation in lectures 90 h
Participation in auditorium classes 90 h
Realization of independently performed tasks 180 h
Preparation for classes 90 h
Examination or Final test 3 h
Additional information
Method of calculating the final grade:

The final grade is the arithmetic mean of the grades obtained from classes and exams, rounded up.
Positive final grade is awarded only when positive results of exams were obtained.

Prerequisites and additional requirements:

Prerequisites and additional requirements not specified

Recommended literature and teaching resources:

J. Stewart, Calculus, Early Transcendentals, 6e, Thomson Brooks/Cole, 2008
T. Jurlewicz, Z. Skoczylas, Algebra liniowa 1, Oficyna Wydawnicza GiS, Wrocław, 2002
W. Krysicki, L. Włodarski, Analiza matematyczna w zadaniach, cz. I i II, PWN, 1993
A. Howard, Calculus with analytic geometry, 3rd ed., John Wiley &Sons, 1989

Scientific publications of module course instructors related to the topic of the module:

Additional scientific publications not specified

Additional information:

None