Module also offered within study programmes:
General information:
Name:
Mathematical Analysis 1
Course of study:
2017/2018
Code:
IES-1-107-s
Faculty of:
Computer Science, Electronics and Telecommunications
Study level:
First-cycle studies
Specialty:
-
Field of study:
Electronics and Telecommunications
Semester:
1
Profile of education:
Academic (A)
Lecture language:
English
Form and type of study:
Full-time studies
Course homepage:
 
Responsible teacher:
dr Mc Inerney Kinga (stolot@agh.edu.pl)
Academic teachers:
dr Mc Inerney Kinga (stolot@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 Knows how to explain mathematical phenomena in an understandable way ES1A_W01 Oral answer
Knowledge
M_W001 Has basic knowledge of calculus, differentiation for functions of a single variable, knows how to calculate approximate value of the function ES1A_W01 Examination,
Test
M_W002 Has knowledge of definite integrals of single variable functions and their applications ES1A_W01 Examination,
Test
M_W003 Has knowledge of number serieses and knows how to apply convergence tests ES1A_W01 Examination,
Test
M_W004 Has knowledge on number series, knows basic convergence tests and how to apply them ES1A_W01 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
Zaj. terenowe
Zaj. warsztatowe
Others
E-learning
Social competence
M_K001 Knows how to explain mathematical phenomena in an understandable way - + - - - - - - - - -
Knowledge
M_W001 Has basic knowledge of calculus, differentiation for functions of a single variable, knows how to calculate approximate value of the function + + - - - - - - - - -
M_W002 Has knowledge of definite integrals of single variable functions and their applications + + - - - - - - - - -
M_W003 Has knowledge of number serieses and knows how to apply convergence tests + + - - - - - - - - -
M_W004 Has knowledge on number series, knows basic convergence tests and how to apply them + + - - - - - - - - -
Module content
Lectures:
Mathematical Analysis 1

The course covers 45h of lectures and 45h of classes.

LECTURES

1. Logic (1 h)
Logical relations and quantifiers. De Morgan’s laws. Necessary and sufficient condition. Contraposition. Theory of sets,
Carthesian product of sets. Upper and lower bound of a set.
2. Elementary functions (3 h)
Definition of a function, domain, codomain, image and preimage of a set. Graph of a function, restriction of a function. Odd and even, periodic, bounded functions. Bijections, injections and monotonic functions. Superposition of the functions, inverse functions.
Polynomials, theorem about division of polynomials, and about roots of a polynomial. Rational functions (also a
homographic function). Exponential and logharitmic functions (as mutually inverse functions). Trigonometric and
cyclometric functions.
3. Sequences and limits (2 h)
Definition of a sequence. Examples of the arithmetic and geometric series, and others. Mathematical induction. Recursive definitions – examples. Properties of sequences (bounddedness, monotonicity). Definition of the limit of a sequence. Arithmetics of the limits. Indeterminate forms. Necessary and sufficient condition of the convergence of a sequence. The Euler number. Theorem about three sequences.
4. Limit of a function and continuity (3 h)
Neighborhood, accumulation points. Heine and Cauchy’s definitions of the limit of a function. One-sided limits. Infinite limits. Algebraic limit theorem. The three functions theorem. Theorem about the limit of a composition. One-sided limits. Definition of a continuous function. One-sided continuity of a function. The composition theorem for continuous functions. The Weierstrass’ and Darboux theorems. Local sign-preserving property for continuous functions.
5. Derivatives of a function (3 h)
Definition of a derivative of a function and its geometrical and physical interpretation. Differential of the function. One-sided derivatives. Continuity of a differentiable function. Arithmetic operations on the derivatives. Derivative of the composition and the inverse function. Derivatives of elementary functions.
6. The fundamental theorems of differential calculus and their applications (2 h)
Calculatung the limits of functions using de l’Hospital rule. Asymptotes of the functions: vertical, horizontal and oblique. Rolle’s and Lagrange’s theorems and their application for determining the monotonicity of functions.
7. Higher order derivatives and the Taylor formula (2 h)
Definition of the n-th derivative. Functions of the C^n class and the C-infinity class. Taylor and Maclaurin formulas and their application to calculations of the approximated values (eg. Approximated value of the Euler number).
8. Local extrema (2 h)
Definition of a local maximum and minimum. Fermat’s theorem. Necessary and sufficient conditions of the existance of local extrema. The global extrema. Optimalization problems.
9. Curve sketching (2 h)
Convexity and concavity of a function and its relation to the second derivative. Inflection points. Checking the properties of the functions and drawing their graphs.

10. Indefinite integrals (7 h)
Definition of the indefinite integral, basic formulas and rules (eg. additivity, scalar multiplication). Remark on nonelementary indefinite integrals. Different methods of integration: by parts, by substitution. Integration of rational functions (resolving the rational function into the prime fractions). Integration of irrational functions (the Euler substitutions). The method of finding unknown coefficients. Integration of trigonometric functions.

11. Definite integrals (3 h)
Riemann definition of the definite integral. Necessary and sufficient condition of the integrability of a function. Additivity and scalar multiplication of the definite integrals. The integral mean value theorem. Function of the upper limit of the definite integral. The relation between the definite and undefinite integral.
12. Improper integrals (2 h)
Definition of the improper integral. Absolute convergence of the improper integral. The comparison test.
13. Geometrical applications of the definite integrals (3 h)
Cartesian and polar coordinates. Calculation of the area of the planar regions bounded by graphs of the functions. Parametrised curves in R^n, calculation of the lenght of the curve. Volume and area of the solid of revolution.
14. Hiperbolic functions (1 h)
Definition, drawing graphs, properties.
15. Number serieses (3 h)
Definition. Convergence and divergence of a series, conditional and absolute convergence. The necessary condition of the convergence of a series. Convergence tests (direct comparison, ratio, root, itegral tests). Arithmetic operations on the serieses. Alternating serieses and Leibniz test.
16. Functional sequences and serieses (2 h)
Functional sequence. Pointwise and uniform convergence of a function sequence. Function serieses. Pointwise, uniform and absolute convergence of a function series. Necessary condition of convergence, convergence tests. Weierstrass convergence test. Theorem about differentiability and integrability of a function series.
17. Power serieses (4 h)
Definition. Abel’s Theorem. Radius of convergence of a power series. Cauchy-Hadamard’s Theorem and d’Alembert’s Theorem. Differenetiation and integration of the power series. Taylor and Maclaurin serieses. An analitic function. Note on functions of a complex variable.

Auditorium classes:
Mathematical Analysis 1

CLASSES:

1. Revision form secondary school, mathematical logic, functions (4 h)
2. Sequences and their limits (3 h)
3. Functions and their limits. Continuity of a function (3 h)
4. Derivatives of single variable functions and their applications (10 h)
5. 1st Class Test (2 h)
6. Undefinite integrals (6 h)
7. Definite integrals and their application (6 h)
8. Number serieses (2 h)
9. Pointwise and uniform convergence of function series (2 h)
10. Calculating the radius of convergence of power series (3 h)
11. 2nd Class Test (2 h)
12. Expanding functions into Taylor series (2 h)

Student workload (ECTS credits balance)
Student activity form Student workload
Summary student workload 199 h
Module ECTS credits 6 ECTS
Participation in lectures 42 h
Participation in auditorium classes 42 h
Realization of independently performed tasks 70 h
Preparation of a report, presentation, written work, etc. 45 h
Additional information
Method of calculating the final grade:

1. The necessariy for obtaining a positive final mark (OK) is positive mark from both classes and the exam. The sufficient condition for being addmited to the exam is a positive mark from the classes.
2. After calculating SW = 0,49SOC+0,51SOE, where SOC is an arithmetic mean of all marks obtained from classes and SOE is an arithmetic mean of all marks obtained at the exams, the final mark (OK) is given on the basis of the algorithm:
if SW >4.75 then OK:=5.0 (bdb) else
if SW >4.25 then OK:=4.5 (db) else
if SW >3.75 then OK:=4.0 (db) else
if SW >3.25 then OK:=3.5 (dst) else OK:=3 (dst)

Prerequisites and additional requirements:

Mathematical knowledge from secondary school.

Recommended literature and teaching resources:

1. J. Bird, „Higher Engineering Mathematics”.
2. B. Demidovitch „Problems in Mathematical Analysis”

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

Additional scientific publications not specified

Additional information:

None