1. SylabUZ
2. Faculty of Mathematics, Computer Science and Econometrics
3. winter term 2019/2020
4. Mathematics - First-cycle studies leading to Bachelor's degree
5. Mathematical Analysis 1

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Mathematical Analysis 1 - course description

Aim of the course

To acquaint students with basic notions of mathematical analysis: convergence of a sequence and series, limit, continuity and derivative of a function, also with connections between these notions.

Prerequisites

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To study the course it is necessary to be familiar with high school mathematics.

Scope

Lecture
I. Real numbers and complex numbers
1. Axioms of real numbers. Infimum and supremum (4 hours)
2. Root of a non-negative number (2 hours)
3. Complex numbers (4 hours)
4. Extended set of real numbers (1 hour)
II. Elementary functions I
1. Polynomials and rational functions. Power functions of a real variable, with rational exponent (1 hour)
2. Trigonometric functions of a real variable. Trigonometric form of a complex number (3 hours)
III. Sequences and series of numbers
1. Sequences of numbers and their convergence. Bounded sequences. Cauchy’s condition (2 hours)
2. Calculating limits of sequences (3 hours)
3. Upper limit and lower limit of a sequence (1 hour)
4. Fundamental properties of series of numbers (3 hours)
5. Series with non-negative terms. Comparison tests. Cauchy’s and d’Alembert criteria (4 hours)
6. Absolute and conditional convergence. Riemann’s theorem (2 hours)
7. Multiplying of series. Mertens’ theorem (2 hours)

IV. Limit and continuity of a function in a single variable
1. Limit of a function (2 hours)
2. Continuity. Intermediate value theorem (2 hours)
3. Global extrema. Extreme value theorem (1 hour)
4. Relationships of limits to continuity (1 hour)
5. Limits of functions of a real variable. One-sided limits (1 hour)
6. Limits of real-valued functions. Squeeze theorem (1 hour)
7. Asymptotes (1 hour)
V. Sequences and series of functions
1. Pointwise and uniform convergence (2 hours)
2. Series of functions. Weierstrass and Dirichlet tests (1 hour)
3. Power series. Cauchy-Hadamard’s theorem (1 hour)
VI. Elementary functions II
1. Exponential functions. Logarithmic functions of a real variable (2 hours)
2. Power functions of a real variable (1 hour)
3. Trigonometric functions and inverse trigonometric functions (2 hours)
VII. Monotonic functions and convex functions
1. Monotonic functions (2 hours)
2. Convex functions (only a brief information; a part of the material, pointed out by the lecturer, should be prepared in student’s own right basing on a literature indicated by the lecturer) (1 hour)
VIII. Elementary differential calculus I
1. Derivative and its interpretation. Differentiability of a function of a single real variable. Fundamental formulas concerning derivatives. Derivatives of elementary functions (2 hours)
2. Mean value theorems. Characterization of monotonicity (2 hours)
3. l’Hôspital’s rule (1 hour)
4. Higher derivatives and Taylor formula (2 hours)

Class
I. Real numbers and complex numbers
1. Using axioms of real numbers in simple proofs (2 hours)
2. Learning basic properties sets of rational and irrational numbers. Determining infima and suprema of sets of real numbers (3 hours)
3. Drawing sets of complex numbers on the plane. Operations in complex numbers. Solving algebraic equations in complex domain (2 hours)
II. Elementary functions I
1. Examples of occurring elementary functions in simple problems outside mathematics (1 hour)
2. Finding the trigonometric form of a complex number. Determining roots of complex numbers (2 hours)
III. Sequences and series of numbers
1. Examining the convergence of sequences of numbers via definition (2 hours)
2. Examining the convergence of sequences of numbers by using Cauchy’s condition (1 hour)
3. Examining the convergence of bounded monotonic sequences (2 hours)
4. Recurrent sequences. Making use of the squeeze theorem (1 hour)
5. Determining upper limits and lower limits (1 hour)
6. Examining the convergence of series of numbers. Using convergence tests (5 hours)
7. Calculating sums of series (1 hour)
8. Calculating Cauchy’s product of series (1 hour)
Colloquium (2 hours)
IV.Limit and continuity of a function in a single variable
1. Examining the existence and determining the limit of a function (4 hours)
2. Checking the continuity of a function (2 hours)

V. Sequences and series of functions
1. Examining the uniform convergence of sequences of functions (2 hours)
2. Examining the uniform convergence of sequences of series of functions (2 hours)
3. Training of using Weierstrass’ test to checking the uniform convergence of series of functions (1 hour)
4. Determining the center and radius of convergence of a power function (3 hours)
VI. Elementary functions II
1. Properties of exponential and trigonometric functions of a complex variable - training of elementary calculating proofs (2 hours)
Colloquium (2 hours)
VII. Monotonic functions and convex functions
1. Examining the convexity of functions via definition (1 hour)
2. Proving inequalities by examining the convexity of a suitable function (1 hour)
VIII. Elementary differential calculus I
1. Calculating derivatives via definition. Checking the differentiability. Finding tangent lines and normal lines to a curve (5 hours)
2. Making use of mean value theorems, verifying the monotonicity of differential functions, proving inequalities (3 hours)
3. Calculating limits of functions by using l’Hôspital’s rule (2 hours)
4. Application of Taylor’s formula to approximating of functions (2 hours)
Colloquium (2 hours)

Teaching methods

Traditional lecture; class where students, leaded by the teacher, solve exercises and discuss; team-work; work over a book; making use of internet.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

1. Verifying the extent of preparation of students and their activity during the classes.
2. Three colloquia with problems of various degree of difficulties, allowing to verify if students attained learning outcomes at the very least.
3. Exam (writ) with indicated point ranges.
The final grade is the arithmetic mean of those of the class and exam. A necessary condition to enter the exam is a positive grade of the classes. A necessary condition to pass the course is a positive grade of the exam.

Recommended reading

1. Witold Jarczyk, Notatki do wykładu z analizy matematycznej, http://www.wmie.uz.zgora.pl/~`wjarczyk/materialy.html
2. Witold Jarczyk, Zadania z analizy matematycznej, http://www.wmie.uz.zgora.pl/~`wjarczyk/materialy.html
3. J. Douglas Faires, Barbara T. Faires, Calculus, Random House, New York.

Further reading

1. Józef Banaś, Stanisław Wędrychowicz, Zbiór zadań z analizy matematycznej, Wydawnictwo Naukowo-Techniczne, Warszawa, 1993.
2. Witold Kołodziej, Analiza matematyczna, Państwowe Wydawnictwo Naukowe, Warszawa, 1986.
3. Walter Rudin, Podstawy analizy matematycznej, Wydawnictwo Naukowe PWN, Warszawa, 2002.

Notes

Modified by dr Robert Dylewski, prof. UZ (last modification: 19-09-2019 13:25)