1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. Mathematical Analysis 1

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Mathematical Analysis 1 - opis przedmiotu

Cel przedmiotu

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. 

Wymagania wstępne

Zapisz zmiany

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

Zakres tematyczny

Lecture

I. Real numbers and complex numbers

1. Axioms of real numbers. Infimum and supremum (4 hours)
2. Root of a non-negative number (1 hours)
3. 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. (2 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 (2 hours)
3. Upper limit and lower limit of a sequence (1 hour)
4.  Fundamental properties of series of numbers (2 hours)
5. Series with non-negative terms. Comparison tests. Cauchy’s and d’Alembert criteria (3 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. Darboux's theorem (1 hours)
3. Global extrema. Weierstrass' 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 (3 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 (1 hours)
2. Power functions of a real variable (1 hour)
3. Trigonometric functions and inverse trigonometric functions (1 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

1. Derivative and its interpretation. Differentiability of a function of a single real variable. Fundamental formulas concerning derivatives. Derivatives of elementary functions (3 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)
5. Local extrema (1 hour)
6. Characterization of the convexity of a function (1 hour)
7.  Relationships of the uniform convergence to differentiating. Differentiation of power series. Taylor series (2 hours)
8. Differentiability of elementary functions (1 hour)
9. Primitive function (2 hours)
10. Algorithm of integrating rational functions (2 hours)
11. Derivative of a function of a complex variable (a brief information) (1 hour)

IX. Applications of differential calculus (the material should be elaborated in a written form by teams of students basing on a literature indicated by the lecturer)

1. Straight-line motion.

2. Applications to geometry

3. Differential and approximate calculation.

4.  Newton method.

5. Applications in economics.

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. Indeterminate forms in the extended set of real numbers (1 hour)

II. Elementary functions I

1. Reminder of the basic properties of trigonometric functions. Their graphs (2 hours)
2. Examples of elementary functions in simple problems outside mathematics (1 hour)

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. Naper-Euler number (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 series of functions (1 hour)
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)
5. Taylor series expansion of functions (2 hours)

Colloquium (2 hours)

Metody kształcenia

Traditional lecture; exercises in which students solve problems and discuss, as well as prepare biographies of mathematicians whose names appear at the lecture; group work completed with a written study; work with a book and with the help of the Internet. If necessary (determined by the order of the Rector of the University of Zielona Góra), classes can be in online form.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

1. Three colloquia with problems of various degree of difficulties, allowing to verify if students attained learning outcomes at the very least.
2. Exam 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.

Literatura podstawowa

1. J. Douglas Faires, Barbara T. Faires, Calculus, Random House, New York, 1988.

2.  J. Douglas Faires, Barbara T. Faires, Calculus of one variable, Random House, New York, 1988.

Literatura uzupełniająca

1. William F. Trench, Introduction to real analysis, Pearson Education, 2013.

Uwagi

Zmodyfikowane przez dr hab. Bogdan Szal, prof. UZ (ostatnia modyfikacja: 13-02-2024 16:55)