Mathematical Analysis I 2400-L1PPAMI
calculus of functions of a single variable, taking into account the economic context.
The first part of the course will be devoted to reviewing and organizing the fundamental mathematical tools that students are expected to have acquired in secondary school. This thematic block will cover the basics of logic, set theory (including Venn diagrams), properties of polynomials (including the existence of real and complex solutions), trigonometric functions, and the principle of mathematical induction.
In the next part, the mathematical apparatus will be expanded to include exponential and logarithmic functions, with reference to issues related to investment income calculations and loan repayments. Averages and relationships between them will also be defined.
The following section will focus on sequences, series, and infinite products. The introduced basic concepts, properties, and computational methods will form an important component of the mathematical toolkit used in subsequent parts of Mathematical Analysis I and II. Examples will include applications in the context of the cobweb model and actuarial calculations (e.g., present values of annuity payments).
The previously introduced topics will allow for defining limits of functions, introducing the concept of continuity, discussing the most important properties of continuous functions, particularly regarding attaining minima and maxima.
The penultimate part of the course will be dedicated to differential calculus of functions of a single variable. Methods of calculating derivatives and their applications—such as computing limits and identifying local and global extrema—will be presented. In this context, Lagrange’s Mean Value Theorem and its consequences will also be discussed. The mathematical tools developed will be applied to selected economic problems (e.g., utility maximization, portfolio optimization).
The final part will focus on the basics of integral calculus for functions of a single variable. Indefinite, definite, and improper integrals will be defined. Basic methods of integration will be introduced, such as integration by parts, substitution, and applications of integrals to economic problems—for instance, calculating total production costs or profit maximization.
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Estimated student workload: 6 ECTS x 25h = 150 hours
• Contact hours (K), Independent work (S):
o Seminar (classes): 60h (K), 0h (S)
o Consultations: 14.5h (K), 0h (S)
o Midterms: 3h (K), 0h (S)
o Final exam: 2.5h (K), 0h (S)
o Preparation for midterms: 0h (K), 8h (S)
o Preparation for exam: 0h (K), 12h (S)
o Preparation for seminars: 0h (K), 30h (S)
o Work with additional materials on the Moodle platform: 0h (K), 20h (S)
Type of course
Learning outcomes
Learning outcomes (codes): K_W04, K_U03, K_K01
After completing the course, the student:
KNOWLEDGE
• Knows basic logical laws, the principle of mathematical induction, and set properties
• Understands the properties of power, logarithmic, polynomial, and trigonometric functions
• Knows the properties of limits of sequences and functions, as well as basic limits related to elementary functions
• Knows the convergence criteria for series
• Understands the definition and geometric interpretation of the derivative and knows the derivatives of elementary functions
• Knows Lagrange’s theorem and its implications for determining function extrema
• Knows the definition of indefinite, definite, and improper integrals, along with their interpretation
SKILLS
• Can reason based on logical laws, the principle of mathematical induction, and facts concerning mathematical objects discussed in class
• Is proficient in using properties of power, logarithmic, polynomial, and trigonometric functions
• Can compute the limits of sequences and functions and apply these skills in verifying the existence of solutions to optimization problems
• Can assess the convergence of series and calculate the sums of selected series relevant to economic applications
• Is proficient in computing the derivatives of single-variable functions, both from the definition and using tables of derivatives
• Can classify critical points of single-variable functions, using i.a. conclusions from Lagrange’s theorem
• Solves optimization problems using differential calculus
• Computes indefinite, definite, and improper integrals using various methods and can apply these skills in economic problems
COMPETENCES
• Demonstrates independence in applying theoretical knowledge from mathematical analysis to solve optimization problems
• Shows responsibility and self-control through the experience of learning in a setting that involves choice in how knowledge is acquired
• Is systematic due to the need to adapt to a course completion structure where the final result includes scores from quizzes, tests, class participation, and the final exam
• Acts with integrity and honesty due to the strict adherence required for completion and examination requirements
Assessment criteria
To pass the course, the student must:
1. Attend the seminar sessions (a maximum of two absences is allowed).
2. The final grade is based on the total number of points earned during the semester (max. 100 points). Points can be earned in the tutorial sessions (max. 20 pts), in two common mid-term tests during the semester (max. 15 + 15 = 30 pts), and on the final exam after the semester ends (max. 50 pts).
To receive a passing final grade, a student must obtain at least 15 points on the final exam.
Default grade thresholds:
o Satisfactory (3.0): 35 pts
o Satisfactory plus (3.5): 45 pts
o Good (4.0): 55 pts
o Good plus (4.5): 65 pts
o Very good (5.0): 75 pts
o Excellent (5.5): 85 pts
3. Points for the tutorials can be earned through quizzes (6 × 2 pts) conducted in each group and for active participation (max. 8 pts).
4. To take the final exam, the student must pass the tutorials, i.e., earn at least a satisfactory grade (20 pts).
5. The exam is held in two sessions: the main exam session and the retake session. The exam, conducted in-person, is written and consists of solving a number of problems covering the full course material.
6. In the case of a justified absence from a test (i.e., a quiz, mid-term, or final exam) due to important circumstances, the student has the right to take the test at a make-up date.
To do this, the student must submit a request for justification:
o For a quiz or mid-term – to the tutorial instructor
o For the final exam – to the Dean of the Faculty of Economic Sciences (WNE)
If no such request is submitted, the student loses the right to a make-up test.
According to the Detailed Study Rules at the Faculty of Economics, the justification request with proper documentation must be submitted no later than 7 days after the date of the exam or 7 days after the reason for the absence has ceased.
Make-up quizzes and mid-terms are arranged and administered by the tutorial instructors. The make-up exam takes place during the next exam session or is scheduled by the Dean of WNE.
7. All of the above rules also apply to upper-year students with conditional pass status.
8. For first-year MSEM students who want to transfer to the Faculty of Economics during the semester, points and grades earned from quizzes and mid-terms at the MIM Faculty cannot be transferred and counted as part of the tutorial and mid-term assessments at the Faculty of Economics.
Bibliography
Basic literature:
M. Krych Analiza matematyczna dla ekonomistów, Wydawnictwa Uniwersytetu Warszawskiego, 2009
Supplementary literature:
R. Antoniewicz, A. Misztal, Matematyka dla studentów ekonomii. Wykłady z ćwiczeniami, WN PWN, Warszawa 2009.
J. Banaś, S. Wędrychowicz, Zbiór zadań z analizy matematycznej, WNT, Warszawa 2006.
T. Bażańska, I. Karwacka, M. Nykowska, Zadania z matematyki, podręcznik dla studiów ekonomicznych, PWN, Warszawa 1980.
Alpha C. Chiang, Podstawy ekonomii matematycznej, Państwowe Wydawnictwo Ekonomiczne, Warszawa 1994.
W. Dubnicki, J. Kłopotowski, T. Szapiro, Analiza Matematyczna. Podręcznik dla ekonomistów, WN PWN, Warszawa 2010.
W. J. Kaczor, M. T. Nowak, Zadania za analizy matematycznej, część 1, 2 i 3, WN PWN, Warszawa 2005.
W. Kołodziej, Analiza matematyczna, WN PWN, Warszawa 2009.
W. Krysicki, L. Włodarski, Analiza matematyczna w zadaniach, część I i II, WN PWN, Warszawa 2008.
K. Kuratowski, Rachunek różniczkowy i całkowy. Funkcje jednej zmiennej, WN PWN, Warszawa 2008.
W. Rudin, Podstawy analizy matematycznej, WN PWN, Warszawa 2009.A. Ostoja-Ostaszewski, Matematyka w ekonomii. Modele i metody, t. 1 i 2, PWN, Warszawa 1996.
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: