Subject 'Mathematics III'

Year:   2015/2016    2016/2017    2017/2018    2018/2019    2019/2020    

The course follows smoothly from the Mathematics II syllabus ending with revision of functions of one real variable and limits. This course starts with revision of basic differential calculus of functions of one and two variables. It then introduces some more advanced integral calculus, and finally it introduces differential equations with applications.
Short description of the course:
- derivative of a function of one real variable, its properties, existence and interpretations;
- differential of a function, derivatives of higher order;
- derivative of a composite function, chain rule, logarithmic derivative, derivative of an inverse and implicit function, l'Hopital's rule;
- geometrical and physical meanings of derivative;
- application of derivative in investigation of functions: intervals of function monotony, extreme points, domains of convexity and concavity, inflection points, asymptotes; extreme value problems;
- function of two real variables, its continuity and limit, partial derivatives;
- antiderivative and indefinite integral of a function of one real variable;
- techniques of integration: linear properties and table of integrals, integration by substitution and by parts, integration of rational and trigonometric functions;
- definite integral and its properties, Newton-Leibniz formula;
- geometrical applications of definite integral: area under the graph of a function, volume of a solid of revolution and computing the arc length of a function;
- differential equation and its solution, Cauchy problem;
- ordinary differential equations of first order: differential equations with separated and separable variables, homogeneous and linear differential equations;
- simple ordinary differential equations of second order;
- some applications of differential equations in natural sciences.

The aim of the course is to provide students with a general overview of the basic themes and issues in differential and integration calculus including differential equations. The material in this course is absolutely fundamental to nearly all areas of natural and engineering sciences. In this course, students develop logical and mathematical thinking ability as well.

By the end of this course, the student should
- know the basic rules for differentiation and be able to find derivatives of elementary and composite functions;
- be able to apply derivatives in investigation of functions and be able to solve extreme value problems;
- be familiar with geometrical and physical interpretations of derivative of a function;
- be familiar with functions of two real variables and be able to find partial derivatives;
- be familiar with the basic techniques of integration and be able to find area under the graph of a function, volume of a solid of revolution and the arc length of a function using integrals;
- be able to solve simple first and second order ordinary differential equations and be able to solve word problems using differential equations..

The course consists of 80 contact hours: lectures, practical lessons, group study.
Independent work is about 76 academical hours: working through the lecture materials, solving practical exercises, e-learning, preparing for the tests.

1. Kangro, G. Matemaatiline analüüs I. Valgus, Tallinn, 1982.
2. Liiva, T. Kõrgem matemaatika. Analüütiline geomeetria. Lineaaralgebra. TTK, Tallinn, 2004.
3. Liiva, T. Kõrgem matemaatika II. Diferentsiaalarvutus. TTK, Tallinn, 2007.
4. Loone, L., Soomer, V. Matemaatilise analüüsi algkursus. TÜ Kirjastus, Tartu, 2007, 2009.
5. Lõhmus, A., Petersen, I., Roos, H. Kõrgema matemaatika ülesannete kogu. Valgus, Tallinn, 1982.
6. Piskunov, N. Diferentsiaal- ja integraalarvutus I. Valgus, Tallinn, 1981.
7. Puusemp, P. Kõrgema matemaatika ülesannete kogu. Polütehniline Instituut, Tallinn, 1988.
8. Pedas, A., Vainikko, G. Harilikud diferentsiaalvõrrandid. TÜ Kirjastus, Tartu, 2011.
9. Reimers, E. Matemaatilise analüüsi praktikum. I, II. Valgus, Tallinn, 1988.
10. Safiulina, E. Integraalarvutus, TTK, Tallinn, 2008.
11.Tammeraid, I. Matemaatiline analüüs I. TTÜ Kirjastus, Tallinn, 2002.
12. E-learning course materials at Moodle environment (http://ekool.tktk.ee).

During the course a student has to pass three tests and to present two homeworks. The tasks of the tests and homeworks are composed on the basis of the standard problems solved at the practical lessons and according to the topics from the course program. The number of tests and homework assignments during the course can be changed by the responsible lecturer.

2019: FOR*  

2018: AT  ET  FOR*  HE  ME  RG  TE  

2017: AT  ET  FOR*  HE  ME  RG  TE  TT  

2016: AT  ET  FOR*  HE  ME  RG  TE  TT  

2015: AT  ET  HE  ME  RG  TE  TT  

* Optional subject