By the successful passing of the exam Mathematics 1, a student will be able to:
1. Handle basic set operations and, on the elementary level, understand the concept of a set.
2. Successfully detect a possibility of an application of the mathematical induction techniques
3. Understand the relations between the sets of natural numbers, integers, rational, real and complex numbers.
4. Calculate the domains of the functions which rise as linear combinations, products, quotients and compositions of elementary functions.
5. Understand the concept of a limit of a function and to calculate elementary limits.
6. To define a derivative of a function and to understand its relation to a tangent of a graph of a function and its relation to the concept of a change of a physical quantity.
7. To calculate the derivative of the functions which are composites or products or quotients of the elementary functions.
8. Use information about the first and the second derivative of a function to understand the behaviour of a function and to draw its graph and use this analysis to
address elementary optimisation problems.
9. Motivate the introduction of the definite integral by the problem of the calculation of the area below the graph of a function and to solve elementary definite and indefinite integrals.
The content of the course:
Lectures:
1. Numbers. Natural numbers. The principle of mathematical induction.
2. Rational numbers. Real numbers.
3. The real line. Complex numbers. Sets. Functions. Domains and Images.
4. The overview of elementary functions.
6. Limit and continuity.
7. The notion of tangent; velocity and derivatives.
8. Studying of a behaviour of a function using derivatives
9. Convexity and concavity
10. Drawing a graph of a function
11. Optimisation problems.
12. Calculation of an area and volume and the notion of integral
13. Integration of elementary functions.
Exercises:
1. Application of the mathematical induction principle.
2. Determination of the infimum and the supremum of a real set.
3. Moivre s formula for powers of complex numbers; de Moivre s formula
4. Calculation of natural domains of combinations of elementary functions.
5. Determination of elementary limits of functions.
6. Derivatives of elementary functions, their products, quotients and compositions.
7. Determination of the intervals of increasing and decreasing of a function, of its convexity and concavity; the asymptotes
8. Solving simple optimisation problems
9. Integration of elementary functions and determination of an area below graph; the volume of rotational bodies.
