Integrální počet

Vztahy:

Podmínkou zápisu na předmět F7ABBITP je, že student úspěšně absolvoval F7ABBLAD nebo získal zápočet a nevyčerpal všechny zkouškové termíny předmětu F7ABBLAD. Předmět F7ABBITP lze klasifikovat až po úspěšné klasifikaci předmětu F7ABBLAD

Úspěšná klasifikace předmětu F7ABBITP je podmínkou pro následnou klasifikaci předmětu F7ABBFVP

Garant předmětu:

Přednášející:

Cvičící:

Předmět zajišťuje:

katedra přírodovědných oborů

Anotace:

The subject is an introduction to integral calculus and integral transforms.

Integral calculus: anti-derivative, indefinite integral, properties and methods of integration (integration by parts and by substitution, partial fractions), definite integral, properties, Newton-Leibnitz fundamental theorem, simple applications of both indefinite and definite integrals, improper integral, solving differential equations (ODEs) (1st order ODEs with separable variables, linear 1st order homogenous as well as non-homogenous ODEs, 2nd order linear homogenous and non-homogenous ODEs with constant coefficients),intro to multiple integrals, particularly double integral and applications.

Integral transforms: Laplace transform and inverse Laplace transform and their application for solving nth order linear ODEs with constant coefficients.

Požadavky:

Form of verification of learning outcomes and other requirements for the student:

A - Compulsory attendance at exercises, absences must be duly excused in advance and subsequently documented e.g. by a medical certificate. Maximum of three properly excused absences. Active participation in the exercises is evaluated (maximum 5 points per semester, and these are added to the exam grade).

Attendance at lectures is not compulsory, but if a student fails to attend a lecture, he/she is obliged to supplement the material by self-study and must come prepared to the exercise.

B - Knowledge in the scope of individual topics of lectures is tested by two semester tests, which students take together in a single term in the middle and at the end of the semester according to the teaching schedule of the subject for the given academic year. Calculators certified for the mathematics graduation exam (i.e., non-programmable, without integrals and equation solving) and a list of formulas that will be included in the test assignment are allowed in the tests.

A condition for the award of credit is the fulfilment of point A and the achievement of at least 50% of the points from both semester tests (each test has a maximum of 40 points, the minimum for successful completion is 40 points in total).

A condition for admission to the examination is the credit entered in KOS. The exam is written only, lasts 120 minutes, calculators and formula lists are allowed, the same as for the mid-semester tests. The exam contains mostly numerical examples supplemented by theoretical subquestions within the of the material covered in lectures. The examination test consists of numerical problems from the material covered in the lectures and exercises supplemented by theoretical sub-questions. The maximum score is 75 points, for successful completion of the examination test, the student must obtain at least half of the points (i.e. 37.5 points). The points from both semester tests will be added to the exam score as follows: points above the mandatory 50% divided by 2 (max. 20 points) and points for activity in the exercises (max. 5 points). The total number of points is therefore (75+20+5) 100

Course grade: A: 100-90, B: 89-80, C: 79-70, D: 69-60, E: 59-50, F: less than 50.

Osnova přednášek:

1. Antiderivative, indefinite integral, basic properties. Table of basic integrals, integration by parts.

2. Method of substitution. Integration of simpler rational functions - decomposition to partial fractions.

3. Integrals of partial fractions. Integration of trigonometric functions, their products and powers.

4. Definite (Riemann's) integral. Geometric and physical meaning. Newton-Leibniz formula. Application of the integration by parts in the calculation of the definite integral

5. Application of the substitution method in the calculation of the definite integral. Further geometric and physical applications.

6, Improper Riemann's integral (due to the function, due to the limit).

7. Mid-term test.

8. 1st order ordinary differential equations (ODE) with separable variables. Initial condition, Cauchy problem. Simple applications.

9. 1st order linear ODE (homogeneous and inhomogeneous). Method of variation of a constant.

10. 2nd order linear ODE with constant coefficients (homogeneous and inhomogeneous). Characteristic equation, fundamental system of solutions of the homogeneous equation, general solution of the homogeneous equation.

11. Particular solution of the inhomogeneous equation, general solution of the inhomogeneous equation. Methods of finding the particular solution: method of estimation and variation of constants.

12. Laplace transform and inverse Laplace transform. Application of the Laplace transform to solution of ODE.

13. Repetition: solution of various types of ODE by various methods. Application s in geometry, physics, chemistry, population dynamics, spread of infection, spread of information, etc.

14. End-term test.

Osnova cvičení:

1. Indefinite integrals, application of formulas from the table of basic indefinite integrals, integration by parts.

2. Method of substitution for indefinite integrals. Decomposition of simple rational functions to partial fractions.

3. Integration of simpler rational functions. Integration of trigonometric functions, their products and powers.

4. Calculation of the definite integral, Newton-Leibniz formula, integration by parts in the definite integral.

5. Application of the substitution method in calculation of the definite integral. Geometric and physical applications of definite integral: area of a plane plate, length of a curve, volume and surface area of a body of revolution, mass, coordinates of the center of gravity, etc.

6. Repetition of the material before the mid-term test.

7. Evaluation of improper integrals.

8. 1st order ordinary differential equations (ODE), solution by separation of variables, physical and other applications.

9. Linear 1st order ODE, method of variation of a constant.

10. Linear 2nd order ODE with constant coefficients.

Characteristic equation, fundamental system of solutions of the homogeneous equation, general solution of the homogeneous equation.

11. General solution of the inhomogeneous equation, methods of finding particular solutions: variation of constants and method of estimation. Physical applications (in mechanics, electrodynamics, etc.).

12. Laplace transform, inverse Laplace transform. Calculation of Laplace and inverse Laplace transforms of various functions.

13. Application of the Laplace transform in solving ODE.

14. Repetition and systematization of knowledge before the exam.

Cíle studia:

The goal of the course is to gain basic knowledge in the fields of integral calculus (indefinite integral, definite Riemann's integral, double integral, applications) and ordinary differential equations (= ODEs) (ODEs with separable variables, 1st and 2nd order linear ODEs with constant coefficients, applications).

Studijní materiály:

Study materials

[1] Neustupa J.: Mathematics 1, skriptum ČVUT, 2004

[2] Bubeník F.: Problems to Mathematics for Engineers, skriptum ČVUT, 2007

[3] Stewart, J. : Calculus, Brooks/Cole, 2012

Poznámka:

The student must have successfully completed F7ABBLAD or received credit and not exhausted all the exam dates for F7ABBLAD as a condition of enrolling in F7ABBITP. The F7ABBITP course can only be classified after the successful classification of the F7ABBLAD course.

Další informace:

Pro tento předmět se rozvrh nepřipravuje

Předmět je součástí následujících studijních plánů:

• Prospectus - bakalářský (!)
• Bakalářský studijní program Biomedicínská technika v aj (povinný předmět)
• Bakalářský studijní program Biomedicínská technika v aj (povinný předmět)