CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2023/2024

# Linear Algebra and Differential Calculus

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Department of Natural Sciences
Synopsis:

Differential calculus consists of: sequences and their limits. Functions of one real variable, their limits,

continuity, derivatives. Local and absolute extrema of a function of one variable, investigations of functions. Taylor-polynomial.

Requirements:

Seminars assesment:

a) Compulsory attendance of all seminars.

b) Activities at seminars will be checked by mini-tests. There will be 8 mini-tests during the semester, evaluated by 5 points each. Total sum MT = 0 - 40 points.

c) Midterm tests

1st midterm test in 7th week and 2nd midterm test in 13th week of the semester.

Midterm test consists of 4 tasks, each task evaluated by 5 points (maximum 20 points at each midterm test).

A student must gain at least 9 points at one midterm test, but minimum gained from both midterm tests must be at least 20 points. So, total sum VT=20-40 points.

Student's grading from seminars transferred for the exam:

MT/8+VT/4=minimum 5 maximum 15 points.

Exam:

Assesment of Seminars registered by repective teacher in KOS.

Exam is only in written form, lasts 90 minutes.

It is forbidden to use a calculator or a mobile phone during the exam.

The exam consists of

a) 7 tasks, evaluated by 10 points each, in total maximum 70 points

b) 5 tests, evaluated by 3 points each, in total maximum 15 points

c) transferred points maximum 15 points

Total maximum 100 points

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

Syllabus of lectures:

1. Number sets, sequences, limit of sequence, convergence of sequence. Functions of one real variable, properties, operations with functions. composed function, inverse function.

2. Limit and continuity of function, rules for calculation of limits, infinite limits, right-hand, left-hand limits.

3. Asymptotes, derivative, rules for calculation, derivative of composite function, inverse function, higher order derivative.

4. Differential of function and its application, properties of a function continuous on a closed interval, L'Hospital rule, implicit functions.

5. Local and global extrema, graph of function.

6. Taylor polynomial, number series, criteria of convergence, sum of series.

7. Gauss elimination method of solution of linear algebraic equation system (LAES). Vector spaces, subspaces, their properties.

8. Linear combinations of vectors, linear (in)dependence of vector system, base and dimension, scalar product.

9. Matrices, rank of matrix, product of matrices, inverse matrix, regular and singular matrices.

10. Permutation, determinant of a square matrix, Sarrus rule, calculation of inverse matrix.

11. Solution of LAES , Frobenius theorem, equivalent systems, structure of general solution of LAES, system with regular matrix, Cramer rule.

12. Coordinates of a vector in given baze. Eigen values and eigen vectors of a matrix. Angle of two vectors, scalar and vector product, application.

13. Some notes to analytical geometry of E2, E3 spaces, conics.

14. Recapitulation.

Syllabus of tutorials:

1. Sequences, limits, elementary functions.

2. Operations with functions, properties, limit of function, continuity.

3. Asymptotes, inverse function, derivative of function.

4. Intervals of monotony, L'Hospital rule for limits.

5. Investigation of function, local and global extrema.

6. Taylor polynomial, number series, convergence. Test 1.

7. Gauss elimination, vector spaces.

8. Linear (in)dependence of vectors, base, dimension.

9. Matrices, inverse matrix, product of matrices.

10. Calculation of determinant, Sarrus rule.

11. LAES solution.

12. Coordinates of vector in given base, eigenvalue and eigen vectors of a square matrix.

13. Analytical geometry in a plane and in a space. Test 2.

14. Revision, credit.

Study Objective:

The goal of study is to get a notion about base of differential calculus and linear algebra and some applications of theory.

Study materials:

L. Gillman, R.H. McDowell: Calculus, Norton, New York, 1973

Note:
Further information:
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2023-08-30
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