Objectives and competences
Objectives:
To know thoroughly the basics of functions of one and more variables, differential and integral calculus and their use in engineering mathematics.
Competences:
• The ability to use the acquired knowledge for computing some quantities in engineering and technics which appear in different basic and applied sciences, for example Mathematical analysis II, Physics, Electrical engineering, Thermodynamics.
• To acquire the basic skills of mathematical modeling of engineering problems related to differential equations.
Content (Syllabus outline)
1. Numerical sets (predicate logic, integer, rational, real, irational, complex numbers).
2. Sequences (definition of a sequence, accumulation point, limit, upper and lower bound).
3. Functions (basic elementary functions, zeros, domain, codomain, inverse of the function, continuity of funkction, limit of function).
4. Derivative (definition, problems of extrems, curvature, higher derivations, connection with physics, L’Hospital’s rule. Taylor’s formula and Taylor’s series).
5. Integral (definition of indefinite integrals and basic properties, integrals of basic elementary functions, special methods of integration, definition of a definite integral, properties of definite integrals, change of variable of integration, improper integrals. applications – surface, rotary body, length of the arc).
5. Differential equations (DE of 1st order, DE of 2nd order with the constant coefficients, Fourier series, Laplace transformation, applications in physics and engineering).
6. Functions of more variables (partial derivatives, total diferential, local extrems, constrained extremes, Taylor series of functions of two variables).
Learning and teaching methods
Lectures, tutorial, coursework assignments.
Intended learning outcomes - knowledge and understanding
Knowledge and understanding:
At the end of the course students are going to:
1. Use basic notions of predicate logic and set theory.
2. Define basic notions concerning real sequences and series and use them at solving different mathematical problems.
3. Apply standard methods for plotting graphs of functions.
4. Apply standard methods for solving basic differential calculus problems (extremes of functions, the tangent and normal lines, incresing and decreasing of functions, curvature, inflection points).
5. Solve the indefinite integral using table of integrals, substitution of new variable and ''per partes'' methods, and the method of partial fractions .
6. Use integral for computing the area of the shape, the area and volume of the rotating body, and the length of the arc and compute the integral with infinite boundaries.
7. Recognize and solve separable, 1st order, homogeneous and Bernoulli DE, and 2nd order linear ODEs with constant coefficients including initial and boundary conditions.
8. Use Laplace transform for solving DE with initial conditions.
9. Describe the origin of partial derivative of functions of several variables, and use it for analysing maxima and minima and constrained optimisation for functions of two variables.
10. Solve simple optimization problem using the theory of functions of more variables.
11. Describe the Taylor expansion of function of one or two variables and Fourier expansion of function of one variable.
Intended learning outcomes - transferable/key skills and other attributes
• Communication skills: unequivocal and precise expression.
• Computational skills: performing a variety of computational operations.
• Problem solving: identifying and solving mathematical problems that occur in energy and that are described by the functions of one or several variables and by differential equations.
Readings
Osnovna/Basic:
- M. Mencinger, Zbirka rešenih nalog iz matematične analize in algebre, FG UM, Maribor, 2006.
- R. Jamnik, Matematika, DMFA Slovenije, Ljubljana, 2008.
Dodatna/Additional:
- I. Vidav, Višja Matematika I, DMFA Slovenije, Ljubljana, 2008.
- E. Kreyszig, Advanced Engineering Mathematics, J. Wiley and Sons, 2011.
Additional information on implementation and assessment Method (written or oral exam, coursework, project):
• Written exam
• Oral exam
• coursework
Notes:The student must finish each part of the exam (written exam, oral exam, coursework) with at least 50%. In the case that a student is offered the opportunity to take a written exam in the form of two midterm exams, each must be graded with at least 30% and both with an average of at least 50% to pass the written part of the exam.
