Single Variable Calculus

10 credits

Syllabus, Bachelor's level, 1MA013

A revised version of the syllabus is available.

Code: 1MA013
Education cycle: First cycle
Main field(s) of study and in-depth level: Mathematics G1F
Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
Finalised by: The Faculty Board of Science and Technology, 19 March 2007
Responsible department: Department of Mathematics

Entry requirements

Basic Course in Mathematics

Learning outcomes

In order to pass the course (grade 3) the student should be able to

give an account of important concepts and definitions in the area of the course;

exemplify and interpret important concepts in specific cases;

formulate important results and theorems covered by the course;

describe the main features of the proofs of important theorems;

express problems from relevant areas of applications in a mathematical form suitable for further analysis;

use the theory, methods and techniques of the course to solve mathematical problems;

present mathematical arguments to others.

Higher grades, 4 or 5, require a higher level of proficiency. The student should be able to solve problems of greater complexity, i.e. problems requiring a combination of ideas and methods for their solution, and be able to give a more detailed account of the proofs of important theorems and by examples and counter-examples be able to motivate the scope of various results. Requirements concerning the student's ability to present mathematical arguments and reasoning are greater.

Content

Functions: monotonicity and inverse. Inverse trigonometric functions. Limits and continuity: notions and rules. The derivative: notions, differentiation rules, the chain rule, the mean value theorem and applications. Extreme value problems. Curve sketching. The integral: definite integral, primitive function, the fundamental theorem of integral calculus. Integration techniques: substitutions, integration by parts, integrals of rational functions. Improper integrals. Applications of integration: area, volume and arc length. Taylor's formula with applications.

Numerical series: convergence, convergence criteria for positive series, absolute convergence.

Convergence criteria for improper integrals. Power series. Ordinary differential equations: existence and uniqueness of solutions. Linear differential equations with constant coefficients. Solvable types of differential equations, integrating factors, variation of parameters.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination at the middle and the end of the course. Moreover, compulsory assignments may be given during the course.