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
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 the concepts of limit, continuity, derivative and integral;

• master the differentiation rules and know how to use the derivative in optimisation problems;

• know a number of standard limits and how to use them in computations;

• know different integration techniques;

• know how to use integrals for the computation of areas, volumes and arc lenghts;

• know some convergence criteria for positive series and the concept of absolute convergence;

• be able to compute Taylor expansions of elementary functions;

• know how to solve linear differential equations with constant coefficients, first order linear differential equations using integrating factors and separable differential equations;

• be able to exemplify and interpret important concepts in specific cases;

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

• be able to present mathematical arguments to others.

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.