Entry requirements

The part Basic Mathematics in the course Introduction to Studies in Mathematics, or Basic Course in Mathematics.

Learning outcomes

On completion of the course, the student should be able to

• give an account of the concepts of limit, continuity, derivative and integral for functions of one single variable and be able to use and calculate expressions that involve these;
• use the concepts in the previous point to analyse the properties of a function, e.g. sketch a graph, and for calculation of areas, volumes and arc lengths;
• calculate the Taylor expansion of elementary functions and be able to use these at applications;
• use some convergence criteria for positive series and be able to use power series and the concept of absolute convergence;
• solve linear differential equations with constant coefficients, first order linear differential equations by means of integrating factor and separable differential equations;
• translate problems from relevant application fields into a form appropriate for mathematical treatment;
• formulate important results and theorems within the field of the course;
• reproduce the fundamental features of the proofs presented in the course and be able to design own simple proofs;
• present mathematical arguments for others.

Content

Functions: monotonicity and inverse. Inverse trigonometric functions. Limits and continuity: notions and rules. The derivative: notions, differential 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. Power series. Ordinary differential equations: linear differential equations with constant coefficients, first order linear differential equations and separable differential equations.

Instruction

Lectures, teaching sessions and calculation exercises.

Assessment

Written examination at the end of the course. The test can be combined with written assignments during the course according to instructions delivered at course start.

Other regulations

The course can not be included in higher education qualification together with any of the courses Single variable calculus, Derivatives and integrals, Series and ordinary differential equations and Analysis for engineers.