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 supremum, infimum, and limits of sequences of real numbers, as well as proofs of relevant theorems;
• give an account of the notion of limits, and compute limits using standard formulas;
• give an account of the notion of continuity and derivative, as well as proofs of relevant theorems;
• use rules of differentiation, and use derivatives to compute extreme values and sketch graphs of functions;
• ​give an account of the Riemann integral of a continuous function and its analytic and geometric meaning, decide convergence or divergence of generalized Riemann integrals, as well as proofs of the basic theorems in integral calculus;
• use variable substitution and integration by parts to compute integrals, and use integrals to compute area, volume and arc length;
• give an account of basic notions, theorems and proofs in the theory of numerical series, Taylor series and power series;
• prove minor results and theorems in differential and integral calculus using fundamental theorems in these subjects.

Content

Real numbers: supremum and infimum, convergence of sequences. Functions: properties of certain functions, monotonicity and inverse, as well as inverses of elementary functions. Limits and continuity: notions, proof of existence and rules. The derivative: notions, differential rules, the chain rule, the mean value theorem with applications, extreme value problems, curve sketching. The integral: primitive function, Riemann integral, the fundamental theorem of integral calculus, integration techniques such as substitutions and integration by parts, improper integrals and their convergence, applications of integration such as area, volume and arc length. Numerical series: convergence, convergence criteria for positive and alternating series, absolute convergence. Taylor's formula with applications and power series.

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.