MATA21 Analysis in One Variable, 15 credits, is a compulsory course for a Bachelor of Science degree in mathematics and physics. The course is given at half-study pace during during the entire semester. The course is given in English during the autumn semester and in Swedish during the spring. The course is usually only given for students admitted to the Bachelor's programme at the Faculty of Science during the autumn semester but can be taken as a stand-alone course during the spring semester.

Course Content
The course treats

The real numbers: axiomatic description and proofs of basic arithmetical rules.
Limits of number sequences: formal definition, proofs and use of arithmetical rules, the Bolzano--Weierstrass theorem.
Limits of functions: formal definition of limits of functions, proofs of the rules of differentiation.
Continuity: definition and basic properties of continuous functions, the intermediate value theorem, the extreme value theorem for continuous functions, uniform continuity.
Derivatives: definition, proofs and applications of computational rules for derivatives, the mean value theorem, optimisation, curve sketching, proof techniques for identities and inequalities.
Primitive functions: proofs and applications of basic computational rules and integration methods such as change of variables, partial integration and integration of elementary functions (trigonometric integrals, rational integrals, partial fraction decomposition).
Definite integrals: definition, integrability of monotonous functions and continuous functions, proof of the fundamental theorem of calculus and applications.
Differential equations: direction field, solution methods for first-order linear or separable differential equations and higher-order linear differential equations with constant coefficients.
Taylor expansions: Taylor's formula, proofs, applications and error term estimates.
Series: proofs and applications of convergence criteria for positive and alternating series.
Improper integrals: proof and applications of convergence criteria for improperintegrals of positive functions

Teaching
The teaching consists of lectures, seminars, exercise classes and mentoring hours. An essential element of the seminars and exercise classes is training in problem solving and oral mathematical communication. A project is included in the course requirements.

Assessment
The examination consists of

written and oral presentations of the project (2 credits)
a written examination possibly together with an oral examination (13 credits)

Regarding examination

Course Literature

Jan-Fredrik Olsen: Don't Panic, A guide to MATA21 Analysis in One Variable, provided by the department. A pdf-version of the book, divided into three parts, is available on the links below:
- Don't Panic, part 0
- Don't Panic, part 1
- Don't Panic, part 2
- Note that the "YouTube"-links in the margins of the textbook are hyperlinks to corresponding videos. All videos have been created specifically for the course material.

Official Course Description

Course Evaluation
Link to course evaluations on the department's website:

Course Evaluations, Mathematics, Faculty of Science