MATA31 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: axioms, examples of proofs of basic arithmetical rules.
The elementary functions, polynomials, rational functions, the exponential function and the natural logarithm, the trigonometric functions and the inverse trigonometric functions; definitions, basic properties, and quantitative approximations using representations in terms of areas and arclengths.
Sequences of numbers and their limits: formal definition of the limit, examples of proofs of their computational rules, visual representation of convergence of recursive sequences, quantitative approximations.
Infinite series: applications and proofs of convergence tests, absolute convergence, quantitative approximations using partial sums and tail estimates.
Functions and their limits: formal definition of the limit, proofs and applications of their computational rules, indeterminate forms and asymptotes.
Continuity: continuity of elementary functions, the intermediate value theorem and the min-max theorem.
Derivatives: definition, proofs and applications of computational rules, differentiation formulas for elementary functions, Rolle’s lemma, the Mean value theorem and L’Hopital’s rule.
Applications of the derivative: optimisation and graph sketching, techniques for establishing identities and inequalities.
Indefinite integrals: proofs and applications of basic computational rules and integration methods, such as change of variables, partial integration and use of partial fraction decomposition.
Definite integrals: Darboux integrability of monotone functions and functions with bounded derivative with related error estimates, the fundamental theorem of calculus, applications to arclength, rotational volumes and surfaces, numerical approximations of definite integrals.
Improper integrals: convergence criteria for improper integrals for positive functions, absolute convergence, comparison to infinite series.
Differential equations: direction fields, analytic solution methods for separable and linear first order differential equations, solution method for linear higher-order differential equations with constant coefficients, numerical approximations of solutions of initial value problems using Euler’s method.
Taylor expansions: Taylor's formula with Lagrange’s formula for the error term, uniqueness theorem for Taylor polynomials, numerical approximations of function values and integrals using Taylor polynomials.

In addition, materials on sets, functions and relations, induction, the binomial theorem, as well as variables, for-loops and if-statements in Python are covered at the beginning of the course.

Teaching
The teaching consists of lectures, seminars, exercise classes and mentoring sessions. An essential element of the seminars is training in problem solving and mathematical communication and assumes students’ active participation. Several compulsory computer-based tests are given during the course. A compulsory assignment providing students with training in mathematical communication in writing as well as a compulsory problem solving assignment are included in the course.

The first part of the course is taught jointly with Algebra and Vector Geometry and Computational Programming in Python, where material on sets, induction, the binomial theorem, functions and relations, as well as variables, for-loops and if- statements in Python, are covered.

Assessment
The examination consists of

written assignment in mathematical communication (1 credit)
oral presentation of problem solving assignment (1 credit)
computer based tests (1 credit)
mid-term written examination (4.5 credits)
final written examination with an optional oral examination (7.5 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.
Adams RA, Essex C: Calculus a complete course, 10th edition, Pearson Canada, 2021, ISBN: 9780135732588
Göran Forsling, Mats Neymark: Matematisk analys, en variabel, Liber förlag, andra upplagan, 2011 (in Swedish) ISBN 978-91-47-10023-1

Official Course Description

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

Course Evaluations, Mathematics, Faculty of Science