Calculus, 12 credits (TAIU10)

Analys i en variabel, 12 hp

Main field of study

Mathematics Applied Mathematics


First cycle

Course type

Programme course


Magnus Berggren

Director of studies or equivalent

Jesper Thorén
Course offered for Semester Period Timetable module Language Campus VOF
6IDAT Computer Engineering, Bachelor of Science in Engineering 1 (Autumn 2018) 0, 1, 2 -, 4, 4 Swedish Linköping o
6IELK Engineering Electronics, Bachelor of Science in Engineering 1 (Autumn 2018) 0, 1, 2 -, 4, 4 Swedish Linköping o
6IKEA Chemical Analysis Engineering, Bachelor of Science in Engineering 1 (Autumn 2018) 0, 1, 2 -, 4, 4 Swedish Linköping o
6IMAS Mechanical Engineering, Bachelor of Science in Engineering 1 (Autumn 2018) 0, 1, 2 -, 4, 4 Swedish Linköping o

Main field of study

Mathematics,Applied Mathematics

Course level

First cycle

Advancement level


Course offered for

  • Bachelor of Science in Computer Engineering
  • Bachelor of Science in Engineering Electronics
  • Bachelor of Science in Chemical Analysis Engineering
  • Bachelor of Science in Mechanical Engineering

Entry requirements

Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.

Intended learning outcomes

That you as a student will learn to feel confident with the mathematical expressions, reasoning and relations from Single Variable Calculus and that it will teach you calculating and problem solving skills needed for your further studies. After a completed course you should be able to:

  • Read and interpret mathematical texts
  • Explain definitions and expressions like local extremes, limits, continuity, derivatives, primitive functions and integrals
  • Explain and use central theorems like The Fundamental Theorem of Calculus, Mean-Value Theorems, The Intermediate-Value Theorem and The Max-Min Theorem.
  • Use mathematical laws for limits of functions, derivatives, antiderivatives and integrals.
  • Perform investigations of functions using derivatives, limits and the properties of basic functions and from this draw conclusions regarding the properties of the functions
  • Use standard techniques to calculate antiderivatives and definite integrals.
  • Express and calculate geometrical quantities like areas of plane regions, arc length, volumes of solids of revolution and areas of solids of revolution.
  • Handle differential equations (first-order separable and first-order linear equations and constant-coefficient equations of higher order.) and integralequations.
  • Explain Taylor’s formula
  • Use Taylorexpansions to approximate functions and investigate limits.
  • Perform checks of results and calculations to verify that they are correct and reasonable

Course content

  • Preparatory course: Equations and systems of equations. Geometric and arithmetic sums. Inequalities. Binomial theorem. Exponential functions and logarithms. Polynomials. Trigonometry and trigonometric functions.
  • Calculus: Real and complex numbers. Induction. Functions of a real variable. Elementary functions. Sequences, limits. Derivatives and continuity. Rules for differentiation. Properties of continuous functions. Study of functions. Primitive functions. Integration and geometrical applications, including area, curve length, areas of rotation and volumes of rotation. Improper integrals. Taylor's formula. Maclaurin expansions of elementary functions with applications to the calculation of limits. Linear ordinary differential equations of first and second order, separable equations.

  • Teaching and working methods

    The course is taken during the first semester of the first year. Teaching is done in lectures and problem classes. The examination consists of two written tests.
    The course runs over the entire autumn semester.


    TEN1Written examination, part 1U,3,4,56 credits
    TEN2Written examination, part 2U,3,4,56 credits
    KTR1Written test0 credits
    KTR2Written test0 credits
    KTR3Written test0 credits
    Passed written test 1 and written test 2 gives a bonus on the first written examination (TEN1). Approved written test 3 gives a bonus to the second written examination (TEN2). The right to count the bonuses from the tests is 12 months from the date of writing.


    F, 3, 4, 5

    Other information

    Supplementary courses: Transform Methods, Discrete Mathematics, Numerical Algorthms

    Subject area


    Disciplinary domain



    Department of Mathematics (MAI)

    Director of Studies or equivalent

    Jesper Thorén


    Magnus Berggren

    Education components

    Preliminary scheduled hours: 202 h
    Recommended self-study hours: 118 h

    Course literature

    Forsling, Göran och Neymark Mats, (2011) Matematisk analys, en variabel Liber


    Forsling, Göran och Neymark Mats, (2011) Matematisk analys, en variabel Liber
    TEN1 Written examination, part 1 U,3,4,5 6 credits
    TEN2 Written examination, part 2 U,3,4,5 6 credits
    KTR1 Written test 0 credits
    KTR2 Written test 0 credits
    KTR3 Written test 0 credits
    Passed written test 1 and written test 2 gives a bonus on the first written examination (TEN1). Approved written test 3 gives a bonus to the second written examination (TEN2). The right to count the bonuses from the tests is 12 months from the date of writing.

    Course syllabus

    A syllabus has been established for each course. The syllabus specifies the aim and contents of the course, and the prior knowledge that a student must have in order to be able to benefit from the course.


    Courses are timetabled after a decision has been made for this course concerning its assignment to a timetable module. A central timetable is not drawn up for courses with fewer than five participants. Most project courses do not have a central timetable.

    Interrupting a course

    The vice-chancellor’s decision concerning regulations for registration, deregistration and reporting results (Dnr LiU-2015-01241) states that interruptions in study are to be recorded in Ladok. Thus, all students who do not participate in a course for which they have registered must record the interruption, such that the registration on the course can be removed. Deregistration from a course is carried out using a web-based form: 

    Cancelled courses

    Courses with few participants (fewer than 10) may be cancelled or organised in a manner that differs from that stated in the course syllabus. The board of studies is to deliberate and decide whether a course is to be cancelled or changed from the course syllabus. 

    Regulations relating to examinations and examiners 

    Details are given in a decision in the university’s rule book:

    Forms of examination


    Written and oral examinations are held at least three times a year: once immediately after the end of the course, once in August, and once (usually) in one of the re-examination periods. Examinations held at other times are to follow a decision of the board of studies.

    Principles for examination scheduling for courses that follow the study periods:

    • courses given in VT1 are examined for the first time in March, with re-examination in June and August
    • courses given in VT2 are examined for the first time in May, with re-examination in August and October
    • courses given in HT1 are examined for the first time in October, with re-examination in January and August
    • courses given in HT2 are examined for the first time in January, with re-examination at Easter and in August.

    The examination schedule is based on the structure of timetable modules, but there may be deviations from this, mainly in the case of courses that are studied and examined for several programmes and in lower grades (i.e. 1 and 2). 

    • Examinations for courses that the board of studies has decided are to be held in alternate years are held only three times during the year in which the course is given.
    • Examinations for courses that are cancelled or rescheduled such that they are not given in one or several years are held three times during the year that immediately follows the course, with examination scheduling that corresponds to the scheduling that was in force before the course was cancelled or rescheduled.
    • If teaching is no longer given for a course, three examination occurrences are held during the immediately subsequent year, while examinations are at the same time held for any replacement course that is given, or alternatively in association with other re-examination opportunities. Furthermore, an examination is held on one further occasion during the next subsequent year, unless the board of studies determines otherwise.
    • If a course is given during several periods of the year (for programmes, or on different occasions for different programmes) the board or boards of studies determine together the scheduling and frequency of re-examination occasions.

    Registration for examination

    In order to take an examination, a student must register in advance at the Student Portal during the registration period, which opens 30 days before the date of the examination and closes 10 days before it. Candidates are informed of the location of the examination by email, four days in advance. Students who have not registered for an examination run the risk of being refused admittance to the examination, if space is not available.

    Symbols used in the examination registration system:

      ** denotes that the examination is being given for the penultimate time.

      * denotes that the examination is being given for the last time.

    Code of conduct for students during examinations

    Details are given in a decision in the university’s rule book:

    Retakes for higher grade

    Students at the Institute of Technology at LiU have the right to retake written examinations and computer-based examinations in an attempt to achieve a higher grade. This is valid for all examination components with code “TEN” and "DAT". The same right may not be exercised for other examination components, unless otherwise specified in the course syllabus.

    Retakes of other forms of examination

    Regulations concerning retakes of other forms of examination than written examinations and computer-based examinations are given in the LiU regulations for examinations and examiners,


    For examinations that involve the writing of reports, in cases in which it can be assumed that the student has had access to other sources (such as during project work, writing essays, etc.), the material submitted must be prepared in accordance with principles for acceptable practice when referring to sources (references or quotations for which the source is specified) when the text, images, ideas, data, etc. of other people are used. It is also to be made clear whether the author has reused his or her own text, images, ideas, data, etc. from previous examinations.

    A failure to specify such sources may be regarded as attempted deception during examination.

    Attempts to cheat

    In the event of a suspected attempt by a student to cheat during an examination, or when study performance is to be assessed as specified in Chapter 10 of the Higher Education Ordinance, the examiner is to report this to the disciplinary board of the university. Possible consequences for the student are suspension from study and a formal warning. More information is available at


    The grades that are preferably to be used are Fail (U), Pass (3), Pass not without distinction (4) and Pass with distinction (5). Courses under the auspices of the faculty board of the Faculty of Science and Engineering (Institute of Technology) are to be given special attention in this regard.

    1. Grades U, 3, 4, 5 are to be awarded for courses that have written examinations.
    2. Grades Fail (U) and Pass (G) may be awarded for courses with a large degree of practical components such as laboratory work, project work and group work.

    Examination components

    1. Grades U, 3, 4, 5 are to be awarded for written examinations (TEN).
    2. Grades Fail (U) and Pass (G) are to be used for undergraduate projects and other independent work.
    3. Examination components for which the grades Fail (U) and Pass (G) may be awarded are laboratory work (LAB), project work (PRA), preparatory written examination (KTR), oral examination (MUN), computer-based examination (DAT), home assignment (HEM), and assignment (UPG).
    4. Students receive grades either Fail (U) or Pass (G) for other examination components in which the examination criteria are satisfied principally through active attendance such as other examination (ANN), tutorial group (BAS) or examination item (MOM).

    The examination results for a student are reported at the relevant department.

    Regulations (apply to LiU in its entirety)

    The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.

    LiU’s rule book for education at first-cycle and second-cycle levels is available at 

    This tab contains public material from the course room in Lisam. The information published here is not legally binding, such material can be found under the other tabs on this page. There are no files available for this course.