Course syllabus adopted 2026-02-19 by Head of Programme (or corresponding).
Overview
- Swedish nameDifferentialkalkyl och skalära ekvationer
- CodeTMV225
- Credits7.5 Credits
- OwnerTKMSK
- Education cycleFirst-cycle
- Main field of studyMathematics
- DepartmentMATHEMATICAL SCIENCES
- GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Course round 1
- Teaching language Swedish
- Application code 44112
- Open for exchange studentsNo
- Only students with the course round in the programme overview.
Credit distribution
Module | Sp1 | Sp2 | Sp3 | Sp4 | Summer | Not Sp | Examination dates |
|---|---|---|---|---|---|---|---|
| 0108 Examination 7.5 c Grading: TH | 7.5 c |
In programmes
Examiner
- Andrii Dmytryshyn
- Associate Professor, Applied Mathematics and Statistics, Mathematical Sciences
Eligibility
General entry requirements for bachelor's level (first cycle)Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements
Specific entry requirements
The same as for the programme that owns the courseApplicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements
Course specific prerequisites
Aim
The aim of the course is to study the theory and applications of differential calculus in mathematical analysis, with a clear connection to analytical and numerical methods for solving scalar equations.The course develops the students ability to use both analytical and numerical methods, and to use programming in Python as a natural tool for modelling and computation. By combining theory, computation, and applications, the course strengthens both theoretical understanding and practical problem-solving skills, and provides a solid basis for further studies in mathematics and engineering.
The student is expected to have basic knowledge of programming in Python, or to acquire such knowledge during the course.
Learning outcomes (after completion of the course the student should be able to)
- Explain basic concepts in set theory and real analysis, such as real numbers, sequences of real numbers, convergence, and simple properties of subsets of R, and use these to analyse simple mathematical problems, as well as give an overview of how they relate to the computer representation of real numbers.
- Understand and use the concept of a function and analyse key properties of functions of one variable, in particular elementary functions, using algebraic and graphical methods.
- Formulate and apply the concepts of limit and continuity for functions of one real variable, and determine and interpret limits using analytical and numerical methods.
- Formulate and apply the concepts of derivative and differentiability, together with basic rules of differentiation, for functions of one real variable, and use derivatives to analyse local and global properties of functions and for linearisation.
- Use Taylor polynomials, power series, and series of real numbers to approximate functions of one real variable, determine limits, and analyse convergence and intervals of convergence.
- Formulate scalar equations as mathematical models for simple engineering problems and solve them using basic numerical methods, and perform a basic analysis of errors and convergence.
- Implement and use numerical algorithms in Python for computations within the scope of the course, and analyse, interpret, and communicate in writing modelling choices, computational results, and limitations in engineering applications.
Content
Real numbers: Set theory and logic; rational and real numbers; sequences and Cauchy sequences; supremum/infimum; open, closed, and bounded sets; computer representation of real numbers (IEEE 754, machine precision).
Functions: The concept of a function; injectivity/surjectivity/bijectivity; inverse and restriction; algebra of functions and composition; polynomials and rational functions; elementary functions (power, exponential, and logarithmic functions, trigonometric functions and their inverses).
Limits and continuity: Definition of limit and continuity; uniform and Lipschitz continuity; symbolic determination of limits (standard limits, rewriting and algebraic manipulation); numerical determination (Richardson extrapolation).
Derivative and linearisation: Definition of the derivative; derivatives of elementary functions; rules of differentiation (sum, product, quotient, chain rule, inverse functions); extrema; the Mean Value Theorem; linearisation and error estimates; numerical differentiation and choice of step size.
Taylor polynomials and series: Taylor and Maclaurin polynomials; determination of limits using big-O notation and LHôpitals rule; series and convergence tests; power series and radius of convergence; Taylor series.
Numerical solution of equations: Equations, roots, and fixed points; the bisection algorithm and the Intermediate Value Theorem; fixed-point iteration and the Banach fixed point theorem; Newtons method; order of convergence and conditions for convergence.
Applications and programming: Python as a computational tool; implementation, testing, and validation; for example, computation of elementary functions, simulation of simple mechanical systems, and chemical reactions.
Organisation
Teaching consists of lectures and exercise classes in smaller groups. More detailed information is provided on the course web page before the start of the course.Literature
S. Larsson, A. Logg, A. Målqvist, MATEMATISK ANALYS & LINJÄR ALGEBRA (I): Differentialkalkyl och skalära ekvationerExamination including compulsory elements
Assessment is by a written examination. Assignments or short quizzes that may give bonus points towards the examination may be included.The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers about disability study support.