Course syllabus for Single variable calculus and analytical geometry

The course syllabus contains changes
See changes

Course syllabus adopted 2024-02-08 by Head of Programme (or corresponding).

Overview

  • Swedish nameEnvariabelanalys och analytisk geometri
  • CodeMVE461
  • Credits6 Credits
  • OwnerTKKMT
  • 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 53134
  • Maximum participants270
  • Open for exchange studentsNo
  • Only students with the course round in the programme overview.

Credit distribution

0124 Examination 6 c
Grading: TH		6 c
  • 29 Okt 2024 pm J
  • 08 Jan 2025 am J
  • 22 Aug 2025 pm J

In programmes

Examiner

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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 above.

Specific entry requirements

The same as for the programme that owns the course.
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites

Prerequisites corresponding to specific entry requirements.

Aim

The purpose of the course is to provide a general knowledge in one variable analysis, analytical geometry and linear algebraic system of equations required in further studies as well as in the future professional career.

Learning outcomes (after completion of the course the student should be able to)

Generally

After this course the student should be able to independently handle differential calculus of elementary functions and analytical geometry in two and three dimensions. The student should be able to solve small linear systems of equations by hand.

This means that the student should know:
  • The concepts domain of definition and domain of values.
  • Composition of functions.
  • The meaning of continuity, and of the basic theorems on continuous functions, including the max/min theorem and the intermediate value theorem.
  • The formal definition of various limits.
  • Computational rules for limits and be able to prove some of
them.
  • Certain standard limits, e.g. limx→0sin(x)/x=1.
  • How to use l'Hospitals rule.
  • The properties and definitions of the inverse trigonometric functions.
  • How to handle compositions of inverse trigonometric functions and trigonometric functions.
  • The domains of definition and values of the natural logarithm and exponential function.
  • That x=eln x om x>0 and att x=ln(ex) for all x, i.e. that these function are each others inverses.
  • The formal definition and significance of the derivative.
  • Rules for differentiation, e.g. the chain, product and

quotients rules. You should also be able to prove some of them.
  • Derivatives of elementary functions and how to differentiate compositions of them. You should also be able to compute derivatives directly from the definition.
  • How to use the derivative to under stand if a function is increasing/decreasing on an interval.
  • The significance of the second order derivative in the connection with concave/convex functions.
  • How to study the sign of the derivative.
  • That the max and min is found in critical points or in end points.
  • Find the local maximum/minimum and absolute max/min of a function.
  • Sketch the graph of a function.
  • Compute linear approximation to a function, estimate the error and find an interval where functions value must belong.
  • Approximate a function with Taylor polynom of grade 2 or 3, estimate the corresponding error in Lagrange form.
  • Apply Taylors polynom and the symbol O(x) to computation of limits. Properties of the O(x) symbol.
  • Writing and interpreting a system of equations in matrix form.
  • Understand elementary row operations.
  • Understand the concept of pivot element, echelon form and reduced echelon form.
  • Understand the meaning of free variables.
  • Solve small linear systems of equations by hand using Gaussian elimination, and determine the number of solutions.
  • Understand the connection between vectors and triples of numbers. Understand the difference and connection between a point in space and the associated position vector. 
  • Understand the definition of addition of vectors. Compute and understand the geometrical significance of dot and cross products. 
  • Understand the correspondence between the geometrical properties of planes/lines and corresponding equation systems.
  • Use the orthogonal projection of a vector along another vector.

Content

  • Theory of elementary functions: trigonometric functions, inverse trigonometric functions, logarithms, exponential functions that serve as the main example for all constructions in one variable analysis
  • Calculation of limits of functions, investigation of functions continuity
  • Derivatives and their properties, application of them in calculations
  • Stationary points, local and absolute maximum and minimum and criteria for them, application to simple functions
  • Inverse functions, their derivatives
  • Taylor's polynomial for elementary functions; application of Taylor's expansion to calculation of limits
  • Scalar product, cross product, scalar triple product, their geometric sense and applications
  • Investigation of geometric objects: points, vectors, lines, planes in terms of their equations and vise versa: formulation and solving geometric problems in terms of equations
  • Approximate iterative methods (subdivision of intervals and Newton's method) for solving non-linear equations.
  • Systems of linear equations, the augmented matrix, method of elimination

Organisation

Instruction is given in lectures and classes. More detailed information will be given on the course web page before start of the course.

Literature

Literature will be announced on the course web page before start of the course.

Examination including compulsory elements

The exam moment is examined with a written exam at the end of the course and has the grading scale U,3,4,5.

During the course there may be tests that generate bonus credits on the exam. Examples of such tests include intermediate tests and hand-in assignments. Information pertaining to the actual course round is provided on the course homepage.

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.

The course syllabus contains changes

  • Changes to course rounds:
    • 2024-05-14: Examinator Examinator changed from Stefan Lemurell (sj) to Johannes Borgqvist (johborgq) by Viceprefekt/adm
      [Course round 1]