Course syllabus for Single variable calculus and analytical geometry

Course syllabus adopted 2021-02-26 by Head of Programme (or corresponding).

Overview

  • Swedish nameEnvariabelanalys och analytisk geometri
  • CodeMVE460
  • Credits7.5 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 53117
  • Maximum participants250
  • Open for exchange studentsNo
  • Only students with the course round in the programme overview.

Credit distribution

0115 Laboratory 1.5 c
Grading: UG		1.5 c0 c0 c0 c0 c0 c
0215 Examination 6 c
Grading: TH		6 c0 c0 c0 c0 c0 c
  • 26 Okt 2021 am J
  • 04 Jan 2022 am J
  • 19 Aug 2022 pm J

In programmes

Examiner

  • Alice Kozakevicius
Go to coursepage (Opens in new tab)

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 basic eligibility.

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 and in general using MATLAB and also handle the interface, plot graphs of functions, solve equations using subdivision of intervals and Newtons method.

This means that the student should know:
  • The concepts domain of definition and domain of values.
  • Composition of functions.
  • The meaning of contunuity.
  • The meaning of the basic theorems on continuous functions: e.g. 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 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 form the definition.
  • The meaning of continuity.
  • The max/min theorem.
  • The intermediate value theorem.
  • How to use the derivative to under stand if a function is increasing/decreasing on an interval.
  • The definition and significance of the derivative.
  • 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.
  • Properties of O(x) symbol.
  • Apply Taylors polynom and the symbol O(x) to computation of limits.
  • 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.
  • 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.
  • Write simple scripts and functions in Matlab: use loops, logical expressions, subroutines, function handles, graphics and text output.
  • Use numerical methods with iterations and solve equations using subdivision of intervals and Newtons method with your own script.
  • Solve linear systems of equations and matrix operations in Matlab.

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
  • Writing simple programs in Matlab, using loops, logic formulas, subprograms, graphics, formatted text output
  • Using approximate iterative methods and the Newton method to solve non-linear equations, and implementation of corresponding algorithms in Matlab
  • Systems of linear equations, the augmented matrix, method of elimination
  • Solving linear systems of equations with Matlab

Organisation

Instruction is given in lectures and classes together with computer sessions using Matlab. More detailed information will be given on the course web page before start of the course. http://www.chalmers.se/math/SV/utbildning/grundutbildning-chalmers/arkitekt-och/kemiteknik http://www.chalmers.se/math/SV/utbildning/grundutbildning-chalmers/arkitekt-och/kemiteknik-med-fysik http://www.chalmers.se/math/SV/utbildning/grundutbildning-chalmers/arkitekt-och/bioteknik

Literature

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

Examination including compulsory elements

More detailed information of the examination will be given on the course web page before start of the course. Examples of assessments are:
 -selected exercises are to be presented to the teacher orally or in writing during the course
 -other documentation of how the student's knowledge develops -project work, individually or in group
 -written or oral exam during and/or at the end of the course
 -problems/exercises are to be solved with a computer and presented in writing and/or at the computer.

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 on educational support due to disability.