Course syllabus

# Endimensionell analys

Calculus in One Variable

## FMAA05, 15 credits, G1 (First Cycle)

## General Information

## Aim

## Learning outcomes

## Contents

## Examination details

## Admission

## Reading list

## Contact and other information

Calculus in One Variable

Valid for: 2016/17

Decided by: Education Board B

Date of Decision: 2016-03-29

Main field: Technology.

Compulsory for: B1, BI1, C1, D1, E1, F1, I1, K1, L1, N1, Pi1, V1, W1

Language of instruction: The course will be given in Swedish

The aim of the course is to give a basic introduction to calculus one variable. Particular emphasis is put on the role that the subject plays in applications in different areas of technology, in order to give the future engineer a good foundation for further studies in mathematics as well as in other subjects. The aim as also to develop the student's ability to solve problems, to assimilate mathematical text and to communicate mathematics.

Knowledge and understanding

For a passing grade the student must

within the framework of the course with confidence be able to handle elementary functions of one variable, including limits, derivatives and integrals of them .

be able to set up and solve some types of linear and separable differential equations that are important in the applications.

be familiar with the logical structure of mathematics, in the way it appears e.g. in plane geometry.

be able to give a general account of and illustrate the meaning of mathematical concepts in calculus in one variable that are used to construct and study mathematical models in the applications.

be able to account for the contents of definitions, theorems and proofs.

Competences and skills

For a passing grade the student must

be able to demonstrate a good algebraic computational ability and without difficulties be able to calculate with complex numbers.

in the context of problem solving be able to demonstrate an ability to independently choose and use mathematical concepts and methods in one-dimensional analysis, and to construct and analyse simple mathematical models.

in the context of problem solving be able to integrate knowledge from different parts of the course.

be able to show capability to explain mathematical reasoning in a structured and logically clear way.

*Part 1.* The number concept. Calculation with fractions.
Inequalities. Square roots. Curves and equations of second degree.
Plane geometry. Analytic geometry. The circle, ellipse, hyperbola.
Arithmetic and geometric sums. The binomial theorem. Modulus of a
number. Trigonometry. Powers and logarithms. The concept of a
function. The properties of the elementary functions: graphs,
formulas. Sequences of numbers. Limits with applications:
asymptotes, the number *e*, series. Continuous functions.
Derivatives: definition and properties, applications. Derivatives
of the elementary functions. Properties of differentiable
functions: the mean value theorem with applications. Curve
sketching. Local extrema. Optimization. Some simple mathematical
models. Problem solving within the above areas.

*Part 2.* Complex numbers and polynomials. The concept of
primitive function. Simple integration methods: partial integration
and change of variable. Partial fractions. Definition of an
integral. Riemann sums. Geometrical and other applications of
integrals. Improper integrals. Differential equations of first
order: linear and with separable variables. Linear differential
equations with constant coefficients. Solving homogeneous and
certain inhomogeneous equations. Applications. The Taylor and
Maclaurin formulae. Expansions of the elementary functions.
Understanding the remainder term. Applications of Maclaurin
expansions. Problem solving within the above areas.

Grading scale: TH

Assessment: Written test in both subcourses, comprising theory and problem solving. The final grade is the integer part of the mean of the two grades of the subcourses (at most 5). Computational ability tests (see subcourse B1 below). Some oral and written assignments. ONLY STUDENTS WHO PASSED THE TESTS OF COMPUTATIONAL ABILITY AND THE FIRST ORAL ASSIGNMENT MAY PARTICIPATE IN THE FIRST WRITTEN TEST. ONLY STUDENTS WHO PASSED THE SECOND ORAL ASSIGNMENT MAY PARTICIPATE IN THE SECOND WRITTEN.

Parts

Code: 0108. Name: Part B1.

Credits: 8. Grading scale: UG. Assessment: Written test comprising theory and problem solving. Computational ability tests must be passed before the examination. One assignment (oral and in writing) must be passed before the examination. Contents: See above, part 1.

Code: 0208. Name: Part B2.

Credits: 7. Grading scale: UG. Assessment: Written test comprising theory and problem solving. One assignment (oral and in writing) must be passed before the examination. Contents: The whole course, but with emphasis on part 2 above.

Code: 0308. Name: Computational Ability Test 1.

Credits: 0. Grading scale: UG.

Code: 0408. Name: Assignment 1.

Credits: 0. Grading scale: UG.

Code: 0508. Name: Assignment 2.

Credits: 0. Grading scale: UG.

Code: 0608. Name: Computational ability test 2.

Credits: 0. Grading scale: UG.

The number of participants is limited to: No

The course overlaps following course/s: FMA410, FMA415, FMA645, FMAA01

- Diehl, S: Inledande geometri för högskolestudier. Studentlitteratur, 2015. Chapters P,T, A.
- Övningar i Inledande geometri för högskolestudier. Studentlitteratur, 2015.
- Månsson, J. och Nordbeck, P.: Endimensionell analys. Studentlitteratur, 2011, ISBN: 9789144056104.
- Övningar i endimensionell analys. Studentlitteratur, 2011, ISBN: 9789144075020.

Course coordinator: Studierektor Anders Holst, Studierektor@math.lth.se

Course administrator: Studerandeexpeditionen, expedition@math.lth.se

Course homepage: http://www.maths.lth.se/utbildning/matematiklth/

Further information: The course Calculus in one variable is taught and examined in two versions, A (=FMAA01) and B (=FMAA05) respectively, depending on the student's program. The goals are the same. The present course description is version B. Before the written retake exams there it will be possible to retake the tests of computational ability and the oral assignments.