Course syllabus

# Endimensionell analys

Calculus in One Variable

## FMAA01, 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: 2014/15

Decided by: Education Board B

Date of Decision: 2014-04-08

Main field: Technology.

Compulsory for: BME1, C1, D1, M1, MD1, N1

Language of instruction: The course will be given in Swedish

The aim of the course is to give a basic introduction to calculus in 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 demonstrate an ability 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 degree 2.
Geometry in the plane. Analytic geometry. The circle, ellipse,
hyperbola. Arithmetic and geometric sums. The binomial theorem. The
modulus of a number. Trigonometry. Powers and logarithms. The
concept of a function. The properties of the elementary functions:
graphs, formulas. Sequences of numbers.

*Part 2.* 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. Complex
numbers and polynomials. The Taylor and Maclaurin formulae.
Expansions of the elementary functions. Understanding the
remainder term. Applications of Maclaurin expansions. Problem
solving within the above areas.

*Part 3.* The concept of primitive function. Simple
integration methods: partial integration and change of variable.
Partial fractions. Definition of an integral. Riemann sums.
Geometric and other applications of integrals. Improper integrals.
Differential equations of first order: linear and with separable
variables. Linear differential equations. Solving homogeneous and
certain inhomogeneous equations. Applications. Problem solving
within the above areas.

Grading scale: TH

Assessment: Written test on each subcourse, comprising theory and problem solving. The final grade is the integer part of a weighted mean (weights 1,1,2) of the three grades on the subcourses (at most 5). Computational ability tests (see subcourse A1 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 A1.

Credits: 5. 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 A2.

Credits: 5. Grading scale: UG. Assessment: Written test comprising theory and problem solving. One assignment (oral and in writing) must be passed before the examination. Contents: See above, part 2.

Code: 0308. Name: Part A3.

Credits: 5. Grading scale: UG. Assessment: Written test comprising theory and problem solving. Contents: The whole course, but with emphasis on part 3 above.

Code: 0408. Name: Computational ability test.

Credits: 0. Grading scale: UG.

Code: 0508. Name: Assignment 1.

Credits: 0. Grading scale: UG.

Code: 0608. Name: Assignment 2.

Credits: 0. Grading scale: UG.

Code: 0708. 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, FMAA05

- Diehl, S: Inledande geometri för högskolestudier. Studentlitteratur, 2010, ISBN: 9789144067612. Chapters P,T, A.
- Övningar i Inledande geometri för högskolestudier. Studentlitteratur, 2010, ISBN: 9789144067865.
- 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 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 A.