(Created 2009-08-11.)
 CALCULUS IN ONE VARIABLE FMA415

Higher education credits: 16,5. Grading scale: TH. Level: G1 (First level). Language of instruction: The course will be given in Swedish. FMA415 overlap following cours/es: FMA410, FMA645, FMAA01, FMAA05, FMA410, FMA645, FMAA01, FMAA05, FMA410, FMA645, FMAA01, FMAA05, FMA410, FMA645, FMAA01 och FMAA05. Compulsory for: BI1. Course coordinator: Director of Studies, Lars-Christer.Böiers, Lars_Christer.Boiers@math.lth.se, Matematik. Assessment: Written test in all subcourses, comprising theory and problem solving. The final grade is the integer part of the mean of the three grades of the subcourses (at most 5). Computational ability tests (see the introductory course below). Some oral and written assignments. Parts: 3. Home page: http://www.maths.lth.se/matematiklth/vitahyllan/vitahyllan.html.

Aim
The course aims at giving a basic treatment of one-dimensional calculus. Particular emphasis is on the role this 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 in problem solving and in assimilating mathematical text and communicating 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.

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.

Skills and abilities
For a passing grade the student must

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

in the context of problem solving be able to show capability independently to 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.

Contents
Part 1. The number concept. Calculation with fractions. Inequalities. Square roots. Curves and equations of second degree. Analytic geometry. The circle, ellipse, hyperbola. Geometry in space. Arithmetic and geometric sums. The binomial theorem. Modulus of a number. Trigonometry. Powers and logarithms. The concept of a function. Properties of elementary functions: curves, 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. Optimisation. Some simple mathematical models. Complex numbers and polynomials. 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. Geometrical 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. The Taylor and Maclaurin formulae. Expansion of elementary functions. The importance of the remainder term. Applications of Maclaurin expansions. Problem solving within the above areas.

Literature
Persson, A. & Böiers, L-C.: Analys i en variabel, chapters 0-9, appendices A-B. Studentlitteratur 2003. ISBN 91-44-02056-2.
Diehl, S: Inledande matematik för högskolestudier. Lund 2008.

Parts

Code: 0108. Name: Introductory Course.
Higher education credits: 6. Grading scale: UG. Assessment: Written test comprising theory and problem solving. Two 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: Differential Calculus.
Higher education credits: 4,5. 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: Integral Calculus.
Higher education credits: 6. Grading scale: UG. Assessment: Written test comprising theory and problem solving. Contents: The whole course, but with emphasis on part 3 above.