(Created 2010-07-25.)

CALCULUS IN ONE VARIABLE | FMA415 |

**Aim**

The course aims at giving a basic treatment of one-dimensional calculus. Particular emphasis is given to 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 to solve problems, to assimilate mathematical text and to communicate mathematics.

*Knowledge and understanding*

For a passing grade the student must

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

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

**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. Optimization. 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 geometri för högskolestudier. Matematikcentrum, Lund 2008.

**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.