(Created 2011-09-01.)

CALCULUS IN ONE VARIABLE | FMAA01 |

**Aim**

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

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 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.

**Contents***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: 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. The Taylor and Maclaurin formulae. Expansions of the elementary functions. The importance of 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.

**Literature**

Diehl, S: Inledande matematik för högskolestudier. Studentlitteratur 2010, ISBN 9789144067612 (Kapitel P,T, A)

Månsson, J. och Nordbeck, P.: Endimensionell analys, Studentlitteratur 2011, ISBN 9789144056104.

and

Matematikcentrum: Övningar i Inledande matematik för högskolestudier. Studentlitteratur 2010 ISBN: 9789144067865

Matematikcentrum: Övningar i endimensionell analys, Studentlitteratur 2011, ISBN 9789144075020.

**Code: **0108.
**Name: **Part A1.

**Higher education 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.

**Higher education 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.

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