(Created 2010-07-25.)

CALCULUS | FMA645 |

**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.

*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 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.

*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 connection with 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 demonstrate an elementary ability to explain the solution to mathematical problems within the course in a well structured way and with clear logic.

**Contents***Algebra:*

Algebraic computations. Functions, equations, inequalities and modulus of a number. Complex numbers. Powers and logarithms. Trigonometry. Curve sketching: elementary functions and second degree curves.

*Calculus 1:*

Complex numbers and polynomials. The concept of a function. Properties of the elementary functions: curves, formulas, limits. Limits with applications: asymptotes, the number e, series. Continuous functions. Derivatives: definition and properties, applications. Differentiation of the elementary functions. Properties of differentiable functions: the mean value theorem with applications. Curve sketching. Local extreme values. Optimisation.

*Calculus 2:*

Primitive functions. Partial integration and change of variable. Partial fractions. Definition of an integral. Methods of integration. Riemann sums. Geometrical and other applications of integrals. Improper integrals. Differential equations of first order: linear and separable. Applications. Linear differential equations of second order: solution of homogeneous and certain inhomogeneous equations, with applications. The Taylor and Maclaurin formulae. Expansion of the elementary functions, with applications.

**Literature**

Dunkels, A m.fl.: Mot bättre vetande i Matematik. Studentlitteratur. ISBN: 9789144322520.

Persson, A & Böiers, L-C: Analys i en variabel. Studentlitteratur 2001. ISBN: 9789144020563.

Övningar till Analys i en variabel, Matematikcentrum, KFS AB Lund.

**Code: **0107.
**Name: **Algebra.

**Higher education credits: ** 3.
**Grading scale: **UG.
**Assessment:** Written test, with grades Passed and Not passed. Some assignments.
**Contents:** See above, Algebra.

**Code: **0207.
**Name: **Calculus 1.

**Higher education credits: ** 6.
**Grading scale: **TH.
**Assessment:** Written test.
**Contents:** See above, Calculus 1.

**Code: **0307.
**Name: **Calculus 2.

**Higher education credits: ** 4,5.
**Grading scale: **TH.
**Assessment:** Written test.
**Contents:** See above, Calculus 2.