The course Applied Mathematics I will most prominently be aimed at levelling out the differences of basic knowledge on subjects that have already been taught in secondary school, with as a side note that in this course, the students will be pushed to acquire more depth and insight, and they are expected to make links between mathematics and other scientific courses, such as chemistry, physics and biology. A great deal of attention is aimed at the filling in of holes in the knowledge from secondary school, in such a way that everybody is able to acquire a same basic utility set when starting the study of applications and the extension to e.g. multiple variables.
The aim is to make the students familiar with techniques from calculus in one variable, which may be of use in a broader (applied) scientific context. As an introduction, the study of the number sets R and C will be given. In calculus, the focus will lie on the study of limits, derivatives and integrals.
In comparison to an average course of mathematics, more attention is being drawn to the recognition of problems and the application of a correct mathematical instrumentarium to solve these, or at least to formulate the problem in a mathematical sound way, so that a further solution (e.g. by computer) is made possible. In other words, the students must familiarize themselves with the formal mathematical language. Besides, the students also have to be able to produce numerical results and interpret these critically. To this end, e.g. numerical methods like the Newton-Raphson theorem will be used. The correct mathematical formulation of concepts will be considered important.
Participation in the lessons implies that the students are expected to participate actively. Among other things, this means that the students will in turn put solutions to exercises on the blackboard. With this, also the end competence is being incorporated that the students must learn to present their results in an organized way before their fellow-students. The students are also expected to present their results in a scientific word processing environment.
I. Complex numbers: basic operations, exponentiation, roots, polar forms, complex polynomial equations
II. Limits amnd continuity: functions, continuity, cyclometric functions, different kinds of limits in R and calculation rules, exponental and logarithmic functions, hyperbolic-goniometric functions, the O-symbol of Landau
III. Derivatives: calculation rules, higher derivatives, extremal analysis, middle value theorems, convexity, asymptotes, graph sketching, method of Newton-Raphson
IV. Primitives: calculation rules, partial integration, substitution, partial fractions, Fuss' rule, primitives of class II and III, recursion
V. Definite integrals: upper, lower and Riemann sums, indefinite integration, numerical integration methods, calculation of areas, surfaces of revolution, arc lengths and complanations