Course syllabus for Introductory course in mathematics

• know the elementary functions, what computational rules applies to them, how they relate, how their graphs can be constructed and be able to use this in solving problems of a fairly complex nature and basic modelling
• understand, be able to define, determine and use various basic properties of real valued functions of one real variable, such as domain/range, increasing/decreasing, even/odd, asymptotes and invertibility.
• understand and be able to define various types of limits of real valued functions of one real variable, know the basic theorems that applies to them and with judgement be able to use these when solving problems.
• understand and be able to define the concept of a continuous real valued function of one real variable, know the basic theorems that applies to them and use these when solving problems. 
• understand and be able to define the concepts of differentiable real valued function of one real variable and the derivative of such a function, know and be able to prove basic theorems that applies to them and with judgement use these when solving problems.
• be able to solve systems of linear equations with several rows and variables using row operations to echelon form and be able to determine the number of solutions to such systems.
• understand and be able to use the complex numbers and the complex exponential function when solving problems.
• be able to use basic concept with respect to linear analytical geometry in three dimensions to determine equations for planes and lines, the distance between such object and area of parallelograms and volume of prallelepipeds.

Differential calculus of functions formed from the elementary functions. General theory of functions. Solution of linear systems of equations using Gauss elimination, and determination of the number of solutions. Vectors and equations for lines and planes in space and calculations using these.

The concepts definition, theorem and proof, and the meaning and use of logical symbols in problem solving and construction of arguments and proofs.

Solution of linear systems of equations using Gauss elimination (row reduction). Determine whether a system has a unique solution, infinitely many solutions or no solutions.

Vector algebra; addition of vectors and multiplication of vector by numbers (scalars) and graphical interpretation of these operations. Length of a vector and normalized vectors, as well as dot and cross products and the geometrical interpretation of these constructions. 

Description of planes and lines in space using equations and normal vectors to planes and direction vectors for lines. Application of dot and cross products to calculations of lengths and distances, e.g. computing the distance between a point and a plane.

The meaning of limits of the form limx±∞f(x), limx±af(x) etc., and the calculation of such limits. The concepts of left and right limits and the definition of a function f(x) being continuous in a point.

Domain and range of a function. Combinations and compositions of functions. Continuity and inverse of a function, and relation to its graph.

The fundamental trigonometric functions sin(x), cos(x) and tan(x), their derivatives and graphs. Values for e.g. sin(a) for certain special angles a, and solution of simple trigonometric equations. Use of trigonometric functions to solve triangles, and trigonometric identities.

Calculations using inverse trigonometric functions and their values for the cases where the answers are known angles. Their definition, domains and ranges, and derivatives and graphs.

Definitions and properties for functions ln(x) and e^x, their graphs and calculations of their derivatives. Limits of functions involving logarithms and exponential functions.

The connection between derivative and directional coefficient of the tangent line and calculations of tangents and normal. Calculations of derivatives (including derivatives of inverse functions) and application of derivatives to determine where a function is increasing/decreasing and convex up/down. Implicit differentiation. The mean value theorem and its application.

Definition of the concepts increasing/decreasing (non-increasing/non-decreasing) and concave up/down functions and their implications. Differentiability implies continuity (but not the other way around) and finding the tangent line to a specified function in a specified point. The max/min theorem and the intermediate value theorems, and their application.

Optimisation of continuous functions on open/closed intervals. Critical points and local extreme points of functions and finding global and local maxima and minima.

Formulation of mathematical models for simple geometrical and physical problems. Solution of the resulting mathematical problems.

Construction of sketch of graph of a function through determining its properties. Vertical, horizontal and oblique asymptotes and finding them.

Non-compulsory  assignments may render bonus points to the exam. Information about this will occur on the course web page at the start of each course occation.