Mathematics for the natural sciences - programme subject in programmes for Specialization in General Studies (MAT3-01)

Mathematics R1

Geometry

The aims of the training are to enable the apprentice to
  • use lines and circles as geometric loci together with congruence and the inscribed angle theorem in geometrical analysis and calculations
  • execute and analyze constructions defined by straight lines, triangles and circles in the plane, with and without the use of dynamic software
  • derive and apply the intersection theorems for the heights, angle bisectors, perpendicular bisectors and medians in a triangle
  • give an account of different proofs for Pythagoras’ equation, in terms of cultural history as well as mathematics
  • visualize vectors in the plane, both geometrically as arrows and analytically in co-ordinate form
  • calculate and analyze lengths and angles to determine the parallelity and orthogonality by combining arithmetical rules for vectors

Algebra

The aims of the training are to enable the apprentice to
  • factorize polynomials with the help of zeros and polynomial division, and use this to solve equations and inequalities with polynomial and rational expressions
  • transform and simplify complex rational functions and other symbolic expressions with and without the use of digital aids
  • derive the basic arithmetical rules for logarithms, and use these and the power rules to simplify expressions and solve equations and inequalities
  • give an account of implication and equivalence, and implement direct and contrapositive proof

Functions

The aims of the training are to enable the apprentice to
  • give an account of the concepts of boundedness, continuity and differentiability, and give examples of functions that are not continuous or differentiable
  • use formulae for the derivative of power, exponential and logarithmic functions, and differentiate composites, differences, products, quotients and combinations of these functions
  • use first derivative and second derivative to elaborate on and discuss the path of functions and interpret the derivatives in models of practical situations
  • draw graphs to functions with and without digital means, and interpret the basic characteristics of a function using the graph
  • find the equation for horizontal and vertical asymptotes to rational functions and draw the asymptotes
  • use vector functions for a parameter presentation of curves in the plane, draw the curve and differentiate the vector function to find velocity and acceleration

Combinatorics and probability

The aims of the training are to enable the apprentice to
  • give an account of the concepts of statistical independence and conditional probability, and derive and apply Bayes' equation for two events
  • elaborate on and discuss combinatoric problems linked to non-random selection with or without replacement and random selection without replacement, and use this to derive rules for calculating probability

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