# 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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