Logarithms as the inverse of exponentiation
The logarithm answers the question: to what power do I raise the base to get this number? With graphs, rules and exponential equations.
Mathematics – grade 12
Studienstufe, second year (S3/S4; at the Gymnasium: grade 12): modules 4 and 5 and one of the elective modules 6 (analytic geometry) and 7 (linear algebra). eA = higher level only. The numbers are the module numbers of the curriculum.
Unit 4Rates of change and totals (exponential functions)
Growth models with exponential functions, Euler’s number, the ln function as the inverse, derivatives and integrals as changes in totals; eA: families of functions, finding zeros numerically.
Inverse functions: reflecting in y = x
Reflect the graph in y = x and the inverse function appears. It only works if no function value occurs twice.
The natural exponential function
Why is the number e so special? For the exponential function, the graph and its derivative are exactly the same.
Newton’s method
Approximate a zero with tangent lines. Each step gets you closer, but sometimes it goes wrong.
Monotonicity, extrema and inflection points
Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
The integral as total change: from velocity to distance
From the speed of a car you get the distance travelled: the area under the curve shows the total change.
Unit 5Applied problems in stochastics
Discrete random variables, the binomial distribution with expected value, standard deviation and sigma rules; eA: hypothesis tests and the normal distribution.
Galton board
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
Binomial distribution
The histogram of the binomial distribution: choose a range and read off the probability directly.
One-sided significance test: type I and II errors
A test decides between two hypotheses. Coloured areas show how often it can be wrong.
Normal distribution
How many bottles are underfilled? Under the bell curve every probability is an area, with the sigma rules as a rule of thumb.
Unit 6Analytic geometry (elective module)
Lines and planes with vectors, plane equations, trace points, relative positions, intersections and angles, normal vectors, reflection; eA: distances, Gaussian elimination.
Dot product and cross product
The dot product tells you the angle between two vectors; the cross product gives a perpendicular vector and an area.
Lines and planes in space
A point travels along a line through space, two vectors span a plane. Parametric, normal and coordinate forms all describe the same thing.
Relative positions and distances in space
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
Unit 7Linear algebra (elective module)
Systems of linear equations with Gaussian elimination, growth and redistribution processes with transition matrices, matrix powers, fixed vectors and long-term behaviour.
No separate simulation is planned for this.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Bildungsplan, Gymnasium, Mathematics, grade 12