Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
Mathematics – grade 12
This refers to the higher level of requirements (“erhöhtes Anforderungsniveau”).
Unit 1Investigating functions – antiderivatives, product and chain rule
1.1Polynomial functions (with parameters)
Investigating families of functions and sketching an antiderivative from a graph.
From a graph to its antiderivative
The height of one graph becomes the slope of another. If you understand bank balances, the antiderivative comes quicker than you think.
1.2The natural exponential function
The function that is its own derivative; product and chain rule, growth and decay.
The natural exponential function
Why is the number e so special? For the exponential function, the graph and its derivative are exactly the same.
1.3Sine and cosine functions
Making the derivatives of sine and cosine plausible graphically.
The derivative function graphically
A tangent slides along the graph and draws its slope as a new graph: the derivative function.
Unit 2Random variables and the binomial distribution
Random variables, expected value and standard deviation, urn models, Bernoulli trials and the binomial distribution.
Binomial distribution
The histogram of the binomial distribution: choose a range and read off the probability directly.
Galton board
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
Unit 3One-sided significance test (binomially distributed variables)
Procedure of a significance test, critical region, type I and type II errors and interpreting results correctly.
One-sided significance test: type I and II errors
A test decides between two hypotheses. Coloured areas show how often it can be wrong.
Unit 4Investigating functions – quotient rule, inverse functions
4.1Rational functions
Differentiating with the quotient rule and investigating simple rational functions.
Poles and asymptotes of rational functions
A zero or a pole, a hole or a leap to infinity? Drag the terms of the fraction and the graph shows the difference.
4.2Square root function
Invertibility via strict monotonicity, square and square root functions as inverses.
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.
4.3The natural logarithm
The natural logarithm as the inverse of the exponential function and its derivative.
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.
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.
Unit 5Foundations of coordinate geometry in space
3D coordinates, adding and scaling vectors, dot product (lengths and angles) and cross product (areas and volumes).
Points and solids in 3D coordinates
Place points in 3D coordinates, rotate the view and calculate distances with Pythagoras' theorem.
Adding and scaling vectors (3D)
Drag vector arrows in space, add them and multiply them by numbers. The calculation is shown alongside in column form.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): LehrplanPLUS, Gymnasium, Mathematics, grade 12