Mathematics – grade 11

Unit 1Special properties of functions

Behaviour as x → ±∞, symmetry, shifting, stretching and reflecting graphs, continuity.

Shifting, stretching and reflecting graphs

Shift, stretch, reflect: every move of the graph leaves its mark in the function's equation.

Grade 10–11Baden-Württemberg: grade 9/10Bavaria: grade 11Berlin: grades 10, 11Brandenburg: grades 10, 11Bremen: grade 10Hamburg: grades 9/10, 11Hesse: grade 11Lower Saxony: grades 9/10, 11North Rhine-Westphalia: grade 11Rhineland-Palatinate: grade 9/10Saarland: grade 11Saxony: grade 10Saxony-Anhalt: grade 10Schleswig-Holstein: grades 10, 11

Limits and asymptotes

What happens to a graph when x gets very large? A point runs outwards and a magnifier shows how close it gets to the asymptote.

Grade 10–12Baden-Württemberg: grades 9/10, 12/13Bavaria: grade 11Berlin: grade 11Brandenburg: grade 11Bremen: grade 10Hamburg: grade 11Hesse: grades 11, 12Lower Saxony: grades 9/10, 11, 12/13North Rhine-Westphalia: grades 11, 12/13Rhineland-Palatinate: grade 11–13Saarland: grade 11Saxony: grades 10, 11/12Saxony-Anhalt: grades 10, 11/12Schleswig-Holstein: grades 10, 11

General sine function

Change the amplitude, period and shift of a sine curve with sliders or by dragging.

Grade 10–11Baden-Württemberg: grade 9/10Bavaria: grades 10, 11Berlin: grade 10Brandenburg: grade 10Bremen: grade 10Hamburg: grades 9/10, 11Hesse: grades 9/10, 11Lower Saxony: grades 9/10, 11North Rhine-Westphalia: grades 9/10, 11Rhineland-Palatinate: grade 9/10Saarland: grades 10, 11Saxony: grade 10Saxony-Anhalt: grade 10Schleswig-Holstein: grade 10

Unit 2Rational functions – limits and asymptotes

Domain, zeros, poles, horizontal, oblique and vertical asymptotes.

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.

Grade 10–13Baden-Württemberg: grade 12/13Bavaria: grades 11, 12Berlin: grade 11Brandenburg: grade 11Bremen: grade 10Hamburg: grade 11Hesse: grade 12Rhineland-Palatinate: grade 11–13Saarland: grade 13Saxony: grade 11/12Saxony-Anhalt: grade 11/12

Unit 3Conditional probability and independence

Conditional probabilities with tree diagrams and two-way tables (e.g. medical tests), independence, questioning data critically.

Conditional probability with medical tests

A medical test is positive, but how likely is it that you are really ill? 1000 people show why the answer is often surprising.

Grade 9–13Baden-Württemberg: grade 9/10Bavaria: grade 11Berlin: grades 9, 10, 11Brandenburg: grades 9, 10, 11Bremen: grade 11/12Hamburg: grades 9/10, 11Hesse: grade 13Lower Saxony: grades 9/10, 12/13North Rhine-Westphalia: grades 9/10, 12/13Rhineland-Palatinate: grade 11–13Saarland: grades 10, 12Saxony: grade 11/12Saxony-Anhalt: grade 11/12Schleswig-Holstein: grade 11

Two-way tables and Venn diagrams

A two-way table, a Venn diagram and a tree diagram show the same data and are linked. Change one and the others follow.

Grade 8–13Baden-Württemberg: grade 9/10Bavaria: grades 9, 11Berlin: grades 9, 11Brandenburg: grades 9, 11Bremen: grade 11/12Hamburg: grades 9/10, 11Hesse: grade 13Lower Saxony: grades 9/10, 12/13North Rhine-Westphalia: grades 9/10, 12/13Rhineland-Palatinate: grades 9/10, 11–13Saarland: grades 10, 12Saxony: grade 11/12Saxony-Anhalt: grades 7/8, 11/12Schleswig-Holstein: grade 11

Unit 4Foundations of differential calculus

4.1Local and global differentiation

Slopes of secants and tangents, average and instantaneous rate of change, the derivative function, equations of tangents.

From secant to tangent

Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.

Grade 10–11Baden-Württemberg: grade 11Bavaria: grade 11Berlin: grades 10, 11Brandenburg: grades 10, 11Bremen: grade 10Hamburg: grades 9/10, 11Hesse: grade 11Lower Saxony: grade 11North Rhine-Westphalia: grade 11Rhineland-Palatinate: grade 11–13Saarland: grade 11Saxony: grade 11/12Saxony-Anhalt: grade 11/12Schleswig-Holstein: grade 11

The derivative function graphically

A tangent slides along the graph and draws its slope as a new graph: the derivative function.

Grade 10–12Baden-Württemberg: grade 11Bavaria: grades 11, 12Berlin: grades 10, 11Brandenburg: grades 10, 11Bremen: grade 10Hamburg: grades 9/10, 11Hesse: grade 11Lower Saxony: grade 11North Rhine-Westphalia: grade 11Rhineland-Palatinate: grade 11–13Saarland: grade 11Saxony: grade 11/12Saxony-Anhalt: grade 11/12Schleswig-Holstein: grades 11, 12

4.2Investigating polynomial functions with calculus

Monotonicity, extrema and inflection points, necessary and sufficient conditions, Newton’s method.

Monotonicity, extrema and inflection points

Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.

Grade 10–12Baden-Württemberg: grade 11Bavaria: grade 11Berlin: grade 11Brandenburg: grade 11Bremen: grades 10, 11/12Hamburg: grades 11, 12Hesse: grades 11, 12Lower Saxony: grade 11North Rhine-Westphalia: grades 11, 12/13Rhineland-Palatinate: grade 11–13Saarland: grades 11, 12Saxony: grade 11/12Saxony-Anhalt: grade 11/12Schleswig-Holstein: grades 11, 12

Newton’s method

Approximate a zero with tangent lines. Each step gets you closer, but sometimes it goes wrong.

Grade 10–12Baden-Württemberg: grade 12/13Bavaria: grade 11Bremen: grade 10Hamburg: grade 12Hesse: grade 11Rhineland-Palatinate: grade 11–13Saarland: grade 11Saxony: grade 11/12Saxony-Anhalt: grade 11/12Schleswig-Holstein: grade 11