General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
Mathematics – grade 11
Introductory phase of upper secondary (here grade 11): curriculum from 2014 (trial phase), written for the eight-year Gymnasium. Some content (sine function, solids, exponential functions) is already covered in grade 10 of the nine-year Gymnasium. The numbers are those of the curriculum.
Unit 1General sine function
Periodicity, the sine function and operations on it: translating, stretching and compressing.
Shifting, stretching and reflecting graphs
Shift, stretch, reflect: every move of the graph leaves its mark in the function's equation.
Unit 2Solid geometry
Prism and cylinder, pyramid and cone, Cavalieri’s principle, hemisphere and sphere.
No separate simulation is planned for this.
Unit 3Exponential functions
Exponential growth, exponential functions of the form x ↦ a · bˣ, logarithms and the laws of logarithms.
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 4Continuity
Limits, local and global continuity, monotonicity, properties of continuous functions.
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.
Unit 5Introduction to differential calculus
Average and local rate of change, differentiability and the derivative; optionally Newton’s method.
From secant to tangent
Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.
The derivative function graphically
A tangent slides along the graph and draws its slope as a new graph: the derivative function.
Newton’s method
Approximate a zero with tangent lines. Each step gets you closer, but sometimes it goes wrong.
Unit 6Polynomial functions
Polynomial functions and their zeros, analysis with the first and second derivative, applications.
Function plotter with custom terms
Up to three of your own functions at once, with a table of values, zeros and intersections. The tool for anything that can be a graph.
Polynomial functions: zeros, multiplicity and end behaviour
Does the graph cross the x-axis or only touch it? The multiplicity of the zero decides, the highest power decides what happens far out.
Monotonicity, extrema and inflection points
Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.
Optimisation problems
Which box holds the most, which can uses the least metal? Five everyday problems show how to find the best solution with the derivative.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Lehrplan, Gymnasium, Mathematics, grade 11