From secant to tangent
Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.
Calculus
Derivatives and integrals made visual: from secant to tangent, from rectangle sums to area and from a graph to its antiderivative.
Link to the German educational standards (KMK)
- L3Structures and functional relationships
- L2Quantities and measurement
- K1Reasoning mathematically
- K4Using mathematical representations
Differential calculus
Grade 10–13
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.
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.
Families of functions and antiderivatives
Grade 11–13
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
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.
Integral calculus
Grade 11–13 · University
The integral as signed area (rectangle sums)
Fill the area under a graph with rectangles. The more rectangles, the more precisely you get the value of the integral.
Integral function and fundamental theorem
The area under a graph grows as you move along. Its rate of change is exactly the function itself: this is the fundamental theorem.
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.
Solids of revolution
The area under a graph rotates about the x-axis and becomes a solid. Its volume is calculated with thin discs.
Further topics (university)
Grade 12–13 · University
Taylor polynomials
Term by term a polynomial hugs a function such as sin x ever more closely. That is pretty much how a calculator does it.
Fourier series
Square, sawtooth and other waves are built from many sine waves. You can also listen to them.