Since this booklet cannot show true animations, one or several selected graphics are shown.

Additional types of visualizations can be found in the documents that constitute the collection and in the description of the commands.

Notices to the programs are in the Manual notebooks Analysis and TrigFunctions.

The alternating harmonic series converges toward log 2.

Examples can be found in the Collection notebooks Sequences and Series.

Notices to the programs are in the Manual notebooks Sequences and Series.

Examples can be found in the Collection notebooks Derivatives and Differential.

Notices to the programs are in the Manual notebook Differentiation.

Analysis of the rational function f(x) = x(x-2)(x+2)/(x2+1):

2

x (-4 + x )

Function: -----------

2

1 + x

Symmetry with respect to the origin

Zeros: {-2, 0, 2}

Minima: {( 0.728786,-1.65111 )}

Maxima: {( -0.728786,1.65111 )}

Points of Inflection: {( -1.73205,0.433013 ), ( 0,0 ),

( 1.73205,-0.433013 )}

Examples can be found in the Collection notebook Analysis.

Notices to the programs are in the Manual notebook Analysis.

Examples can be found in the Collection notebook Integration.

Notices to the programs are in the Manual notebook Integration.

Notices to the programs are in the Manual notebook ODEs.

This is the direction field for the equation y' = -x/y.

These are the solutions of y y' + x = 0.

2 2

Solutions: {-Sqrt[-x + C[1]], Sqrt[-x + C[1]]}

((1 - Sqrt[5]) x)/2 ((1 + Sqrt[5]) x)/2

Solutions: {E C[1] + E C[2]}

This is a graphic with solutions of cos(3/2 t) = x'' + x in phase space.

3 t

4 Cos[---]

2

Solutions: {C[2] Cos[t] - ---------- - C[1] Sin[t]}

5

Parabolas, ellipses, and hyperbolas as the locus of points that satisfy certain distance constraints can be visualized by an animation.

Reflection properties of these conic sections are also illustrated.

Examples can be found in the Collection notebooks ConicSections, Reflections, and Classification.

Notices to the programs are in the Manual notebook ConicSections.

This is the identity function f(z) = z.

The function f(z) = 1/z^2 has a double pole at z = 0.

Examples can be found in the Collection notebook ComplexFunctions.

Notices to the programs are in the Manual notebook ComplexFunctions.

If a two-dimensional linear map has two real-valued eigenvectors, any point can be written in terms of these and each component can be mapped separately. In an animation, this process can be performed for each point of the unit circle.

M = 1.83333 0.166667

0.333333 1.66667

Eigenvalues: {2., 1.5}

Eigenvectors: {{0.707107, 0.707107}, {-0.447214, 0.894427}}

Examples can be found in the Collection notebook LinearMaps.

Notices to the programs are in the Manual notebook LinearMaps.

Examples can be found in the Collection notebook Conformal.

Notices to the programs are in the Manual notebook ComplexMap.

The curve traced by a point with a distance of 0.8 from the center of a circle (radius 1) which rolls in a circle with radius 7/2 is a curtate hypocycloid.

Examples can be found in the Collection notebooks RollingCircles, Cycloids, Hypocycloids, and Epicycloids.

Notices to the programs are in the Manual notebook RollingCircles.

The creation process of such surfaces can be shown by an animation.

Examples can be found in the Collection notebook Revolution.

Notices to the programs are in the Manual notebook Revolution.

Examples can be found in the Collection notebooks Polyhedra and PolyPictures.

Notices to the programs are in the Manual notebook Polyhedra.

Examples can be found in the Collection notebook Icosahedra.

Notices to the programs are in the Manual notebook Icosahedra.

Examples can be found in the Collection notebook MinimalSurfaces.

Notices to the programs are in the Manual notebook MinimalSurfaces.

Examples can be found in the Collection notebook Chaos.

Notices to the programs are in the Manual notebook Chaos.

Up to Illustrated Mathematics