100 guided animations · 10 topic units · Free to explore
Make a prediction, change a value, and explain the result. Work through each unit in order or choose the topic you want to understand today. Every investigation has controls, a worked explanation, and a question to test your reasoning.
Your collection at a glance
Contraction estimates
Investigations 1–10
Smoothing a weak derivative
Investigations 11–20
Heat semigroup modes
Investigations 21–30
Variational energy
Investigations 31–40
Coercivity in a finite model
Investigations 41–50
Fourier incompressibility projection
Investigations 51–60
Critical norm scaling
Investigations 61–70
Rare events and convergence
Investigations 71–80
Conditional expectation by groups
Investigations 81–90
Beta–Bernoulli updating
Investigations 91–100
Related investigations use the same mathematical model with different questions and starting values. Reset restores the values used in that investigation’s worked answer. Read its assumptions before applying a model to a new situation.
100 animations · Showing 1–20
Graduate · Contraction estimates
Contraction estimates · Investigation 1 of 100
Contraction estimates: investigate fast positive contraction with contraction slope q = 0.2; constant b = 1.
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Contraction estimates · Investigation 2 of 100
Contraction estimates: investigate slow positive contraction with contraction slope q = 0.8; constant b = 1.
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Contraction estimates · Investigation 3 of 100
Contraction estimates: investigate fast alternating contraction with contraction slope q = -0.2; constant b = 1.
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Contraction estimates · Investigation 4 of 100
Contraction estimates: investigate slow alternating contraction with contraction slope q = -0.8; constant b = 1.
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Contraction estimates · Investigation 5 of 100
Contraction estimates: investigate a constant map with contraction slope q = 0; constant b = 2.
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Contraction estimates · Investigation 6 of 100
Contraction estimates: investigate a zero fixed point with contraction slope q = 0.5; constant b = 0.
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Contraction estimates · Investigation 7 of 100
Contraction estimates: investigate a negative fixed point with contraction slope q = 0.5; constant b = -1.
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Contraction estimates · Investigation 8 of 100
Contraction estimates: investigate a negative slope and offset with contraction slope q = -0.5; constant b = -2.
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Contraction estimates · Investigation 9 of 100
Contraction estimates: investigate an initial point already fixed with contraction slope q = 0.5; constant b = 1.5.
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Contraction estimates · Investigation 10 of 100
Contraction estimates: investigate a small negative fixed point with contraction slope q = -0.4; constant b = -0.5.
Explore animationGraduate · Smoothing a weak derivative
Smoothing a weak derivative · Investigation 11 of 100
Smoothing a weak derivative: investigate a broad rounded corner with smoothing width ε = 1; corner location b = 0.
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Smoothing a weak derivative · Investigation 12 of 100
Smoothing a weak derivative: investigate a narrow rounded corner with smoothing width ε = 0.1; corner location b = 0.
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Smoothing a weak derivative · Investigation 13 of 100
Smoothing a weak derivative: investigate a nearly sharp corner with smoothing width ε = 0.05; corner location b = 0.
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Smoothing a weak derivative · Investigation 14 of 100
Smoothing a weak derivative: investigate translate the corner right with smoothing width ε = 0.2; corner location b = 1.
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Smoothing a weak derivative · Investigation 15 of 100
Smoothing a weak derivative: investigate translate the corner left with smoothing width ε = 0.2; corner location b = -1.
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Smoothing a weak derivative · Investigation 16 of 100
Smoothing a weak derivative: investigate moderate smoothing at a shifted point with smoothing width ε = 0.5; corner location b = 0.5.
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Smoothing a weak derivative · Investigation 17 of 100
Smoothing a weak derivative: investigate negative half-unit corner with smoothing width ε = 0.5; corner location b = -0.5.
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Smoothing a weak derivative · Investigation 18 of 100
Smoothing a weak derivative: investigate small smoothing and positive shift with smoothing width ε = 0.15; corner location b = 0.75.
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Smoothing a weak derivative · Investigation 19 of 100
Smoothing a weak derivative: investigate small smoothing and negative shift with smoothing width ε = 0.15; corner location b = -0.75.
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Smoothing a weak derivative · Investigation 20 of 100
Smoothing a weak derivative: investigate compare a wide off-center transition with smoothing width ε = 0.8; corner location b = 0.25.
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