cc4b7674-0d53-4bfe-850e-c6611bf7544a 8 4 5 decimal

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INFINITESIMAL ARTEFACT:
Some functions will superpose  exactly 
1/x, scale 2, 1/x² scale 4, 
1/(1+x²) scale 4 ... artefact
interesting sin(x), cos(x) scale 1
skewed  1/(1+x²) ... h ±0.5, scale at will. h 0.001 : scale 4 : 12 mat x f 11 x 2^+ / : x h Φ x h - f - x f + x f + x h - f - (x * scale h * - / : 31 mat

x f x h Φ 2 1 sys

Algo style Memo formula of incremental analysis of this demo example h 0.0001 : (increment ≡ x h f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x h f D1 x 2 h * - f x 2 h * + f - 4 h * / - : (1rst Derivative ≡ x h f D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / : (2nd Derivative ≡ x h f h_int h x h + f x f - (* 2 x f * + (x * : (Integral Curve [antiderivative] ≡ x h f h_cumul 2 h x h + f x f - (* 2 x f * + (x 2^* (* : (Cumulative Integration ≡ 61 line

plot compactum plot x f x h f h_int x h f D1 x h f D2 0.25 x h f D2 x diff (* XY 0.5 x h f h_cumul * 71 sys :

Axes system [0, 0] x y XY x y / - :

plot

Infinitesimal representation x h Infinitesimal x h + f x h - f + 2 / ≡

Infinitesimal representation x h ∞simal x h + f x h - f + 2 / : h 0.001 : x f 11 x 2^+ / : 12 mat x c 11 - x ≤ (x 1 ≤ (& cases : error (-h²/2)*2nd Deriv 41 line

x f x c * x h ∞simal 2 1 sys x f x h ∞simal -

x Δ x f x h ∞simal - :

x Δ ( x c * h 2 ^ 2 - / x f x 2 diff * 2 1 sys