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Interactive 3D/Taylor Series
sin(x) ≈ x/1 + -x^3/6...
Function
Expansion Point a
a = 0.00
-3.13.1
Order n
n = 3
18
Metrics
f(a): 0.0000
T3(a): 0.0000
max |error|: 14.8744
Key Idea
Taylor series represent any smooth function as an infinite polynomial. Near the expansion point a, low orders work well. Far away, you need more terms. Neural nets approximate functions the same way - with learned basis functions instead of polynomials.
Try: sin(x) at a=0, order 1→8. Watch the orange dashed line match the blue over a wider range with each added term.

Taylor Series - Interactive Visualization

Taylor series approximate any smooth function as a polynomial. The n-th order Taylor expansion around point a matches the function's value and first n derivatives there. This visualization lets you see how well the approximation fits for different orders (1–8) and see the error ribbon shrink as n increases. Taylor series underlie Adam optimizer analysis, activation function approximations, and sensitivity analysis.

  • Choose from sin, exp, ln, cos, 1/(1+x²) functions
  • Drag expansion point a and watch the approximation re-center
  • Increase order from 1 to 8 and see error ribbon shrink
  • See the exact polynomial terms shown in the HUD
  • Connect to Adam optimizer derivation and function approximation

Part of the EngineersOfAI Interactive 3D - free interactive visualizations covering every major concept in machine learning and AI engineering. Hover any element for a plain-English explanation. No code required.