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Taylor Series Approximation

Pubblico: UniversityCategoria: Math

Descrizione

Demonstrates how the Taylor series builds increasingly accurate polynomial approximations of cos(x) around x=0 by successively adding terms. Each new term is animated in a new color, revealing how the approximation extends further from the center.

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Taylor Series Approximation

Description

Demonstrates how the Taylor series builds increasingly accurate polynomial approximations of cos(x) around x=0 by successively adding terms. Each new term is animated in a new color, revealing how the approximation extends further from the center.

Phases

# Phase Name Duration Description
1 Setup 5s Title, axes, plot cos(x) in white
2 T₀ = 1 6s Plot constant term, explain 0th-order approximation
3 T₂ = 1 - x²/2 7s Add quadratic term, new color, better fit near origin
4 T₄ = 1 - x²/2 + x⁴/24 7s Add quartic term, fits further out
5 T₆ = add -x⁶/720 7s Add sixth-degree term, excellent fit over wider range
6 T₈ 6s Add 8th-degree term, near-perfect over [-π, π]
7 Formula display 8s Show general Taylor series formula, error term O(x^n)
8 Summary 4s All approximations overlaid, convergence message

Layout

+--------------------------------------------------+
|  Title: "Taylor Series Approximation"            |
+--------------------------------------------------+
|                                                  |
|   Axes with cos(x)           | Formula panel    |
|   (white curve)              | (right side)     |
|                              |                  |
|   T₀ (red)                   | cos(x) = Σ       |
|   T₂ (orange)                | (-1)^n x^(2n)    |
|   T₄ (yellow)                | ─────────────    |
|   T₆ (green)                 |    (2n)!         |
|   T₈ (teal)                  |                  |
|                              | Current term     |
|                              | highlighted      |
+--------------------------------------------------+

Area Descriptions

  • Left 60%: Axes from -2π to 2π showing cos(x) and successive Taylor polynomials
  • Right 40%: Taylor series formula with current partial sum highlighted
  • Bottom: Degree label and approximation quality description

Assets & Dependencies

  • Fonts: LaTeX
  • Manim version: ManimCE 0.19.1

Notes

  • cos(x) Taylor series centered at 0: Σ (-1)^n x^(2n)/(2n)!
  • Terms: 1, -x²/2!, x⁴/4!, -x⁶/6!, x⁸/8!
  • Colors cycle: RED → ORANGE → YELLOW → GREEN → TEAL
  • Use axes x_range = [-2π, 2π], y_range = [-1.5, 1.5]
  • Clip polynomial plots to avoid large values far from origin