∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
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Chapter 5 Theory

Taylor & Maclaurin Series Approximations

Polynomial clones matching higher-order derivatives at a target point

Increase polynomial degree N from 1 to 9 and watch the Taylor polynomial hug sin(x), e^x, or cos(x) across a widening radius of convergence.

1. Why Taylor Series Work

Taylor's theorem matches not just position f(a), but velocity f'(a), acceleration f''(a), and jerk f'''(a). Dividing by n! cancels out the factorial factors produced by repeated differentiation of x^n.

sin⁡(x)=x−x33!+x55!−x77!+…\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots

Quick Reference & Cheat Sheet

Standard Maclaurin Expansions
ex=∑xnn!,cos⁡x=∑(−1)nx2n(2n)!,11−x=∑xne^x = \sum \frac{x^n}{n!}, \quad \cos x = \sum (-1)^n\frac{x^{2n}}{(2n)!}, \quad \frac{1}{1-x} = \sum x^n
Analytic basis functions for numerical scientific computing.