Polynomial Approximation: A Comprehensive Guide

Polynomial approximation uses polynomials to represent complex functions, simplifying computations in numerical analysis. Polynomials are easy to evaluate, differentiate, and integrate, making them ideal for approximating functions like \( e^x \), \( \sin(x) \), or non-elementary ones.

This MathMultiverse guide explores Taylor series, a key method, with examples, visualizations, and applications in computational mathematics.

Taylor Series

Approximates \( f(x) \) around \( a \):

\[ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n \]

Maclaurin series (\( a = 0 \)). Remainder:

\[ R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} (x - a)^{n+1} \]

Examples

\( e^x \) at \( x = 0 \)

Derivatives: \( f^{(n)}(x) = e^x \), \( f^{(n)}(0) = 1 \).

\[ e^x = \sum_{n=0}^\infty \frac{x^n}{n!} \]

For \( x = 1 \), \( e \approx 2.7183 \):

  • \( n=1 \): \( 1 + 1 = 2 \), error \( \approx 0.7183 \)
  • \( n=4 \): \( 1 + 1 + 0.5 + 0.1667 + 0.0417 \approx 2.7083 \), error \( \approx 0.01 \)

\( \sin(x) \) at \( x = 0 \)

Derivatives cycle: \( 0, 1, 0, -1 \).

\[ \sin(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} \]

For \( x = \frac{\pi}{6} \), \( \sin\left(\frac{\pi}{6}\right) = 0.5 \):

  • \( n=1 \): \( 0.5236 \), error \( \approx 0.0236 \)
  • \( n=3 \): \( 0.5236 - 0.0239 \approx 0.4997 \), error \( \approx 0.0003 \)

\( \ln(1 + x) \) at \( x = 0 \)

Derivatives: \( f^{(n)}(x) = \frac{(-1)^{n-1} (n-1)!}{(1 + x)^n} \).

\[ \ln(1 + x) = \sum_{n=1}^\infty \frac{(-1)^{n-1} x^n}{n} \]

For \( x = 0.5 \), \( \ln(1.5) \approx 0.4055 \):

  • \( n=4 \): \( 0.5 - 0.125 + 0.0833 - 0.0625 \approx 0.3958 \), error \( \approx 0.0097 \)

Taylor Approximation of \( e^x \)

Approximating \( e^x \) with Taylor series terms at \( x = 0 \).

Applications

Physics: Motion

Small oscillations, \( \sin(\theta) \approx \theta \):

\[ \frac{d^2\theta}{dt^2} + \frac{g}{L} \theta = 0 \]

Signal Processing

Linearize \( e^{j\omega t} \):

\[ e^{j\omega t} \approx 1 + j\omega t \]

Numerical Solvers

Newton’s method:

\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]

Computer Graphics

Bezier curves:

\[ B(t) = \sum_{i=0}^n \binom{n}{i} (1 - t)^{n-i} t^i P_i \]