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Understanding Taylor's Remainder Theorem

Feb 25, 2025

Calculating Maximum Error Using Taylor's Remainder Theorem

Key Concepts

  • Taylor's Remainder Theorem: Used to calculate the maximum error in an approximation.
  • Formula: [ R_n(x) = \frac{f^{(n+1)}(z) \cdot (x-c)^{n+1}}{(n+1)!} ] where (z) is a number between (x) and (c).

Key Steps and Definitions

  1. Identify Variables:

    • (n) is the degree of the polynomial used in the approximation.
    • (x) is the value at which the approximation is evaluated.
    • (c) is the center of the Taylor expansion.
    • (z) is a number between (x) and (c) that maximizes the (n+1) derivative.

  2. Determine Values:

    • Example 1:
      • Approximate (\ln(1.1))
      • (n = 4), (x = 1.1), (c = 1), (x-c = 0.1)
      • (z = 1) since lower (x) gives higher (f^{(5)}(z)) when in the denominator.
    • Example 2:
      • Approximate (\sqrt{1.2})
      • (n = 2), (x = 1.2), (c = 1), (x-c = 0.2)
      • (z = 1) by similar reasoning.

Calculation Process

  • Finding Derivatives:

    • Original Function: (f(x))
    • Take derivatives up to (n+1).
    • Example 1: (f(x) = \ln(x)), derivatives up to (f^{(5)}(x)).
    • Example 2: (f(x) = x^{1/2}), derivatives up to (f^{(3)}(x)).

  • Evaluate Remainder:

    • Plug in values for derivatives, (x), (c), and factorial.
    • Example 1: [ R_4(1.1) = \frac{24 \cdot (0.1)^5}{5!} = 2 \times 10^{-6} ]
    • Example 2: [ R_2(1.2) = \frac{3/8 \cdot (0.2)^3}{3!} = 5 \times 10^{-4} ]

Comparing Errors

  • Exact Error:
    • Calculate true value and polynomial approximation.
    • Example 1: (\ln(1.1) = 0.09531), polynomial evaluates to (0.095308), error is (1.846 \times 10^{-6}).
    • Example 2: (\sqrt{1.2} = 1.09545), polynomial evaluates to (1.095), error is (4.45 \times 10^{-4}).
    • The exact error is less than the maximum error calculated.

Conclusion

  • Taylor's Remainder Theorem provides a method to calculate the maximum error in polynomial approximations.
  • Exact error calculated by comparing actual value against polynomial approximation and is typically less than the maximum error.

These notes summarize the process of calculating maximum and exact errors using Taylor's Remainder Theorem, providing examples and detailed steps for understanding and application.

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