Calculating Maximum Error Using Taylor's Remainder Theorem

Introduction

• Topic: Calculating maximum error of an approximation using Taylor's Remainder Theorem.
• Key formula: Remainder = (f^{(n+1)}(z) \cdot (x-c)^{n+1} / (n+1)!)

Key Terms and Concepts

• n: Last term in the polynomial (e.g., n = 4 in the approximation of ln(1.1)).
• x: Value being approximated (e.g., ln(1.1)).
• c: Center of the approximation (e.g., for ln(1.1), if x = 1.1, c = 1 because 1.1 - 1 = 0.1).
• z: A number between x and c such that the nth derivative is maximized.

Problem 1: Approximating ln(1.1)

1. Determine values:
   • n = 4
   • x = 1.1
   • c = 1
   • z = smallest value between x and c (here, z = 1).
2. Calculate derivatives:
   • Start with original function: (f(x) = \ln(x))
   • First derivative: (1/x)
   • Second derivative: (-1/x^2)
   • Third derivative: (2/x^3)
   • Fourth derivative: (-6/x^4)
   • Fifth derivative: (24/x^5)
3. Calculate maximum error:
   • Use z = 1 to get max value for fifth derivative: (24)
   • Calculate: (r_4(1.1) = 24 \cdot 0.1^5 / 5!)
   • Result: Maximum error = (2 \times 10^{-6})
4. Calculate exact error:
   • Exact error = ln(1.1) - polynomial evaluated at 1.1
   • Exact value: (1.846 \times 10^{-6})
   • Maximum error > Exact error.

Problem 2: Approximating (\sqrt{1.2})

1. Determine values:
   • n = 2
   • x = 1.2
   • c = 1
   • z = smallest value between x and c (here, z = 1).
2. Calculate derivatives:
   • Original function: (f(x) = \sqrt{x} = x^{1/2})
   • First derivative: (1/2 \cdot x^{-1/2})
   • Second derivative: (-1/4 \cdot x^{-3/2})
   • Third derivative: (3/8 \cdot x^{-5/2})
3. Calculate maximum error:
   • Use z = 1 for max third derivative value: (3/8)
   • Calculate: (r_2(1.2) = (3/8) \cdot 0.2^3 / 3!)
   • Result: Maximum error = (5 \times 10^{-4})
4. Calculate exact error:
   • Exact error = (\sqrt{1.2} -) polynomial evaluated at 1.2
   • Exact value: (4.45 \times 10^{-4})
   • Maximum error > Exact error.

Conclusion

• Taylor's Remainder Theorem helps determine the maximum error for approximations.
• Exact error is calculated by finding the difference between the actual value and the polynomial approximation.
• Maximum error is generally higher than the exact error, serving as a safe estimate.