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Math / Calculus / Undergraduate / mcd01

Taylor Approximation with a Remainder Bound

Connect local polynomial approximations with a defensible error estimate.

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5 questions0m 0s

Prerequisites

Derivatives through third order, factorials, and absolute-value inequalities.

Learn the skill

A quadratic Taylor polynomial at a is f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2. If |f'''| <= M throughout the segment from a to x, the absolute remainder is at most M|x-a|^3/6. The bound is not the actual error.

Worked example

For f(x)=x^3 at a=1, P2(1+h)=1+3h+3h^2. The exact remainder is h^3, which also meets the third-derivative bound with M=6.

Model and Assumptions

Approximate f(x)=ln(1+x) about a=0 for 0 <= x <= 0.2. Its derivatives at zero are f(0)=0, f'(0)=1, f''(0)=-1. On this interval f'''(x)=2/(1+x)^3 is positive and at most 2.

1. Which quadratic polynomial follows from these derivatives?
2. What is the polynomial estimate at x=0.2?
3. Which absolute error bound follows at x=0.2?
4. How does the estimate compare with the exact value at positive x here?
5. By what factor does this bound shrink when x is halved, retaining M=2?

Further inquiry

Derive P2 and its remainder bound. Repeat the bound on [-0.2,0], choosing a valid derivative maximum there, and explain both the changed bound and the sign of the remainder.

Review criteria

  • Use a bound valid on the whole segment, not just at zero.
  • Obtain M=125/32 and an endpoint bound of 1/192 on the negative interval.
  • Distinguish a signed remainder from its absolute bound.