Taylor polynomials and error budgets

What this concept is

A Taylor polynomial at a uses successive derivatives: f(a)+f′(a)(x−a)+f″(a)(x−a)²/2 and higher terms. If the next derivative is bounded by M along the interval, the degree n remainder has magnitude at most M|x−a|^(n+1)/(n+1)!.

Why it matters here

Approximation needs an error statement and an interval. Floating-point roundoff, truncation and model mismatch are different errors.

Concrete example

At zero, exp(x) has quadratic approximation 1+x+x²/2. At x=0.1 this is 1.105. Since exp(x)<2 on [0,0.1], the error is at most 2·0.1³/6≈0.000334. This bound is conservative and does not say the error equals the bound.

Common confusion / misconception

An infinite Taylor series is not automatically equal to its function everywhere. This module requires finite polynomial bounds, not a complete theory of series convergence.

How to use it

Specify center, degree, interval and derivative bound before choosing precision. A central difference [f(x+h)−f(x−h)]/(2h) cancels even Taylor terms; under a bounded third derivative its truncation error is at most M h²/6. Tiny h amplifies evaluation roundoff when subtracting nearly equal outputs, so reducing h indefinitely is not reliable.

Check yourself

How does halving h change the central-difference truncation bound? Why might the observed error eventually increase?

Mini drill or application

Bound the error of 1+x for exp(x) at x=0.05 using exp(x)<2 on [0,0.05]. Answer: at most 0.0025. Compare with the quadratic bound and state the extra derivative information required.

Read this only if stuck

OpenStax selective support. Use the reading guide for the section sequence and practice contract. Record your reasoning before consulting solutions.