The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.
Finite approximations, exact remainders and models
Choose a substitution and evaluate a stated truncation.
Calculate the signed remainder and absolute error.
Check a claimed accuracy and interpret units and percentage changes.
Write \(F(x)=P(x)+R(x)\), where \(P\) retains the requested terms. The signed remainder is \(R=F-P\); the absolute error is \(|R|\). If \(R>0\), \(P\) is low. A finite expansion has no convergence condition such as \(|x|<1\). Bounds and models here are algebraic applications; calculus is not needed.