Mr. Integral
Math notes/Calculus III/Optimization in Several Variables

Multivariable Taylor Approximation and Error — Cheat sheet

CHEAT SHEET   ·   FREE

Choose “Save as PDF” in the print dialog.
On this page

Core rule

P2=f(a)+∇f(a)Th+12hTH(a)h,∣f−L∣≤M2∥h∥2.P_2=f(a)+\nabla f(a)^Th+\tfrac12h^TH(a)h,\quad |f-L|\le\tfrac M2\|h\|^2.

Watch for

Include both mixed terms. A rigorous error bound requires a derivative bound along the entire input segment.