textbook-theory · English
Difference Equations to Differential Equations: Section 3.1 Best Affine Approximations
Section 3.1 of Dan Sloughter's open calculus textbook introduces the concept of best affine approximations to functions, rigorously defining the tangent line as the unique affine function T whose approximation error R(h) is o(h) — that is, goes to zero faster than h itself. The section develops the formal definitions of o(h) and O(h) (little-o and big-O notation), works through several concrete examples including f(x) = sqrt(x) and f(x) = x^2, and culminates in the formal definition of the derivative as the slope of the best affine approximation.
calculusaffine approximationstangent linesderivatives