Calculus I · 2A · exploration
Differentiability, Little-o Notation, and the Best Local Linear Model
Explore differentiability, little-o notation, and the best local linear model in an optional advanced note connecting Calculus I to later analysis and mathematical theory.
Section overview
Optional advanced explorationsWhat this section is building
Explore differentiability, little-o notation, and the best local linear model in an optional advanced note connecting Calculus I to later analysis and mathematical theory.
Differentiability is a local linear approximation property with consequences beyond computation.
Return here after the core path is secure, and connect each abstraction to a concrete derivative example.
Collecting formal language without linking it to the local linear model it describes.
Differentiability as Controlled Error
The ordinary definition
can be rearranged into
where
Analysts write , read "little-o of ." The notation means that becomes insignificant compared with . Thus
For at ,
The linear part is , and the remainder is . Since , the quadratic remainder is genuinely smaller than the first-order change.
Read this graph as text
The nonlinear remainder shrinks faster than the input step. For f(x)=x 2 at a=3 , the exact change is 6h+h 2 . The tangent model keeps 6h ; the discarded remainder h 2 becomes small not only in absolute size, but also relative to h . Near h=0 , the gold curve falls toward zero much faster than the blue V-shaped graph. Dividing the remainder by the step gives h 2/|h|=|h| , which also approaches zero. That relative comparison, rather than the mere fact that both quantities are small, is what little- o notation records.
The visual uses labeled positions, solid and dashed line styles, and written descriptions so the nonlinear remainder shrinks faster than the input step does not depend on color.
Why it matters: This figure turns the symbolic statement r(h)=o(h) into a comparison of scales. Students often hear that the tangent error is "small" without learning what kind of smallness differentiability requires. The figure must make clear that the remainder is negligible relative to the first-order input displacement, not merely that it tends to zero.
For f(x)=x² at a=3, the exact change is 6h+h². The tangent model keeps 6h; the discarded remainder h² becomes small not only in absolute size, but also relative to h.
Why this definition generalizes
In several variables, division by a vector makes no sense. The error formulation survives: a function is differentiable when it equals a constant plus a linear map plus an error small relative to the input displacement. The derivative becomes a matrix or linear transformation.
For at , expand and identify the constant, linear, and remainder terms.
Show that the remainder divided by tends to zero.
Explain why at cannot have one linear coefficient that makes the relative error vanish from both sides.
advanced-little-o-01For at , identify the remainder after the linear term.
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