Math · Calculus · Concept
Build a Taylor approximation
Match a function’s value and successive derivatives with a polynomial centered at a.
Match more than the value
A constant matches the height at a; adding a linear term also matches slope. A quadratic can match curvature as well. Powers of (x−a) make these requirements easy to separate because higher powers and their lower derivatives vanish at a.
Where the factorial comes from
Differentiating (x−a)^k exactly k times gives k!, so dividing by k! sets the kth derivative at a to f⁽ᵏ⁾(a). This construction is local; increasing degree does not promise accuracy everywhere.
An error bound needs more information
If f has n+1 continuous derivatives on the interval between a and x and |f⁽ⁿ⁺¹⁾|≤M there, Taylor’s remainder is bounded by M|x−a|ⁿ⁺¹/(n+1)!. A sampled maximum error is useful feedback but is not this certified bound.
Common mistakes
- The expansion center need not be zero.
- A Taylor series can fail to represent a function outside its convergence interval.
Work through an example
Build the degree-two approximation at zero and bound its error.
Approximate e^0.2 with a quadratic →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.