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Taylor's theorem [4] [5] [6] ... This version covers the Lagrange and Cauchy forms of the remainder as special cases, and is proved below using Cauchy's mean value ...
As shown below, the second-derivative test is mathematically identical to the special case of n = 1 in the higher-order derivative test. Let f be a real-valued, sufficiently differentiable function on an interval I ⊂ R {\displaystyle I\subset \mathbb {R} } , let c ∈ I {\displaystyle c\in I} , and let n ≥ 1 {\displaystyle n\geq 1} be a ...
A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
The special case of calculus in three dimensional space is often called vector calculus. Introduction ... using Taylor's theorem to construct the remainder: ...
Taylor's theorem; Rules and identities ... The reciprocal rule may be derived as the special case where ... rule and can be derived using the fundamental theorem of ...
The above is obtained using a second order approximation, following the method used in estimating the first moment. It will be a poor approximation in cases where () is highly non-linear. This is a special case of the delta method.
Linear approximations in this case are further improved when the second derivative of a, ″ (), is sufficiently small (close to zero) (i.e., at or near an inflection point). If f {\displaystyle f} is concave down in the interval between x {\displaystyle x} and a {\displaystyle a} , the approximation will be an overestimate (since the ...
This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula. The chain rule is also valid for Fréchet derivatives in Banach spaces. The same formula holds as before. [8] This case and the previous one admit a simultaneous generalization to Banach manifolds.