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The second derivative implied by a parametric equation is given by = = = (˙ ˙) ˙ = ˙ ¨ ˙ ¨ ˙ by making use of the quotient rule for derivatives. The latter result is useful in the computation of curvature .
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
Parametric continuity (C k) is a concept applied to parametric curves, ... zeroth, first and second derivatives are continuous : 0-th through -th ...
Consider a parametric curve ... The second example is the inverse tangent function ... Considering a second derivative of ...
Second derivative; Implicit differentiation; ... (This can arise, for example, if a multi-dimensional parametric curve is defined in terms of a scalar variable, and ...
The derivative of ′ is the second derivative, denoted as ″ , and the derivative of ″ is the third derivative, denoted as ‴ . By continuing this process, if it exists, the n {\displaystyle n} th derivative is the derivative of the ( n − 1 ) {\displaystyle (n-1)} th derivative or the derivative of order ...
The second fundamental form of a general parametric surface S is defined as follows. Let r = r(u 1,u 2) be a regular parametrization of a surface in R 3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of r with respect to u α by r α, α = 1, 2.
Let γ be as above, and fix t.We want to find the radius ρ of a parametrized circle which matches γ in its zeroth, first, and second derivatives at t.Clearly the radius will not depend on the position γ(t), only on the velocity γ′(t) and acceleration γ″(t).