Question: Two vector-valued functions r 1 (s) and r 2 (s) are said to agree to order 2 at s 0 if ri (So) = r(so),

Two vector-valued functions r₁(s) and r₂(s) are said to agree to order 2 at s₀ if

r(So) = r(So), r(so) = r(so), r(so) = r (so) Let r(s) be an arc length parametrization of a curve C, and let

ri (So) = r(so), r (so) = r(so), r(so) = r2(so) Let r(s) be an arc length parametrization of a curve C, and let P be the terminal point of r(0). Let y(s) be the arc length parametrization of the osculating circle given in Exercise 80. Show that r(s) and y(s) agree to order 2 at s = 0 (in fact, the osculating circle is the unique circle that approximates C to order 2 at P).

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