Explore the radius of curvature of curves. There can be many circles that are tangent to a curve at a particular point, but there is one that provides a “best fit” (Figure 13). This circle is called an osculating circle of the curve. We define it formally in Section 13.4. The radius of the osculating circle is called the radius of curvature of the curve and can be shown to be

Consider the ellipse 9x² + y² = 36.
(a) Compute the radius of curvature in terms of x and y.
(b) Compute the radius of curvature at (−2, 0), (1, −3√3), and (0, 6). Sketch the ellipse, plot these three points, and label them with the corresponding radius of curvature.

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r = (1 + (dy/dx)²)3/2 |d²y/dx²|