8. In this exercise, we make more precise the sense in which the osculating circ

Question

8. In this exercise, we make more precise the sense in which the osculating circle is the circle which best approximates a plane curve at a point. o By translating and rotating our coordinate system, we can always arrange that the point is (0.0) and that the curve is y = f(x) with f,(0) = 0 and f"(0) > 0, (We are assuming that the curvature at the point is nonzero.) o Let y - g(x) be the bottom half of the circle of radius r which is centred at (0,r) 0, then Show that if () and g(x) have the same second order Taylor approximation at r is the radius of curvature of y = f(x) at x-0.

Explanation / Answer

Let f(x) and g(x) have the same second order Taylor approximation at x=0.

The two curves y=f(x) and y=g(x) will meet at a value x=0 if f(0)=g(0).

Then they will have the same tangent f'(0)=g'(0).

                                                           f''(0)=g''(0)

This means that thecurvatuves are equal.

Then they have the same curvatuve at x=0.

Thus the property desires for a polynomial function that approximates f(x) at x=0.

Since y=g(x) be the bottom half of the circle of radius r and centered at (0,r).

Then r is the radius of curvature of y=f(x) at x=0.

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