Question

Let y = x 2 + 3 be a curve in the plane. (a) Give a...

Let y = x 2 + 3 be a curve in the plane.

(a) Give a vector-valued function ~r(t) for the curve y = x 2 + 3.

(b) Find the curvature (κ) of ~r(t) at the point (0, 3). [Hint: do not try to find the entire function for κ and then plug in t = 0. Instead, find |~v(0)| and dT~ dt (0) so that κ(0) = 1 |~v(0)| dT~ dt (0) .]

(c) Find the center and radius of the circle of curvature at the point (0, 3) on the curve.

(d) Write the equation for the circle of curvature at the point (0, 3) and sketch a picture of the curve and the circle of curvature together.

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