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The parametric equations z = 6cos t and y = 2 sin t, 0 < t < 2Ï€, generate the ellipse shown in the following figure. The curvature of a curve is a quantity that measures how much a curve is bending at a point. When the curve is described parametrically in terms of a parameter t, the curvature in terms of this same parameter is given by the formula:

Îº = (32 + 12)^(3/2)

where the dots indicate derivatives with respect to t, so dÂ²x/dtÂ². Use the formula to find the curvature of the ellipse at the points where the curve intersects the positive x and y-axes. Does the curve support your calculations? That is, does the curve bend more at the point where the curvature is larger?

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Show that the curvature function of the parametrization $\mathbf{r}(t)=$ $\langle a \cos t, b \sin t\rangle$ of the ellipse $\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1$ is$$\kappa(t)=\frac{a b}{\left(b^{2} \cos ^{2} t+a^{2} \sin ^{2} t\right)^{3 / 2}}$$

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The curvature of a plane parametric curve x = f(t), y = g(t) is given by ðœ… = |áº‹â€‰y double dot âˆ’ áºâ€‰x double dot| / (áº‹^2 + áº^2)^(3/2), where dots indicate derivatives with respect to t. Find the curvature of the following curve: x = a cos(ðœ”t), y = b sin(ðœ”t).

03:34

Show that the curve with parametric equations $ x = \sin t $, $ y = \cos t $, $ z = \sin^2 t $ is the curve of intersection of the surfaces $ z = x^2 $ and $ x^2 + y^2 = 1 $. Use this fact to help sketch the curve.

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