91Ó°ÊÓ

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=e^{-x^{2}},(1,1 / e) $$

What condition leads to a graph that is symmetric with respect to the following? (a) \(x z\) -plane (b) \(y\) -axis (c) \(x\) -axis

A ship is sailing due south at 20 miles per hour. A man walks west (i.e., at right angles to the side of the ship) across the deck at 3 miles per hour. What are the magnitude and direction of his velocity relative to the surface of the water?

Find the velocity \(\mathbf{v}\), acceleration \(\mathbf{a}\), and speed \(s\) at the indicated time \(t=t_{1}\). $$ \mathbf{r}(t)=t^{6} \mathbf{i}+\left(6 t^{2}-5\right)^{6} \mathbf{j}+t \mathbf{k} ; t_{1}=1 $$

Find the general equation of a central ellipsoid that is symmetric with respect to the following: (a) origin (b) \(x\) -axis (c) \(x y\) -plane

Find the symmetric equations of the tangent line to the curve with equation $$ \mathbf{r}(t)=2 \cos t \mathbf{i}+6 \sin t \mathbf{j}+t \mathbf{k} $$ at \(t=\pi / 3\).

Find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\)

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=\tan x,(\pi / 4,1) $$

Julie, flying in a wind blowing 40 miles per hour due south, discovers that she is heading due east when she points her airplane in the direction \(\mathrm{N} 60^{\circ} \mathrm{E}\). Find the airspeed (speed in still air) of the plane.

Make the required change in the given equation. \(r^{2}+2 z^{2}=4\) to spherical coordinates