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hajavaF

Answered 2021-09-15
Author has **25074** answers

asked 2021-09-12

Surface s is a part of the paraboloid \(\displaystyle{z}={4}-{x}^{{2}}-{y}^{{2}}\) that lies above the plane \(z=0\).\((6+7+7=20pt)\)

a) Find the parametric equation \(\displaystyle\vec{{r}}{\left({u},{v}\right)}\) of the surface with polar coordinates \(\displaystyle{x}={u}{\cos{{\left({v}\right)}}},{y}={u}{\sin{{\left({v}\right)}}}\) and find the domain D for u and v.

b) Find \(\displaystyle\vec{{r}}_{{u}},\vec{{r}}_{{v}},\) and \(\displaystyle\vec{{r}}_{{u}}\cdot\vec{{r}}_{{v}}\).

c) Find the area of the surface

asked 2021-09-07

The plane \(\displaystyle{x}+{y}+{2}{z}={18}\) intersects the paraboloid \(\displaystyle{z}={x}^{{2}}+{y}^{{2}}\) in an ellipse. Find the points
on this ellipse that are nearest to and farthest from the origin.

asked 2021-09-01

Write an equation in the indicated coordinate system. Also write a parametric equation
and a vector equation for the following equation

\(\displaystyle{x}^{{{2}}}+{y}^{{{2}}}={100}\in{R}^{{{2}}}\), polar coordinates

\(\displaystyle{x}^{{{2}}}+{y}^{{{2}}}={100}\in{R}^{{{2}}}\), polar coordinates

asked 2021-09-05

Find a set of polar coordinates for the point for which the rectangular coordinates are given. Choose the polar coordinates where r is positive and \(\displaystyle{0}\le\theta\le\pi{\left({5}\sqrt{{3}},{5}\right)}\)

asked 2021-09-05

Hence, write the other two polar coordinates that represent the point.

asked 2021-09-12

Determine the x-y coordinates of the points where the following parametric equations will have horizontal or vertical tangents
\(\displaystyle{x}={t}^{{3}}-{3}{t}\)

\(\displaystyle{y}={3}{t}^{{2}}-{9}\)

\(\displaystyle{y}={3}{t}^{{2}}-{9}\)

asked 2021-09-07

Find the area of the surface.

The part of the paraboloid

\(\displaystyle{z}={1}−{x}^{{2}}−{y}^{{2}}\)

that lies above the plane

\(\displaystyle{z}=−{6}\)

The part of the paraboloid

\(\displaystyle{z}={1}−{x}^{{2}}−{y}^{{2}}\)

that lies above the plane

\(\displaystyle{z}=−{6}\)