?Describe and sketch the surface. \(z=-\sqrt{x}\)________________Equation

(a) What does the equation \(y=x^{2}\) represent as a curve in \(\mathbb{R}^{2}\) ? (b) What does it represent as a surface in \(\mathbb{R}^{3}\) ? (c) What does the equation \(z=y^{2}\) represent? ________________ Equation Transcription: ?2 ?3 Text Transcription: y=x^2 R^2 R^3 z=y^2

(a) Sketch the graph of \(y=e^{x}\) as a curve in \(\mathbb{R}^{2}\). (b) Sketch the graph of \(y=e^{x}\) as a surface in \(\mathbb{R}^{3}\). (c) Describe and sketch the surface \(z=e^{y}\) ________________ Equation Transcription: ?2 ?3 Text Transcription: y=e^x R^2 R^3 y=e^x z=e^y

Describe and sketch the surface. \(x^{2}+z^{2}=4\) Equation Transcription: Text Transcription: x^2+z^2=4

Describe and sketch the surface. \(y^{2}+9 z^{2}=9\) ________________ Equation Transcription: Text Transcription: y^2+9z^2=9

Describe and sketch the surface. \(x^{2}+y+1=0\) ________________ Equation Transcription: Text Transcription: x^2+y+1=0

Describe and sketch the surface. \(z=-\sqrt{x}\) ________________ Equation Transcription: Text Transcription: z=-sqrt x

Describe and sketch the surface. \(x y=1\) ________________ Equation Transcription: Text Transcription: xy=1

Describe and sketch the surface. \(z=\sin y\) Equation Transcription: Text Transcription: z=sin y

Write an equation whose graph could be the surface shown.

Write an equation whose graph could be the surface shown.

(a) Find and identify the traces of the quadric surface \(x^{2}+y^{2}-z^{2}=1\) and explain why the graph looks like the graph of the hyperboloid of one sheet in Table 1. (b) If we change the equation in part (a) to \(x^{2}-y^{2}+z^{2}=1\), how is the graph affected? (c) What if we change the equation in part (a) to \(x^{2}+y^{2}+2 y-z^{2}=0\)? Equation Transcription: Text Transcription: x^2+y^2-z^2=1 x^2-y^2+z^2=1 x^2+y^2+2y-z^2=0

(a) Find and identify the traces of the quadric surface \(-x^{2}-y^{2}+z^{2}=1\) and explain why the graph looks like the graph of the hyperboloid of two sheets in Table 1. (b) If the equation in part (a) is changed to \(x^{2}-y^{2}-z^{2}=1\), what happens to the graph? Sketch the new graph. ________________ Equation Transcription: Text Transcription: -x^2-y^2+z^2=1 x^2-y^2-z^2=1

Use traces to sketch and identify the surface. \(x=y^{2}+4 z^{2}\) ________________ Equation Transcription: Text Transcription: x=y^2+4z^2

Use traces to sketch and identify the surface. \(4 x^{2}+9 y^{2}+9 z^{2}=36\) Equation Transcription: Text Transcription: 4x^2+9y^2+9z^2=36

Use traces to sketch and identify the surface. \(x^{2}=4 y^{2}+z^{2}\) Equation Transcription: Text Transcription: x^2=4y^2+z^2

Use traces to sketch and identify the surface. \(z^{2}-4 x^{2}-y^{2}=4\) Equation Transcription: Text Transcription: z^2-4x^2-y^2=4

Use traces to sketch and identify the surface. \(9 y^{2}+4 z^{2}=x^{2}+36\) Equation Transcription: Text Transcription: 9y^2+4z^2=x^2+36

Use traces to sketch and identify the surface. \(3 x^{2}+y+3 z^{2}=0\) Equation Transcription: Text Transcription: 3x^2+y+3z^2=0

Use traces to sketch and identify the surface. \(\frac{x^{2}}{9}+\frac{y^{2}}{25}+\frac{z^{2}}{4}=1\) Equation Transcription: Text Transcription: x^2/9+y^2/25+z^2/4=1

Use traces to sketch and identify the surface. \(3 x^{2}-y^{2}+3 z^{2}=0\) Equation Transcription: Text Transcription: 3x^2-y^2+3z^2=0

Use traces to sketch and identify the surface. \(y=z^{2}-x^{2}\) Equation Transcription: Text Transcription: y=z^2-x^2

Use traces to sketch and identify the surface. \(x=y^{2}-z^{2}\) Equation Transcription: Text Transcription: x=y^2-z^2

Match the equation with its graph (labeled I-VIII). Give reasons for your choice. \(x^{2}+4 y^{2}+9 z^{2}=1\) Equation Transcription: Text Transcription: x^2+4y^2+9z^2=1

Match the equation with its graph (labeled I-VIII). Give reasons for your choice. \(9 x^{2}+4 y^{2}+z^{2}=1\) Equation Transcription: Text Transcription: 9x^2+4y^2+z^2=1

Match the equation with its graph (labeled I-VIII). Give reasons for your choice. \(x^{2}-y^{2}+z^{2}=1\) Equation Transcription: Text Transcription: x^2-y^2+z^2=1

Match the equation with its graph (labeled I-VIII). Give reasons for your choice. \(-x^{2}+y^{2}-z^{2}=1\) Equation Transcription: Text Transcription: -x^2+y^2-z^2=1

Match the equation with its graph (labeled I-VIII). Give reasons for your choice. \(y=2 x^{2}+z^{2}\) Equation Transcription: Text Transcription: y=2x^2+z^2

Match the equation with its graph (labeled I-VIII). Give reasons for your choice. \(y^{2}=x^{2}+2 z^{2}\) Equation Transcription: Text Transcription: y^2=x^2+2z^2

Match the equation with its graph (labeled I-VIII). Give reasons for your choice. \(x^{2}+2 z^{2}=1\) Equation Transcription: Text Transcription: x^2+2z^2=1

Match the equation with its graph (labeled I-VIII). Give reasons for your choice. \(y=x^{2}-z^{2}\) Equation Transcription: Text Transcription: y=x^2-z^2

Sketch and identify a quadric surface that could have the traces shown. Traces in \(x = k\) Traces in \(y = k\) Equation Transcription: Text Transcription: x = k y = k

Sketch and identify a quadric surface that could have the traces shown. Traces in \(x = k\) Traces in \(y = k\) Equation Transcription: Text Transcription: x = k y = k

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(y^{2}=x^{2}+\frac{1}{9} z^{2}\) Equation Transcription: Text Transcription: y^2=x^2+1/9z^2

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(4 x^{2}-y+2 z^{2}=0\) Equation Transcription: Text Transcription: 4x^2-y+2z^2=0

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(x^{2}+2 y-2 z^{2}=0\) Equation Transcription: Text Transcription: x^2+2y-2z^2=0

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(y^{2}=x^{2}+4 z^{2}+4\) Equation Transcription: Text Transcription: y^2=x^2+4z^2+4

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(x^{2}+y^{2}-2 x-6 y-z+10=0\) Equation Transcription: Text Transcription: x^2+y^2-2x-6y-z+10=0

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(x^{2}-y^{2}-z^{2}-4 x-2 z+3=0\) Equation Transcription: Text Transcription: x^2-y^2-z^2-4x-2z+3=0

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(x^{2}-y^{2}+z^{2}-4 x-2 z=0\) Equation Transcription: Text Transcription: x^2-y^2+z^2-4x-2z=0

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(4 x^{2}+y^{2}+z^{2}-24 x-8 y+4 z+55=0\) Equation Transcription: Text Transcription: 4x^2+y^2+z^2-24x-8y+4z+55=0

Graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface. \(-4 x^{2}-y^{2}+z^{2}=1\) Equation Transcription: Text Transcription: -4x^2-y^2+z^2=1

Graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface. \(x^{2}-y^{2}-z=0\) Equation Transcription: Text Transcription: x^2-y^2-z=0

Graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface. \(-4 x^{2}-y^{2}+z^{2}=0\) Equation Transcription: Text Transcription: -4x^2-y^2+z^2=0

Graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface. \(x^{2}-6 x+4 y^{2}-z=0\) Equation Transcription: Text Transcription: x^2-6x+4y^2-z=0

Sketch the region bounded by the surfaces \(z=\sqrt{x^{2}+y^{2}}\) and \(x^{2}+y^{2}=1\) for \(1 \leqslant z \leqslant 2\). Equation Transcription: 1 ? z ? 2 Text Transcription: z=sqrt x^2+y^2 x^2+y^2=1 1 leqslant z leqslant 2

Sketch the region bounded by the paraboloids \(z=x^{2}+y^{2}\) and \(z=2-x^{2}-y^{2}\) Equation Transcription: Text Transcription: z=x^2+y^2 z=2-x^2-y^2.

Find an equation for the surface obtained by rotating the curve \(y=\sqrt{x}\) about the x -axis. Equation Transcription: Text Transcription: y=sqrt x

Find an equation for the surface obtained by rotating the line \(z=2 y\) about the z -axis. Equation Transcription: Text Transcription: z=2y

Find an equation for the surface consisting of all points that are equidistant from the point \((-1,0,0)\) and the plane \(x=1\). Identify the surface. Equation Transcription: Text Transcription: (-1,0,0) x=1

Find an equation for the surface consisting of all points \(P\) for which the distance from \(P\) to the \(x\) -axis is twice the distance from \(P\) to the \(yz\) -nlane Identify the surface. Equation Transcription: Text Transcription: P x yz

Traditionally, the earth’s surface has been modeled as a sphere, but the World Geodetic System of 1984 (WGS-84) uses an ellipsoid as a more accurate model. It places the center of the earth at the origin and the north pole on the positive z-axis. The distance from the center to the poles is \(6356.523 km\) and the distance to a point on the equator is \(6378.137 km\). (a) Find an equation of the earth’s surface as used by WGS-84. (b) Curves of equal latitude are traces in the planes \(z = k\).What is the shape of these curves? (c) Meridians (curves of equal longitude) are traces in planes of the form \(y = mx\). What is the shape of these meridians? Equation Transcription: Text Transcription: 6356.523 km 6378.137 km z = k y = mx

A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet [see Figure 12(b)]. The diameter at the base is \(280 m\) and the minimum diameter, \(500 m\) above the base, is \(200 m\). Find an equation for the tower. Equation Transcription: Text Transcription: 280 m 500 m 200 m

Show that if the point \((a, b, c)\) lies on the hyperbolic paraboloid \(z=y^{2}-x^{2}\), then the lines with parametric equations \(x=a+t, y=b+t, z=c+2(b-a) t\) and \(x=a+t\), \(y=b-t, z=c-2(b+a) t\) both lie entirely on this paraboloid. (This shows that the hyperbolic paraboloid is what is called a ruled surface; that is, it can be generated by the motion of a straight line. In fact, this exercise shows that through each point on the hyperbolic paraboloid there are two generating lines. The only other quadric surfaces that are ruled surfaces are cylinders, cones, and hyperboloids of one sheet.) Equation Transcription: Text Transcription: (a,b,c) z=y^2-x^2 x=a+t,y=b+t,z=c+2(b-a)t x=a+t y=b-t,z=c-2(b+a)t

Show that the curve of intersection of the surfaces \(x^{2}+2 y^{2}-z^{2}+3 x=1\) and \(2 x^{2}+4 y^{2}-2 z^{2}-5 y=0\)lies in a plane. Equation Transcription: Text Transcription: x^2+2y^2-z^2+3x=1 2x^2+4y^2-2z^2-5y=0

Graph the surfaces \(z=x^{2}+y^{2}\) and \(z=1-y^{2}\) on a common screen using the domain \(|x| \leqslant 1,2,|y| \leqslant 1,2 \mid\) and observe the curve of intersection of these surfaces. Show that the projection of this curve onto the xy -plane is an ellipse. Equation Transcription: |x|?1,2,|y|?1,2 Text Transcription: z=x^2+y^2 z=1-y^2 |x|leqslant 1,2,|y|leqslant 1,2