How to draw hyperbolic paraboloid

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Current Score : – / 25 Due : Saturday, January 28 2017 11:55 PM MST

1. –/1 pointsSEssCalcET2 10.6.AE.001.

Video Example

EXAMPLE 1 Sketch the graph of the surface

SOLUTION Notice that the equation of the graph doesn't involve y. This means that any vertical plane with equation (parallel to the xz-plane) intersects the graph

in a curve with equation So these vertical traces are parabolas.

The figure shows how the graph is formed by taking the parabola in the xz-plane and moving it in the direction of the ---Select--- . The graph is a

surface, called a parabolic cylinder, made up of infinitely many shifted copies of the parabola. Here the rulings of the cylinder are parallel to the ---Select--- .

10.6 Homework- Cylinders & Quadric Surfaces (Homework)

Salem Almarar Mat 267, section 10687, Spring 2017 Instructor: Sergey Nikitin

WebAssign

z = 8x2.

z = 8x2

y = k z =

. z =

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2. –/1 pointsSEssCalcET2 10.6.AE.005.

Figure 1 Video Example

EXAMPLE 5 Sketch the surface

SOLUTION The traces in the vertical planes are the parabolas

which open upward. The traces in are the parabolas

which open downward. The horizontal traces are

a family of hyperbolas. We draw the family of traces in Figure 2, and we show how the traces appear when placed in their correct planes in Figure 3. In

Figure 1 we fit together the terms to form the surface a hyperbolic paraboloid. Notice that the shape of the surface near the origin resembles that of a saddle. This surface will be investigated further in a later section when we discuss saddle points.

Figure 2

Figure 3

z = 5y2 − 5x2.

x = k z =

, y = k z =

,

= k,

z = 5y2 − 5x2,

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3. –/1 pointsSEssCalcET2 10.6.003.

Describe the surface.

sphere

ellipsoid

hyperboloid

circular cylinder

elliptic cylinder

hyperbolic cylinder

parabolic cylinder

elliptic paraboloid

Sketch the surface.

x2 + z2 = 3

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4. –/1 pointsSEssCalcET2 10.6.005.

Describe the surface.

cone

ellipsoid

hyperboloid

elliptic cylinder

hyperbolic cylinder

parabolic cylinder

elliptic paraboloid

hyperbolic paraboloid

Sketch the surface.

z = 4 − y2

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5. –/1 pointsSEssCalcET2 10.6.011.

Use traces to sketch the surface.

Identify the surface. parabolic cylinder

elliptic paraboloid

elliptic cone

ellipsoid

elliptic cylinder

hyperboloid of one sheet

hyperboloid of two sheets

hyperbolic paraboloid

x = y2 + 2z2

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6. –/1 pointsSEssCalcET2 10.6.013.

Use traces to sketch the surface.

Identify the surface. elliptic cylinder

elliptic cone

hyperboloid of one sheet

hyperbolic paraboloid

ellipsoid

parabolic cylinder

hyperboloid of two sheets

elliptic paraboloid

7. –/1 pointsSEssCalcET2 10.6.016.

Use traces to sketch the surface.

x2 = y2 + 7z2

4x2 + 7y2 + z = 0

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Identify the surface. elliptic cylinder

hyperboloid of one sheet

elliptic cone

elliptic paraboloid

hyperboloid of two sheets

hyperbolic paraboloid

parabolic cylinder

ellipsoid

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8. –/1 pointsSEssCalcET2 10.6.018.

Use traces to sketch the surface.

Identify the surface. parabolic cylinder

hyperboloid of two sheets

elliptic paraboloid

ellipsoid

hyperbolic paraboloid

elliptic cone

hyperboloid of one sheet

elliptic cylinder

2x2 − 8y2 + z2 = 8

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9. –/1 pointsSEssCalcET2 10.6.020.

Use traces to sketch the surface.

Identify the surface. ellipsoid

elliptic paraboloid

parabolic cylinder

hyperbolic paraboloid

elliptic cone

hyperboloid of two sheets

hyperboloid of one sheet

elliptic cylinder

x = 2y2 − 2z2

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10.–/1 pointsSEssCalcET2 10.6.502.XP.

Match the equation with its graph.

11.–/1 pointsSEssCalcET2 10.6.504.XP.

Match the equation with its graph.

9x2 + 4y2 + z2 = 1

−x2 + y2 − z2 = 1

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12.–/1 pointsSEssCalcET2 10.6.506.XP.

Match the equation with its graph.

13.–/1 pointsSEssCalcET2 10.6.023.

Consider the equation below.

Reduce the equation to one of the standard forms.

Classify the surface. ellipsoid

elliptic paraboloid

hyperbolic paraboloid

cone

hyperboloid of one sheet

hyperboloid of two sheets

Sketch the surface.

y2 = x2 + 2z2

x2 + 6y − 6z2 = 0

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14.–/1 pointsSEssCalcET2 10.6.025.

Consider the equation below.

Reduce the equation to one of the standard forms.

Classify the surface. hyperbolic paraboloid

4x2 + y2 + 4z2 − 4y − 24z + 36 = 0

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elliptic paraboloid

hyperboloid of two sheets

elliptic cylinder

hyperboloid of one sheet

parabolic cylinder

circular cone

ellipsoid

Sketch the surface.

15.–/1 pointsSEssCalcET2 10.6.028.

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Consider the equation below.

Reduce the equation to one of the standard forms.

Classify the surface. elliptic paraboloid

hyperboloid of one sheet

circular cone

hyperbolic paraboloid

ellipsoid

hyperboloid of two sheets

parabolic cylinder

elliptic cylinder

Sketch the surface.

x2 − y2 + z2 − 2x + 2y + 4z + 2 = 0

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16.–/1 pointsSEssCalcET2 10.6.510.XP.

Find an equation for the surface obtained by rotating the line about the x-axis.

x = 2y

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17.–/1 pointsSEssCalcET2 10.6.032.

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-plane.

Identify the surface. cone

hyperboloid of two sheets

elliptic cylinder

hyperbolic paraboloid

hyperboloid of one sheet

ellipsoid

parabolic cylinder

elliptic paraboloid

18.–/1 pointsSEssCalcET2 10.6.512.XP.

A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet. The diameter at the base is 260 m and the minimum diameter, 500 m above the base, is 220 m. Find an equation for the tower. (Assume the center is at the origin with axis the z-axis and the minimum diameter is at the center.)

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19.–/1 pointsSEssCalcET2 10.6.TEC.A.002.

Concept

Instructions

Simulation

Exercise

In Part 1 of this module, you can move vertical and horizontal planes parallel to the coordinate planes through a quadric surface. The intersections of these planes with the surface form traces, which are shown both on the surface and on a 2D graph as you might draw on paper. You will be asked to investigate the families of traces in the exercises.

Part 2 shows planes parallel to the coordinate planes and the traces of a surface they contain, but you will not see the surface itself. By moving the planes and watching the shapes of the traces, you can visualize the shape of the surface.

Part 1: Select one of the first two surfaces from the pull-down menu. The surface is graphed on the left. You will see a vertical plane, x = n, slicing through the surface. Its intersection with the surface forms a trace of the surface which is also drawn on a 2D graph at the right. Drag the slider to change the value of n and move the plane through the surface. The trace will change shape accordingly. You can also click on the slider bar at a desired value or click on a number above the bar. When you have finished investigating the traces in click the check box for

and then to see additional families of traces of the surface.

Part 2: Select the third or fourth surface from the pull-down menu. You will see three planes graphed: in red, in green, and in blue. Drag the slider to change the values of a, b, and c, thereby moving the planes through the space. You will see traces of a surface appear on the planes. Try moving just one plane through the space and then look at two or all three in intermediate positions. Can you see the shape of the surface?

x = n, y = n z = n

x = a y = b z = c

Click here to access the TEC simulation.

Traces of surface

Determine the equation for the family of traces in

x2 + y2 − z2 = 1:

x = n.

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20.–/1 pointsSEssCalcET2 10.6.TEC.A.014.

Concept

Instructions

Simulation

Exercise

In Part 1 of this module, you can move vertical and horizontal planes parallel to the coordinate planes through a quadric surface. The intersections of these planes with the surface form traces, which are shown both on the surface and on a 2D graph as you might draw on paper. You will be asked to investigate the families of traces in the exercises.

Part 2 shows planes parallel to the coordinate planes and the traces of a surface they contain, but you will not see the surface itself. By moving the planes and watching the shapes of the traces, you can visualize the shape of the surface.

Part 1: Select one of the first two surfaces from the pull-down menu. The surface is graphed on the left. You will see a vertical plane, x = n, slicing through the surface. Its intersection with the surface forms a trace of the surface which is also drawn on a 2D graph at the right. Drag the slider to change the value of n and move the plane through the surface. The trace will change shape accordingly. You can also click on the slider bar at a desired value or click on a number above the bar. When you have finished investigating the traces in click the check box for

and then to see additional families of traces of the surface.

Part 2: Select the third or fourth surface from the pull-down menu. You will see three planes graphed: in red, in green, and in blue. Drag the slider to change the values of a, b, and c, thereby moving the planes through the space. You will see traces of a surface appear on the planes. Try moving just one plane through the space and then look at two or all three in intermediate positions. Can you see the shape of the surface?

x = n, y = n z = n

x = a y = b z = c

Click here to access the TEC simulation.

Traces of surface C:

Describe the family of traces in the plane x = a. For x = a, the traces are circles.

For x = a, the traces are straight lines.

For x = a, the traces are hyperbolas.

For x = a, the traces are ellipses.

For x = a, the traces are parabolas.

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21.–/1 pointsSEssCalcET2 10.6.TEC.A.018.

Concept

Instructions

Simulation

Exercise

In Part 1 of this module, you can move vertical and horizontal planes parallel to the coordinate planes through a quadric surface. The intersections of these planes with the surface form traces, which are shown both on the surface and on a 2D graph as you might draw on paper. You will be asked to investigate the families of traces in the exercises.

Part 2 shows planes parallel to the coordinate planes and the traces of a surface they contain, but you will not see the surface itself. By moving the planes and watching the shapes of the traces, you can visualize the shape of the surface.

Part 1: Select one of the first two surfaces from the pull-down menu. The surface is graphed on the left. You will see a vertical plane, x = n, slicing through the surface. Its intersection with the surface forms a trace of the surface which is also drawn on a 2D graph at the right. Drag the slider to change the value of n and move the plane through the surface. The trace will change shape accordingly. You can also click on the slider bar at a desired value or click on a number above the bar. When you have finished investigating the traces in click the check box for

and then to see additional families of traces of the surface.

Part 2: Select the third or fourth surface from the pull-down menu. You will see three planes graphed: in red, in green, and in blue. Drag the slider to change the values of a, b, and c, thereby moving the planes through the space. You will see traces of a surface appear on the planes. Try moving just one plane through the space and then look at two or all three in intermediate positions. Can you see the shape of the surface?

x = n, y = n z = n

x = a y = b z = c

Click here to access the TEC simulation.

Traces of surface D:

Describe the family of traces in the planes x = a.

For x = a, the traces are ellipses for a = 0 and are pairs of lines when a ≠ 0.

For x = a, the traces are ellipses for a ≠ 0 and are pairs of lines when a = 0.

For x = a, the traces are parabolas for a ≠ 0 and are pairs of lines when a = 0.

For x = a, the traces are hyperbolas for a ≠ 0 and are pairs of lines when a = 0.

For x = a, the traces are hyperbolas for a = 0 and are pairs of lines when a ≠ 0.

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22.–/1 pointsSEssCalcET2 10.6.TEC.B.001.

Concept

Instructions

Simulation

Exercise

23.–/1 pointsSEssCalcET2 10.6.TEC.B.002.

Concept

Instructions

Simulation

Exercise

The six basic types of quadric surfaces are graphed. Each equation has constants a, b, and c that can be varied. Try to determine the effect each of these constants has on the shape of the surface.

Select one of the six basic types of quadric surfaces from the pull-down menu. The equation is shown at the right. Drag the slider handles, click on a slider bar, or click on a number above a slider to adjust the values of the constants a, b, and c in the equation. You may rotate the axes with the mouse for different views of the surface.

Click here to access the TEC simulation.

For the given quadric surface, describe the effect that changing the value of a, b, and c has on the shape of the surface. Ellipsoid

This answer has not been graded yet.

The six basic types of quadric surfaces are graphed. Each equation has constants a, b, and c that can be varied. Try to determine the effect each of these constants has on the shape of the surface.

Select one of the six basic types of quadric surfaces from the pull-down menu. The equation is shown at the right. Drag the slider handles, click on a slider bar, or click on a number above a slider to adjust the values of the constants a, b, and c in the equation. You may rotate the axes with the mouse for different views of the surface.

Click here to access the TEC simulation.

For the given quadric surface, describe the effect that changing the value of a, b, and c has on the shape of the surface. Cone

This answer has not been graded yet.

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24.–/1 pointsSEssCalcET2 10.6.TEC.B.003.

Concept

Instructions

Simulation

Exercise

25.–/1 pointsSEssCalcET2 10.TF.017.

Determine whether the statement is true or false.

The set of points is a circle.

True

False

The six basic types of quadric surfaces are graphed. Each equation has constants a, b, and c that can be varied. Try to determine the effect each of these constants has on the shape of the surface.

Select one of the six basic types of quadric surfaces from the pull-down menu. The equation is shown at the right. Drag the slider handles, click on a slider bar, or click on a number above a slider to adjust the values of the constants a, b, and c in the equation. You may rotate the axes with the mouse for different views of the surface.

Click here to access the TEC simulation.

For the given quadric surface, describe the effect that changing the value of a, b, and c has on the shape of the surface. Elliptic Paraboloid

This answer has not been graded yet.

(x, y, z) | x2 + y2 = 81

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