2013 10 17 quadric surfaces

7/27/2019 2013 10 17 Quadric Surfaces

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NAME DATE BAND

QUADRIC SURFACESMULTIVARIABLE CALCULUS |PACKERCOLLEGIATE INSTITUTE

Types of Quadric Surfaces:

Ellipsoid Hyperboloid of One Sheet Hyperboloid of Two Sheets2 2 2

2 2 21

x y z

a b c

2 2 2

2 2 21

x y z

a b c

2 2 2

2 2 21

x y z

a b c

Elliptic Cone Elliptic Paraboloid Hyperbolid Paraboloid2 2

2

2 20

x yz

a b

2 2

2 20

x yz

a b

2 2

2 20

y xz

b a

1. Individually identify each of these figures. Ignore the letters at the end of the equation.(a) 22 2

4

yz x (b) 22 2

41

yz x (c) 2 2

4 9

x yz

(d) 2 24 9

y xz [*] (e) 22 2

41

zx y (f) 22 2

41

zx y

2. We are going to analyze the heck out of surface (d). First, you are going to take slices along the z-axis andsketch the level curves for various z-values.

(a) First, write down the equation for the level curves for various z-values. Make sure you rewrite/rearrange

the equation so it looks like it is a standard conic form for easy graphing. Write down what the curve is an

ellipse? hyperbola? Parabola? Circle? If there is nothing in the slice, explain why not!

When z=-2, the graphical slice I am going to see is a _____________ and has this equation:

When z=-1, the graphical slice I am going to see is a _____________ and has this equation:

When z=0, the graphical slice I am going to see is a _____________ and has this equation:


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(b) Now graph what all the level curves would look like superimposed:

Next to each level curve on the graph, write z=blah so we can tell the height of the curve.

(c) Now see if your graph matches what WinPlot gives you for the level curves (remember the process? 3D

Graph Implicit LevelAuto) To see all the level curves superimposed, dont forget to press see all!

Is your hand drawn graph a rough approximation of what you see in WinPlot?

x

y


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(d) Now we are going to take slices of this surface along the x-axis

When x=-2, the graphical slice I am going to see is a _____________ and has this equation:

When x=-1, the graphical slice I am going to see is a _____________ and has this equation:

When x=0, the graphical slice I am going to see is a _____________ and has this equation:



(e) Now we are going to take slices of this surface along the y-axis

When y=-2, the graphical slice I am going to see is a _____________ and has this equation:

When y=-1, the graphical slice I am going to see is a _____________ and has this equation:

When y=0, the graphical slice I am going to see is a _____________ and has this equation:




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(f) Practice graphing level curves on Winplot. See what the level curves look like when you take x-slices, and

superimpose them Do that again for the y-slices. Draw a rough sketch of what you see

x-slices sketch: y-slices sketch:

3. If we take slices in the x-, y-, and z-directions, what sorts of conics will we get?

(a) 22 24

yz x

x-direction:

y-direction:

z-direction:

(b) 22 24

1y

z x

x-direction:

y-direction:

z-direction:

(c) 2 24 9

x yz

x-direction:

y-direction:

z-direction:

(d) 2 24 9

y xz [*]

x-direction:

y-direction:

z-direction:

(e) 22 24

1z

x y

x-direction:

y-direction:

z-direction:

(f) 22 24

1z

x y

x-direction:

y-direction:

z-direction:


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4. Can you picture what these various surfaces are going to look like, now that you know what their sliceslook like? Practice using WinPlot to see if what youre imagining matches up with reality!

2

2 2

4

yz x

2

2 2

41

yz x

2 2

4 9

x yz

2 2

4 9

y x

z

[*]

2

2 2

41

z

x y

2

2 2

41

z

x y

5. When you finish, identify the following surfaces:(a)

2 2 23 4 12 12 0x y z

(b)2 2

4 4 0x y z

Then work on the following problems: Section 11.7#1, 5, 7, 9, 15**, 17**, 21** and Section 11.6#32, 33

** dont sketch identify the surface and then use WinPlot to graph the surface. Make a rough sketch of what

you see.

2013 10 17 quadric surfaces

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