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# cylinder

Three dimensional coordinate systems level three.
in the previous video we started to
describe basic equations
in R cubed. Specifically describing equations
on a three-dimensional Cartesian
coordinate system
in this video we will continue describing basic equations
in three-dimensional space. Let’s start by
graphing the equation y=x
in R-squared. Since this graph will
provide us with a starting point
to successfully graph it in R-cubed. For
starters,
this equation is represented by a line
in R squared
and has the form y=mx + b
where the slope is equal to one and the
y-intercept
is equal to zero. So we typically graph these
equations by first plotting the
y-intercept
in this case it’s equal to zero, then we use
the slope
which in this case is equal to one, to
generate a second point.
recall that a slope of positive one
means that we move in the positive
y direction
one unit and move in the positive x-direction
one unit. Remember the slope can be
interpreted
as the rise over the run. Once we have our two points we go ahead and connect them
obtaining the following straight curved. Now
if we were to graph this equation in R-cubed
we would obtain the same curve in the
xy-plane
this curve has a y-intercept of zero and a slope of
positive one, but notice that the
variable z is not mentioned in the
equation
meaning we have not specified z in any
way
so we must assume that z can take on
any value
this means that any particular value of z
will get a copy of this line
both the positive values of z such as
z=1
z=2, z=3, and so on and negative values of z
such as z=-1, z=-2,
and z=-3 and so on. In this
sense the graph represents a vertical
plane
that lies over the line given by the
equation y=x
in the xy-plane. Notice that we started by
drawing the curve
on the plane that contained the variables
for which the equation was defined
in this case the equation was defined
but the variables x and y
so we first graphed it on the XY plane, then
we drew copies of this curve
on both the positive and negative
direction of the missing variable
which was not expressed in the equation in this case the variable that was not expressed
was z
remember variable that are not explicitly
expressed in the equation
are assumed to attain any value unless mentioned otherwise.
we will go over equations in which the
variables are restricted
in a later video. Alright, lets spice
things up a bit
and graph the equation z=-2y + 1.
once again, this equation has the form
y=mx + b, but instead we have the
equation
z=my + b, where z is the dependent
variable
and y is the independent variable, we
essentially treat the equation
just like we did in the previous example,
in the xy-plane we are actually going to
graph the curve on the yz-plane
as follows, here we have a
two-dimensional representation
of the yz-plane we’re going to graph the
curve in this plane
and then translate it into R-cubed, so
what used to be the y-intercept which in
this case is now the z-intercept on the
yz-plane
so we plot the first points at z=1, then we use the slope,
in this case its negative two, which can be
interpreted
as negative two over one or in this case the rise over run
and use it to generate a second point, so
from the z-intercept
we go ahead and move two units down along the negative z-axis
and move one unit to the right along the
positive y-axis
then we connect the points with a curve as
follows
alright, having graphed the equation in the
yz-plane
we’re ready to translate it into the yz-plane of a three-dimensional
Cartesian coordinate system
its the same idea, but now we need to be
able to navigate
along the yz-plane of this
three-dimensional coordinate system
the yz-plane is located here, the
positive y-axis is located here
and the positive z-axis is here, so we
literally draw the same curve
we just graphed in R-squared onto the yz-plane
of R-cubed as follows
next we notice the variable x is not
expressed in the equation
and there are also no restrictions this
means that x can take on any value
as a consequence, at any particular value of
x
there will be a copy of this line both
on the positive values of x such as
x=1, x=2, and x=3,
and so on, and the negative values of x such as
x=-1, x=-2, x=-3,
and so on, in this sense the graph
represents a plane
that intercepts the z-axis at z=1
and has a negative slope the plane is
inclined
with a slope of negative two over one
alright, lets try the next example. This one
requires graphing an equation
on the xz-plane. Let’s graph the equation
2z + 1=3x, notice that this equation is not in slope
intercept form so we first need to arrange
the equation into the slope intercept
form
of a line, notice that all the concepts
you learned in your previous math classes
come back in some shape way or form
this will be true as we dive deeper into multivariable calculus
okay, in order to write the equation in
slope intercept form
we first need to solve for the dependent
variable in this case the dependent
variable
is z and the independent variable is x
this is similar to the way we solved for
y when dealing with an equation
in the xy-plane back in your pre-calculus
class
so we go ahead and subtract one from both
sides
then we divide both sides by two, doing that
we obtain the equation z equals three half x minus one half.
let’s go ahead and graph this equation
directly into the three-dimensional coordinate
system
you first need to graph the curve on the xz-plane so looking at the equation
we see that the z-intercept is negative
one half
so we first plot this point as follows
then we noticed that the slope is
positive three halves
so we use this slope to generate an
additional point, so starting from the
z-intercept
we move three units towards the positive
z-axis and two units towards the positive
x-axis, then we go ahead and connect a
curve connecting both points
as follows. Having created our first curve
we go ahead and make copies of this curve
at positive values of y
and at negative values of y, remember
when a variable is not expressed
in an equation and no restrictions are
mentioned the variable is free to take
on any value
lastly, we connect all the curves together
and obtain the following plane
that has a z-intercept of negative one half and is also inclined
with a slope of positive three halves
alright, these three examples were
designed to illustrate the fact that you
can use all your knowledge and
experience from graphing equations
in a two-dimensional coordinate system and
translate them into three dimensions
it might take you some time to get used to but like in all your math classes practice is key
in developing this skill. In a later video
we will actually develop a more powerful way
of describing planes
in 3D space. For now, these examples were
designed so you can start obtaining an idea
of how to navigate through this
three-dimensional coordinate system
alright, lets try graphing something other
than planes
let’s give it a shot and graph the following
equation.
x squared plus y squared equals four. Notice that the variables
that are explicitly expressed are the
variables x and y.
so we are going to first graph this equation in R squared
specifically on the xy-plane. Recall from your studies of precalculus that this
equation represents a circle
centered at the origin with radius two, so
its graph is represented in R-squared
as follows. Now if we want to graph this
equation in R-cubed
we go ahead and graph this curve on the xy-plane as follows.
next we notice that the variable z is
not expressed in the equation
remember this means that every single
value of z both positive and negative
will have copies of this curve which has the
shape of a circle
finally we connect the curves with
rulings as follows
this equation represents a cylinder
centered in the z-axis
so in R-squared the equation
represents a circle
and in R-cubed it represents a cylinder once again
this example illustrates the importance
of specifying what coordinate system
we are referring to, since it ultimately
determines the graph the equation
aright, let’s end the video by graphing
one final equation
lets graph the equation z equals x squared
notice that the variables z and x are expressed in this equation
so we’re going to go ahead and graph them
on the xz-plane
in addition, notice that this equation
looks like the equation of a parabola
of the form y equals x squared but in this case
the dependent variable is z and the
independent variable is x
other than the variable z, this equation
pretty much represents a parabola
in the xz-plane and is graphed as follows
this is how this equation will look like
in R-squared
next we go ahead and graph the same curve
on the xz-plane
of R-cubed as follows
notice that the variable that is not
expressed is y
this means that every single value of y
will have a copy of this curve
since its free to take any value of y, in
no restrictions have been imposed on the
variable y. Next we go ahead
and connect the curves as follows, doing
that we obtain the following surface
this surface is called a parabolic
cylinder
we will study this surface and many more
in a later video
for now, keep in mind that equations in R-squared are usually
represented by curves on a plane and
equations in R-cubed
are usually represented by a surface but
not always
they can also be represented by curves
in space
as we will see in a later video. Alright
in our next video we will start
exploring common formulas associated
with a three-dimensional coordinate system

Three-Dimensional Coordinate systems (Level 10)
In this final video on three dimensional coordinate
systems, we will go ahead and learn how to
graph equations and inequalities that are
restricted to a given interval. Recall from
your prior math classes that the graph of
a function can be restricted to certain values
of its domain and range by specifying the
restriction using inequalities. For example
if we were the graph the equation y equals
x squared in R-squared it will look like
this, it’s essentially a parabola that attains
a minimum value at x=0. Now, say that I only
want to graph this equation starting at x=1 to positive infinity, as follows
the way we mathematically express this graph is
by denoting an inequality that describes this
restriction, in this case we want to graph
the equation y equals x squared when x is
greater than or equal to 1, if we don’t want
to include the value at x=1 then we write
x>1, then the closed circle is changed to
an open circle to let us know that we do not
include the value at x=1. In the same manner
we can restrict the graph of an equation in
three dimensional space.
Let’s go over the first example.
Example 1: Describe the region of R-cubed
represented by the equation.
y equals 0 when z is greater than or equal to 0.
Recall from the previous videos that the equation
y=0 represents the xz-coordinate plane and
is graphically represented as follows, next
we go ahead and apply the restriction in this
case we only want that part of the graph where
the values of z are greater than or equal to zero,
this means that we only want that part of
the graph that is located on the regions where
the value of z are positive in this case
everything above the xy-plane. So our graph
will look like this, notice that there was
no restrictions on the values of x remember
this means that the values of x are free to
attain any value. If on the other hand we
than or equal to 0, then we would need to
erase that part of the graph y=0 where the
values of x are negative as follows, because
we only want that portion of the graph where
the values of z and x are positive. In this
is the region that is graphed it’s essentially
the first quadrant of the xz-coordinate plane
if we were to look at it from this direction
Alright, let’s try the next example:
Example 2: Describe the region of R-cubed
represented by the equation.
x squared plus y squared equals 4, when z is between negative 2 and 2, inclusive.
When dealing with these type of problem it’s
always a good idea to ignore the restrictions
for now and graph the equation as if there
was no restrictions. Recall that this equation
represents a cylinder in R-cubed, the variables
expressed in the equation are x and y so the
expression x squared plus y squared equals
4 represents a circle in the xy-plane, in
addition, since we are graphing this equation
without any restrictions we create a copy
of this circle on every single value of z
as follows.
So technically this cylinder extends all the
way to positive infinity in the positive z
direction and negative infinity in the negative
z direction. Next lets go ahead and apply
the restrictions, notice that the inequality
describes an interval in this case the acceptable
values of z that this equation can attain
are all the values between negative 2 and
positive 2 inclusive, This means that our
cylinder will only have copies of the circular
trace between these values of z as follows,
so we now have a cylinder that has a height
of 4 units. What if we wanted to graph only
the left half of the cylinder? What restriction
would I have to add? In this case we only want
that part of the graph where the values of
y are negative so we need to add the following
inequality y is less than or equal to 0. If
I want the right half then I include the inequality
y is greater than or equal to zero, notice
that in both cases we are including 0 if you
don’t want to include zero then we remove
the equal sign and add dashed lines on the
sides of the cylinder as follows.
As you just saw we can control what portion
of the graph we want to focus on by restricting
certain values with inequalities.
Alright, the next types of graphs we will focus on
are the graphs of inequalities themselves.
Recall from your previous math classes that
when you graph an inequality in R-squared
we are trying to find a specific region whose
points satisfy the inequality. For example
say we want to graph the inequality y is greater
than or equal to x squared, the first step
is to graph the inequality as if it were a
regular equation, as follows, next we need
to determine the region above or below the
graph that satisfies the inequality, in this
case we want to shade the region were the
values of y are greater than the values of
x squared, this means that we need to shade the region
above the graph as follows. Recall that this
region determines a set of points that satisfies
the inequality so if you were to choose any
point on this region it should satisfy the
inequality in other words you should obtain
a true statement when you substitute the coordinates
of any point located in this region into the
expression. Notice that the set of points
that will make this inequality a true statement
also includes the points located on the parabola
itself, if we were to change the inequality
to y>x squared then we would graph the function
using dashed lines this means that we don’t
include the points located on this function itself.
Now let’s go ahead and graph some inequalities
in R-cubed.
Example 3: Describe the region of R-cubed
represented by the inequality.
x is greater than 3
Alright, the first step is to graph the inequality
as if it were an equation so we are essentially
graphing the equation x=3, recall that this
equation represents a plane that is parallel
to the yz-coordinate plane and located 3 units
in the positive x direction, Next we go ahead
and apply the inequality, in this case the
values of the x have to be greater than 3,
this region consist of all the points in front
of the plane x=3 not including those points
on the plane. Alright let’s try the next example.
Example 4: Describe the region of R-cubed
represented by the inequality.
Here we have an inequality that represents
an interval. The way we deal with these types
of inequalities is by graphing the function
in the middle of the inequality by equating
it with the left value and the right value,
so we first want to graph z=0 and z=6,
These equations represent planes that are
parallel to the xy-plane.
So we go ahead and graph them.
Next we apply the
inequality, so the region represented by the
inequality are all the points between these
two planes including the points on the bottom
plane but not including the points located
on the top plane.
Alright, let’s try the next example:
Example 5: Describe the region of R-cubed
represented by the inequality.
x squared plus z squared is less than or equal to 9
As always let’s go ahead and graph the inequality
as if it were an equation, this equation represents
a cylinder with circular traces of radius
3 located in the xz-plane, and centered in
the y-axis. Next, lets apply the inequality,
we want to shade that part of R-cubed where
x squared + z squared is less than or equal to 9, another
way of representing this inequality is by
taking the square root of both sides as follows, so we essentially want the set of all points
whose distance from the y-axis is at most
3, these points are located inside of the
cylinder we also want to include those points
that are located on the cylinder.
Alright, lets try the next example:
Example 6: Describe the region of R-cubed
represented by the inequality.
Alright, once gain we have an inequality defined
by a given region, in this case the middle
function represents a sphere and the numerical
values represent the smallest and largest
value of the radius that the sphere can attain.
So let’s go ahead and graph the equation x
squared plus y squared plus z squared equals
1 and the equation x squared plus y squared
plus z squared equals 4 we essentially have
two concentric spheres centered on the origin.
Next, let’s go ahead and apply the inequality,
lets rewrite the inequality by taking square
root though out as follows. So we
want the set of all points that are at least
1 and at most 2 from the origin, in other
words we want to shade the region that is
between these two spheres. Including the points
that are located on the surface of the sphere
of 1.
Alright, say that we only want the upper half
of these spheres in other words all the points
that are above the xy-plane how can we accomplish
this? We can denote these points by including
an additional inequality in this case we want
all the points that are above the xy-plane
or all the points that contain a positive
z-coordinate, so we go ahead and include the
inequality z is greater than or equal to 0,
so we can restricted the values of z to take
on positive values only. Doing that we would obtain
the following graph.
Alright, Let’s go over the final example.
Example 7: Write an inequality to describe
the region:
The region inside a cube centered in the origin
with side length equal to 4.
In this problem we are asked to come
up with the inequalities that describe the
given region. We basically want the set of
points that are located inside a cube of side
length equal to 4. We can start by generating
each of the faces of the cube since each face
represents a plane. The top and bottom face
can be obtained by graphing the equation z=2 and z=-2, as follows, in addition since
we want all the points that are between these
two planes we want to set up an inequality
that represents the region between these two
planes we express this region as follows,
next we want an expression that defines the
sides of the cube, the sides that are parallel
to the xz-plane are represented by the equation
y=-2 and y=2 respectively and the inequality
that defines the region between the planes
is expressed as follows, and finally the front
and back faces of the cube are parallel to
the yz-plane and can be obtained by graphing
the equation x=2 and x=-2 respectively
and the region between these two planes can
be expressed as follows, so the inequalities
that define the set of all points inside a
cube centered on the origin with side length
equal to 4 are defined by these three inequalities.
Alright, now that we have laid the foundations
in navigating and maneuvering a three dimensional