We're asked to graph each
space curve given by the vector functions.
We will be using the 3D Calc Plotter JavaScript App
from the Exploring Multivariable Calculus Site.
As we graph these space curves, we should be thinking
about what it is about the equations of the x, y, and z
components that make the curves behave as they do.
Let's graph our first space curve.
Once the app is open, this is always
the default surface that is graphed.
To clear the plot, we can click this button here,
but because we'll be entering space curves, let's close
this entire window by checking the x on the right.
And now to the right of Add to Graph,
we click this arrow and select Space Curve r[t].
Now we enter the x, y, and z
components of each vector function.
For the first space curve, x[t] is equal
to four times cosine t, y[t] is equal to
four times sine t, and z[t] is equal to t.
Here we can change the interval of t,
change the number of steps, and down at the bottom,
we can adjust the window manually.
We'll talk about this animation tool in just a moment.
To graph the first space curve, we click the box on the left
and notice how the space curve is graphed on the right.
Let's change this space curve to a solid color.
To do this, we check the box Use Constant Primary Color.
Notice now the space curve is red.
If we don't like red, we can click
the red bar and change the color.
We can see we need to adjust the window.
We can do this manually down here,
or zoom in or zoom out using the + or - on top.
Let's zoom out by clicking the -.
Now we have a nice graph of the space curve.
This is a helix.
Now let's talk about this animation tool.
We can show the vectors that trace out the space curve.
To do this, we first click this circular gear.
And let's go ahead and show the trace point, show the trace
vector, show the velocity vector, which is a tangent vector,
but let's uncheck Show Acceleration Vector.
Let's close this, and now if we click Animate,
that was a little fast, let's go back.
Also notice how we need to see more of the curve.
The interval for t changed from negative 16 to 16,
so let's click Graph so we see more of the space curve.
And now if we drag this blue point, we can see
the blue vector that traces out the space curve.
We can also see the black tangent vector or velocity vector.
Again, this space curve is a helix.
Let's graph our second space curve,
which is given by x[t] equals cosine t,
y[t] equals sine t, and z[t] equals cosine four t.
Click Graph, we can see from the graph,
we'll need to zoom in by clicking the +.
Now we have a nice graph of our second space curve,
but notice how it's a little jagged.
We can make this more smooth by
changing the number of steps.
So let's change the number of steps to 200.
And because we zoomed in, the interval of t has changed.
Let's change the interval of t from negative 10 to 10.
And let's press Graph, now we have
a much better graph of the space curve.
Let's show this space curve traced out.
Starting on the left we'll drag the blue point to the right.
Again, the blue vector is tracing out the space curve, and
the black vector is the tangent vector, or velocity vector.
Let's graph our third space curve,
which is given by x[t] equals t times cosine t,
y[t] equals t times sine t, and z[t] equals t.
Let's click Graph, let's zoom out by clicking -,
and now we have a nice graph of our third space curve.
If we do want to save this graph, we can click
these three horizontal bars, click File,
click Save the Plot as a PNG.
If we want to see more of the space curve,
again we can change the interval of t, shown here.
And we can trace out the space curve
again using the animation tool.
Notice right now, the terminal point of this blue vector
is not on the space curve, and that's because
the interval of t changed when we zoomed out.
So we need to click Graph again to see more of the graph.
Our next space curve is x[t] equals t,
y[t] equals t, and z[t] equals t times sine t.
Click Graph, looks like we have a pretty good window.
Here's the fourth space curve.
And again, we can trace it out using the animation tool.
Now our last space curve is given by x[t] equals sine t,
y[t] equals cosine t, and z[t] equals sine t.
Let's go ahead and click Graph, and let's zoom in.
Notice how we have a circle in space.
When we zoomed in, the interval of t changed,
so if we want to trace the entire space curve,
we need to change the interval of t.
Let's change the interval of t from negative seven to seven,
click Graph again, now we can trace the space curve.
I hope you found this helpful.