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# The view of the trefoil knot shown in Figure 8 is accurate, but it doesn't reveal the whole story. Use the parametric equations$$x = (2 + \cos 1.5t) \cos t$$$$y = (2 + \cos 1.5t) \sin t$$$$z = \sin 1.5t$$to sketch the curve by hand as viewed from above, with gaps indicating where the curve passes over itself. Start by showing that the projection of the curve onto the $xy$-plane has polar coordinates $r = 2 + \cos 1.5t$ and $\theta = t$, so $r$ varies between 1 and 3. Then show that $z$ has maximum and minimum values when the projection is halfway between $r = 1$ and $r = 3$. When you have finished your sketch, use a computer to draw the curve with viewpoint directly above and compare with your sketch. Then use the computer to draw the curve from several other viewpoints. You can get a better impression of the curve if you plot a tube with radius 0.2 around the curve. (Use the tubeplot command in Maple or the tubecurve or Tube command in Mathematica.)

Vector Functions

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##### Lily A.

Johns Hopkins University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

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### Video Transcript

The problem is use parametric equations. Ex according to classical Sign one point up to your society. Isaac, go to classical science One point up, Tio have society c is equal to sign one point after just sketch the curve. I hand as wayward from a half with caps indicating wise curve has it over itself standing by showing that the protection of the curve on X Y plain has polar coordinates are record two two plus call sign one point after on DH there has the quantity and then show that the has maximum and minimum. Wallace won. The protection is halfway train. One, two, three first Look at this graph. Figure eight. Yeah, So figure eight. This's a graph. Now we're sketched this graph by hand, so it looks like tres there the projection of the curve onto X y plain. So behalf if they use polar coordinates Are you going y squared? Negative. X square plus y square. With this. This can't you two plus sign one point five. Here. We used the fact scientists going plus scientist square using one. And they are you sicko to have tended to steer as you could. Why over axe this isn't a good tenant. Become a lot. Did I sink too? You? So are you. Virus Train one hundred three. Then showed that the hats. Mike's mom. One minimum values. Why? The projection is halfway between are you could want on our equal to three. Look at it. The equation of C. C You sneak off to sign one point t one assign one point five two. Yes, your conscience zero with no sign. One point lt has maximum minimum values Want or negative one In this case, our physical too. This's halfway between. I put one on article two three and now we use a computer to gather graph with different points. Look up here. This is a graph. So B changes were points.

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#### Topics

Vector Functions

##### Lily A.

Johns Hopkins University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

Lectures

Join Bootcamp