I feel very confused when teaching students how to sketch a parametric curve like

$$x(t)=e^{-t}t; y(t)=t^{2}+t.$$

Here students are supposed to know derivatives, increasing-decreasing,...

In literature, I haven't found any example for this type of problems.

  • 1
    $\begingroup$ There are several videos online illustrating techniques for sketching parametric curves. E.g., Krista King. Also webpages, e.g., shmoop.com. $\endgroup$ – Joseph O'Rourke Oct 2 '19 at 11:39

In general there is no standard method for sketching parametric curves unless the paremeterization has some particular form (which the example considered here does not, so far as I can tell).

Sketching a parametric curve is like graphing a function. Actually, it directly generalizes graphing a function, because the graph of $f(t)$ is the trace of the parameterized curve $(t, f(t))$. To graph $f$, one finds its zeros, its asymptotic behavior, its critical points, etc. To sketch a parameterized curve, one does the same with each coordinate function.

In this example $(x(t), y(t)) = (e^{-t}t, t^{2} + t)$ one should focus on the simple and qualitative features. First, when $t = 0$ the curve passes through the origin. When $t \to \infty$, $x(t) \to 0$ and $y(t) \to + \infty$, while when $t \to -\infty$, $x(t) \to -\infty$ and $y(t) \to + \infty$. In this case $y(t)$ has a unique critical point, when $t = -1/2$, which is a local minimum, and $x(t)$ has a unique critical point, when $t = 1$, which is a local maximum. Also $x(t)$ is positive for positive $t$, and $y(t)$ changes sign at $t = 0$ and $t = -1$. Thus the curve passes through the origin, $(-e, -1)$, $(-e^{1/2}/2, -1/4)$, and $(e^{-1}, 1)$, crossing the horizontal axis at $(-e, -1)$, staying always above $(-e^{1/2}/2, -1/4)$ staying always to the left of $(e^{-1}, 1)$, and staying always to the right of the vertical axis when $t > 0$, and one knows how it behaves as $t \to \pm \infty$, one knows that $x(t) > 0$ when $t > 0$. All this information is enough to make a qualitatively correct sketch of the curve.

In general there is not much else one can do but:

  • Look at where the curve goes for particular values of $t$.
  • Find the asymptotic behavior of each coordinate function.
  • See where the coordinate functions change signs.
  • See where the coordinate functions have critical points, and find the types of these critical points.
  • Look at convexity of the coordinate functions.
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When I introduce parametric curves in Calculus 3, I like to bring my Etch A Sketch to class. It is a drawing toy from about 1960. The knob on the left encodes $x(t)$, and the knob on the right $y(t)$. A coordinated person can compose a circle. This proves to be a memorable segue to a short but longer discussion about Lissajous figures. There are many tactile ways to experience Lissajous figures, and good apps on the web to explore them.

The Etch A Sketch seems rather silly and retro, but when I meet former students a few years after the class, they always remember this. One told me that when he took a physics class a year later and used and oscilloscope in a lab, he immediately thought of the Etch A Sketch and understood the oscilloscope device.

Etch a Sketch picture from Wiki

Of course, in class one needs to move on quickly to the calculus of parametric curves with things like $\frac{dy}{dx}=\frac{\frac{dy}{dx}}{\frac{dx}{dt}}$ and $ds=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\, dt$. But there is lots of fun physics lurking under parametric curves, so taking some time to plant a future memory connection is fruitful.

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This is a standard part of the calculus course. Most books will have an explanation, examples, and problems. (Which is how to learn the topic, or any topic: explanation (or general case), examples, drill.)

I looked at a year old text (Granville) and it has this:

http://djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf (page 79-83).

But so does Thomas Finney, Swokowski, etc. Just go to the library and browse some different texts. (Of the three I hold, I like the 80s Thomas Finney the best...most applied. But Granville emphasizes curve tracing more and Swokowski emphasizes ideas about smoothness and uniqueness more.)

Perhaps you need to sit down and work some of the problems yourself, to get more comfortable teaching it.

P.s. Not sure what you mean by "literature" (on teaching or on the topic, itself). But if you're looking for applications, the common one is kinematics (e.g. artillery shell physics problems).

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I once taught as shown below (before computer graphics were widely available), but have stopped. Beyond making a table of points and connecting the dots and knowing a few basic examples (circle, cartesian graph, some polar-coordinate graphs), such intricate skill with sketching parametrized curves seems to add little to the education of the students (imo & my department's).

Anyway, there's a 2 x 2 grid of graphs, starting from the top left and proceeding clockwise: y vs. t, y vs. x, t vs. x, t vs. t. Note that the t-axis in the top-left is horizontal and in the bottom-right is vertical. Then pick a t value in the bottom left. Use a ruler or the edge of a piece of paper to find the x, y coordinates in the bottom-right, top-left plots (respectively), and transfer them to the top-right.

enter image description here


Mathematica code for the graphics (may be run on wolframcloud.com, but dynamic updating is run server-side and performs slowly in a browser):

ClearAll[x, y, t, pr, plot];
With[{x = Exp[t] t, y = t^2 + t, tdom = {t, -2, 2}},
 plot[a_, b_, t0_] := 
  ParametricPlot[{a, b}, tdom, PlotRange -> {pr[a], pr[b]}, 
   GridLines -> {{a /. t -> t0}, {b /. t -> t0}},
   GridLinesStyle -> LightRed,
   Epilog -> {Red, Point[{a, b} /. t -> t0]}];
 pr[t] = {-2, 2}; pr[x] = {-1, 3}; pr[y] = {-1, 3};
     xt = plot[x, t, t0],
     ty = plot[t, y, t0],
     tt = plot[t, t, t0],
     xy = plot[x, y, t0]},
      Inset[ty, {0, 6}],
      Inset[tt, {0, 0}],
      Inset[xy, {6, 6}],
      Inset[xt, {6, 0}]
   {t0, -2, 1}]
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I find it very useful to draw random functions of $x$ vs $t$ and $y$ vs $t$ on two distinct sets of axes, and then have the students try to draw the parameterized curve in the $xy$ plane. This forces them to think about what they are doing in a very direct way, without any algebra or function notation getting in the way. I spend a considerable amount of time on this activity (1 or 2 50 minute lessons if possible, and I include such a question on the exam).

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