Purposes:
To experiment with a beautiful example of geometry in nature;
To study an important family of polar curves.
Prerequisites:
The helper application tutorial for your computer algebra system;
An understanding of semilog (and log-log) graphing;
An understanding of polar coordinates and their relationship to Cartesian coordinates;
Some familiarity with parametric representations of curves;
The Chain Rule.
Equipment:
A printer -- useful, but not required.
We have included this module in our Multivariable Calculus collection because that's where our students take up polar and parametric representations. However, students who already know about these representations could do this module in Differential Calculus.
We recommend that the instructor provide a printed copy of the figure in Part 3 for each student team, although it is possible to take the measurements on a monitor screen. We also recommend metric rulers, if available. Students using English units tend to get wrapped up in sixteenths and thirty-seconds rather than using short decimal approximations.
A caution: There is a tendency for several of the smaller radii in different directions to come out the same -- that is, to have three or four data points on a horizontal line. This is not "wrong," in spite of the visual evidence that r is growing -- it's just an artifact of the relatively imprecise measurements at the start. For the larger measurements, this is not a problem. Students should be encouraged to determine their parameters from measurements in which they have more confidence.
Students will undoubtedly notice that we have provided a complete set of data. However, our experience has been that they prefer to use their own data -- and they certainly get more out of the module if they have that sense of "ownership" of the problem.
Another caution: Students who have used logarithmic plotting to find formulas will generally remember -- or be forced to remember, if their formula doesn't produce a reasonable graph -- that the semilog or log-log plot is showing them logarithms of the numbers in the data. Thus, in Part 3 they have to use logarithms to find the slope of the line as one of their parameters. However, they may be confused about whether to use logs base 10 or base e. The logarithmic plotting routines always use base 10 -- but it doesn't matter as far as the linear shape of the graph is concerned. It does matter when you exponentiate, however -- the bases have to match. In Maple, for example, this point is obscured -- students type log, which means ln to Maple, and when they define a function by exp, it comes out right. On the other hand, in Mathcad, log and ln are distinct functions, so one must either use ln to get the slope and exp to define the function or use log and 10^. If there is a mismatch, it shows up in step 5 of Part 3 as a reasonable-looking curve that just doesn't line up with the data points.
Our editor, Jerry Porter, has observed that calculating the angle between radius and tangent by using the dot product is a nice linear algebra problem. That could also be done in a multivariable calculus course, where dot product is a standard topic. We have not included this in the module because our students do this module before any vectors appear in the calculus course, and few if any have taken or are taking a linear algebra course. However, this would make a nice follow-up homework or test problem once the dot product has been introduced.
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