Question
Graph several members of the family of curves $ x = \sin t + \sin nt $, $ y = \cos t + \cos nt $, where $ n $ is a positive integer. What features do the curves have in common? What happens as $ n $ increases?
Step 1
These equations represent a family of curves in the plane, where each member of the family is determined by a different value of $ n $. Show more…
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$$ \begin{array}{l}{\text { Graph several members of the family of curves }} \\ {x=\sin t+\sin n t, y=\cos t+\cos n t \text { where } n \text { is a positive }} \\ {\text { integer. What features do the curves have in common? What }} \\ {\text { happens as } n \text { increases? }}\end{array} $$
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Graph several members of the family of curves $x=\sin t+\sin n t, y=\cos t+\cos n t,$ where $n$ is a positive integer. What features do the curves have in common? What happens as $n$ increases?
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Investigate the family of curves given by $f(x)=x^{n} e^{-x}$ where $n$ is a positive integer. What features do these curves have in common? How do they differ from one another? In particular, what happens to the maximum and minimum points and inflection points as $n$ increases? Illustrate by graphing several members of the family.
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