Analyzing a Smooth Curve for -π < t < π

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The discussion focuses on determining the smoothness of the curve defined by R(t) = (4sin^3(t), 4cos^3(t)) for -π < t < π. Participants clarify that while sine and cosine functions are inherently smooth, the smoothness of R(t) must be assessed by examining its derivatives. Concerns are raised about points where the denominator of any derived fraction could approach zero, potentially indicating discontinuities. The definition of a smooth curve is reiterated, emphasizing that it should not have any abrupt changes or corners. Overall, the thread seeks to clarify the conditions under which R(t) remains a smooth curve within the specified interval.
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Homework Statement



Determine where r(t) is a smooth curve for -pi <t<pi
R(t)= (x(t),y(t))=(4sin^3(t), 4cos^3(t))

Homework Equations





The Attempt at a Solution



To be honest I have no idea where to start. I know what a smooth function is but my understanding is that the sin(t) and cos(t) functions over all of t are smooth. No corners.
Any starting help would be appreciated.
 
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Yes, but you are NOT asked if sine and cosine are smooth- you are asked if F is smooth. What happens if the denominator of a fraction goes to 0? What fraction is involved here?
 
The function is not continuous at that particular point that makes the denominator go to zero.
Perhaps we could rewrite in the complex plane?
 
No, it is not necessary to work with the complex plane. What is the definition of "smooth curve"?
 
As someone on mathstackexchange said, a smooth curve is a curve with no stubble, like this: :bugeye:
 

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