Polar-coordinate Problem

natHenderson

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I don't even know where to begin with this problem, can anybody help?
 

Subhotosh Khan

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I don't even know where to begin with this problem, can anybody help?
If I were to do this problem, I would first use wolframalpha.com to plot:

r = f(Θ) = Θ^3 - 6*Θ^2 + 9*Θ

That will let you analyze the position of the particle.

If you get stuck somewhere, come back and tell us exactly where you are stuck....
 

Otis

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Hi natH. The distance from the Origin to a point on the polar curve is r. We're told that r-values are the same as f-values. We're told that t is the same number as θ. Therefore, distance r is a function of time:

r(t) = t^3 - 6t^2 + 9t

Do you know how to use calculus, to find the intervals where distance r(t) is increasing, while 0 ≤ t ≤ 2pi seconds?

😎
 

natHenderson

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Ok, so I figured out how to find the interval where it's increasing by using the derivative, is D the correct answer?
 

Otis

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… is D the correct answer?
Yes. A function increases when its first derivative (rate-of-change, or slope) is positive.

The first derivative of r is an easily-factored quadratic polynomial.

So, it's easy to determine intervals where the parabola lies above the horizontal axis. That serves as a check, for answer D.

😎
 
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