Question

how can you the parameterization of a circle to findi its maxima and minima extrema?

Answer 1

Not sure about this, but in the case of a unit circle where
x=cos(t)
y=sin(t)
with 0<=t<=2pi
I would determine the maximum and minimum of y=sin(t) then compute the corresponding x values. i.e.
y'=cos(t)
Set cos(t)=0 then determine which value of t fulfills that.
Thus t=pi/2 and t=3pi/2
Now plug in these t values into the x and y equations to get the maximum and minimum respectively.

Again, I'm not sure whether this method extends to other parametrizations, such as a circle in 3D, not centered at the origin.

Answer 2

my circle is at the origin with a radius of 5. when i create an equation, what do i do with it?

Answer 3

circle is:
\[x^{2} + y^{2} = 25\]
max: (0,5)
min: (0,-5)

Answer 4

For \(x=r\cos t,y=r\sin t\) we can determine \(\dfrac{dy}{dx}\) as follows:$$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{r\cos t}{-r\sin t}=-\cot t$$To find extrema, we solve for where our curve is locally flat:$$\frac{dy}{dx}=0\\\cot t=0\implies\cos t=0\\\cos t=0\implies t=\frac\pi2+\pi n\text{ for }n\in\mathbb{Z}$$

Answer 5

For even \(n\) we have maxima, for odd \(n\) minima: