1. Find the limit as . Does anyone see anything that doesnt make it zero easily enough? Ive tried a few, but maybe I'm missing something.
2.Find the absolute extremes of
Someone help me set this up?
3.Evaluate: Integral of region C Mdx + Ndy for from (0,0) to (2pi,0).
Looks like unit circle. I dont know how to set it up.
We must have input our posts around the same time, because yea I saw what that path makes it.
So for #2, I got critical points of (0,0) (0,1) (0,-1) (0,-2) (2,1) (2,0). But do I disregard (0,-1)&(0,-2) since they are outside my boundary?
Either way, the values I got are 0,4,6. So would 0 be my minimum and if so in my answer would I say that the function has absolute min. at both (0,1) and (0,0). Or would 4 be my min at (2,0)?
And 6 would be my abs. max. at (2,1)?
Does this look right?
is a contour, so I suspect it is a contour integral that's wanted.
The contour, , is a single arch of a cycloid.
Find and in terms of from the contour equations.
Take and from the contour equations and plug them into the equations for and . The resulting integral looks like it would be messy.
Alternative: Use Green's Theorem:
, where is a closed path around region, .
Break up into path as defined above, and path , which goes from to along the -axis. along . Also, along . So the integral along contributes nothing to the integral around .
Now, compute: .
This will also take some work. Good Luck.
Not thinking on #2, whoops.
So idk if this is right, but now I have (-2,1)(-2,-1)(2,-1)(2,1)(-2,2)(2,-2)(1/2,-1)(-1/2,1) as my points to check. Does this look correct?
If so, my values output are 6,2,-1/2, and for (-2,2)&(2,-2)= 0 . But I would exclude those last two values?
Someone check my work?
How do we find the extrema for over some bounded closed region, Yes, look for critical points.
Critical points occur at locations within the interior of where and , or where these partial derivatives do not exist. They also occur along the boundary of , where directional derivatives along that boundary are zero, or don't exist. Take all of that into account, and find the minimum & maximum values of at all of the critical points.
. Setting this equal to zero gives: . . Setting this equal to zero gives: . This gives a critical point at the origin.
On the boundary : , so . So, is a critical point. Similarly, is also a critical point.
On the boundary : , so . So there are no critical points here, except at the end points.
Critical points are thus:
Evaluate at each of these points to find the extrema.