Notes:
Stuff
 Achieve due on Wed.
 Group work Friday TUT.
Random Notes
Constrained Optimization.
 Finding extrema of $f(x,y)$ where $(x,y)$ are subject to the constraint $g(x,y)=K$.
 We can imagine plotting the level curves of $f$ and tracing out the single level curve $g$ (cuz we want the level curve where $g=K$).
 The points where the curves of the two functions intersect are points that satisfy the constraint as well as lie on $f$.
 Let’s say we want to maximize $f$. So we want our $g$ curve to come as close as possible to the center of $f$‘s level curves.
 This happens at the point where the two curves are tangential.
 Because at that point of intersection, there’s no way for $g$ to get closer to $f$ without breaking out of the $g$ curve.
 If we were minimizing, we would look for a point furthest away from the center. But the lowest possible would still be a tangential point of contact.
 Since tangential, the directional derivatives of both curves would be parallel (or antiparallel) (since directional derivatives are perp to level curves).
 We include the second case because the constraint may be satisfied at a point which is at the local extrema of $g$.
 $λ$ is allowed to be zero cuz the point may be over $f$‘s local extrema.
Double Integrals
 Single: area; double: volume; n: n+1 dim analog.
 Along with splitting the curve into n dxs, we also split the region into dys. da = dx.dy
 Volume = area . height
 If we have a nice rectangular region, it’s basically like two nested forloops. The order of iteration (across columns or down rows) doesn’t matter. We can swap dx, dy and swap the limits.
 But with other shapes, it depends.
Type 1 Region
 Defined by two vertical lines, and two horizontal curves (in the xy plane view).
 Here we split xaxis into dxes first, cuz x always ranges from a to b.
 Each vertical split is then split into n dys. The limits of y are the two curves defining the region.
 Since we are iterating over x first, it’s the outer integral.
 Here we can’t simply swap. Swapping would be much more complex here.
Type 2 Region

Same as above, but sideways.

Defined by two horizontal lines, and two vertical curves (in the xy plane view).

Now we first iterate over y (outer integral) and for each horizontal segment, we chop it into square elements.

It doesn’t need to have a straight line to be defined as type 1/2 region. Sketch the region, use logic to split it up into better shapes then integrate over them to find volumes.