Given r=f(θ), we are generally not concerned with r′=f′(θ); that describes how fast r changes with respect to θ
You’re using the derivative of a polar equation as the basis for what a tangent line is. But as the textbook explains, that doesn’t give you a tangent line or describe the slope at that point. I never bothered defining what “tangent” means, but since this seems so important to you why don’t you try coming up with a reasonable definition?
I think we fundamentally don’t agree on what “tangent” means. You can use
x=f(θ)cosθ, y=f(θ)sinθ to compute dydx
as taken from the textbook, giving you a tangent line in the terms used in polar coordinates. I think your line of reasoning would lead to r=1 in polar coordinates being a line, even though it’s a circle with radius 1.
Given r=f(θ), we are generally not concerned with r′=f′(θ); that describes how fast r changes with respect to θ
I think this part from the textbook describes what you’re talking about
Instead, we will use x=f(θ)cosθ, y=f(θ)sinθ to compute dydx.
And this would give you the actual tangent line, or at least the slope of that line.
Polar Functions and dydx
We are interested in the lines tangent a given graph, regardless of whether that graph is produced by rectangular, parametric, or polar equations. In each of these contexts, the slope of the tangent line is dydx. Given r=f(θ), we are generally not concerned with r′=f′(θ); that describes how fast r changes with respect to θ. Instead, we will use x=f(θ)cosθ, y=f(θ)sinθ to compute dydx.
From the link above. I really don’t understand why you seem to think a tangent line in polar coordinates would be a circle.
A straight line in polar coordinates with the same tangent would be a circle.
I’m not sure that’s true. In non-euclidean geometry it might be, but aren’t polar coordinates just an alternative way of expressing cartesian?
Looking at a libre textbook, it seems to be showing that a tangent line in polar coordinates is still a straight line, not a circle.
The tangent of all points along the line equal that line
Looks like the back of someone’s head to me, but the picture is kind of blurry
Organic Maps (the app) lets you download maps of various areas ahead of time
food pyramid
You’re really showing your age with this one. The food pyramid got replaced 13 years ago
I don’t know about a distro, but it looks like they are still maintaining an HTPC app for windows, mac and linux. The linux one looks like it was last updated about 2 weeks ago
I know there were a bunch of subreddits with bots that automatically commented Are you looking for this specific thing? on every post, and a lot of posts were answered correctly by those bots.
Not all of it, obviously. But if you want someone else to, you should consider not making them search through a different website to try to find it.
You could at least make it easy and post a link to the pdf
What, like the pinned post on the smyths reddit page?
Looking at the Food & Drink section, it doesn’t look like you can get much more than coffee and meat with monero, so I’m really not sure how you’re supposed to “be your own bank” by using it and still have access to food.
Alright, so monero is good for coffee, maple syrup, and meat bars (?!). I don’t currently eat a lot of any of those things, so I’m not sure how this will help me with my groceries.
Another big example is any web-browser that isn’t safari. That should be changing soon/recently (see here https://www.theverge.com/2024/1/25/24050478/apple-ios-17-4-browser-engines-eu) but that’s only because Apple was forced to.
Unless I’m mistaken, they’re talking about Nausicaä of the Valley of the Wind (1984)
Even the first season had Amon, the guy that wanted equality between benders and non-benders. At one point we’re even shown that power was cut to a predominantly non-bender neighborhood, and when people went outside to protest to get their power turned back on, they were all rounded up and arrested. Afterwards, when Korra goes and tries to get the people that were arrested set free, she’s told
All equalist suspects are being detained indefinitely. They’ll be freed if and when the task force deems them no longer a threat.
Just in case it wasn’t clear enough by that point that non-benders were treated as second class citizens.
Looks like this one https://en.wikipedia.org/wiki/Superpermutation based on the content of the wikipedia article