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[Calculus/Analysis] Question about Problem 7a from Chapter 3 of Spivak's Calculus

2022.01.25 17:32 _Radivoy [Calculus/Analysis] Question about Problem 7a from Chapter 3 of Spivak's Calculus

Problem statement:
Prove that for any polynomial function f, and any number a, there is a polynomial function g, and a number b, such that f(x) = (x - a)g(x) + b Hint: a formal proof is possible by induction on the degree of f.
I could prove the basis for induction (so proving for first degree polynomials), but got stuck after that. The solution manual says that (after supposing that the proposition is true for polynomials functions of degree k) let's suppose f(x) is a polynomial function of degree k + 1: (_ followed by stuff is supposed to be indexing)
f(x) = a_(k+1)*x^(k+1) + ... + a_1*x + a_0
Then f(x) - a_(k+1)*(x-a)^(k+1) has a degree less then or equal to k.
So we can state the following:
f(x) - a_(k+1)*(x-a)^(k+1) = (x-a)g(x) + b
Then
f(x) = (x-a)[g(x) + a_(k+1)*(x-a)^k] + b

I get the gist of the proof but I have some questions:

  1. The only reason we're subtracting a_(k+1)*(x-a)^(k+1) from f(x) and not a_(k+1)*x^(k+1), is that in this case we can later factor out (x-a) to get the desired equation, right? a_(k+1)*x^(k+1) still brings down the degree to <= k, but it does not allow for factoring.
  2. In the final equation, g(x) + a_(k+1)*(x-a)^k stands in place of the g(x) from the original proposition. So technically, g(x) + a_(k+1)*(x-a)^k should be considered a "new" polynomial function which satisfies the last induction step?
Thanks in advance.
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2022.01.25 17:32 wiredmat Lost the Turbo 'Kick' - but still have power?

Lost the Turbo 'Kick' - but still have power? Hi, I know this sounds strange and yes I don't know anything about cars.
Bought my GTI 2021 Autobahn Manual in May of last year. Used it daily until the end of July without any issues. I then noticed that in any gear, at any speed I never got the 'Kick' out of the engine ( the so-called Turbo lag a second or two after putting your foot down on the accelerator). My young boys constantly complained about this 'Kick' from the back seats and also stopped their complaints.
So I can step on the gas peddle and there is power to get the full RPMs out of each gear (gradual like in none turbo cars) but again no turbo 'Kick' at any RPM. My none automotive mind tells me there is an issue and it should be fixed before it causes any further issues. Also no dash board lights or strange noises coming from the car at all. It performs like it has from new, minus the 'Kick' from the turbo.
I've taken my wife, my father and some neighbors out in my car to a long stretch of road just outside of my subdivision where you can push 2nd & 3rd gears to there full. Everyone says the same thing "where is the turbo". Again the car has power, its not choking or anything, it can gradually get to top speed just no 'Kick'.
I did manage to get my car back to the dealer just before Christmas (I work outside of the country and was away for five months after I noticed this issue). They tested the car and said there was nothing wrong with it. I insisted they were wrong and took the mechanic and service manager out for a spin around the block. They kept saying "See it has power", but as I said over and over. I know it has power, that isn't the issue, the issue is no 'Kick'. Then came the usual dealer BS and much of it conflicting about what was wrong or how I was wrong, including that it must be the change to 'winter gas' causing this issue. In July that would really be something ...
Is it possible that the flap of the turbo charger could be open and thus air is always rushing in through to the engine? I've read that the usual issues with turbos is this flap not opening causing the engine to choke. Again I don't know anything about cars, but anyone except the dealers people, that have been in my car says there is something wrong.
I'm currently writing a letter to VW head office to see if they can help. And as we are just coming out of Lock-Down again (yes I know Socialist Canada) the boys are back to school and I can get out to other mechanics to have a look at this.
did get winter tires before the first snow fall
Any other suggestion?
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2022.01.25 17:32 Zorrm Requesting /r/puzzle in order to bring together online puzzles (word and number games) into a community that pushes physical, piece puzzles as well as things like Rubik's cubes, as well as drive more attention towards the competitive world in those realms.

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2022.01.25 17:32 mka4X Are you a fan of music festivals?

View Poll
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2022.01.25 17:32 uditgala047 I have a perfect girlfriend but then also I look at porn.....need help from you guys

So basically....i wanna say that i hv a perfect nd healthy relationship....at the beginning i spent 1 month without PMO but one day i looked at it nd since then i hv started watching it more nd i masturbated more....i ssly don't want this addiction to fu*k my relationship nd im feeling so miserable nd idk wht to do....i hv blocked the adult content on my phone but when i get massive urges i just unblock it nd hv a peek....
People who are reading this pls help me....m trying to win over my addiction since 3 yrs nd m feeling so miserable i m feeling so negative pls give me some advice so that i can achieve my goal of 90 days of noPMO....
Thanks for reading this till here :-)
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2022.01.25 17:32 kookookool Shapovalov is the Machine Gun Kelly of tennis.

bratty and spoiled, unlikeable, drops disses that have a few good bars but ultimately trash, given the L by the guy considered the goat by many.
no?
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2022.01.25 17:32 babyodathefirst Thoughts on Young Sheldon?

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2022.01.25 17:32 TeacupGrad His demanding job is making me SO anxious

Hi all!
I (30F) have been seeing him (36M) for six weeks.
We met through a mutual friend one night, clicked immediately, he came on really strong- he reached out to see me five days in a row! It felt a bit like a dream, as I don’t often click with people (I have been single for more than two years after a traumatic breakup).
Here’s the kicker- the last couple weeks he has seen me only once or so per week, and although he texts everyday, it’s just a few quick texts.
He has an extremely demanding job, one that comes with incredibly long/ varied hours. He’s only been on his own in the job for six months, and he is stressed. Like, doesn’t sleep through the night, mumbles about work when he does sleep for an hour or two, etc. I cannot imagine that kind of stress, and I’ve tried to help in the ways I know how- rubbing his back to help him fall asleep, waking him up when he has nightmares and talking with him through the night, I even made him cookies a few weeks ago when he had an especially rough day at work.
I recognize that my feelings/ life need to take a bit of a back seat. I swear I do understand that… but my anxiety is absolutely killing me. We had a big tearful conversation about this last week, and he explained that he was able to see me so much when we met due to it being a slow period at work, and now it’s back to normal, and that isn’t a reflection of how much he likes me, but sometimes he needs a bit of time alone to decompress.
My friends think I should call it off. The thought makes me really upset, but I’m also upset being in this situation- I want to spend time with him, and I hate feeling anxious / insecure.
To his credit, he has been keeping me in the loop via text since our conversation (e.g. this morning I got a text explaining that he was working all night due to an emergency and is just going to bed). I think it’s kind that he’s trying, but that doesn’t help the fact that I was miserable last night because I hadn’t heard from him all day.
Am I the insane person here? Has anyone else navigated dating someone with a demanding job? Tips and tricks?
Thanks in advance for any helpful words :).
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2022.01.25 17:32 Due-Percentage69 I am something of a mistaker myself

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2022.01.25 17:32 Vast_Garage7334 List of independent board game publishers

Hello,
I'm interested in compiling a list of unique board game publishers that break the mold of mass market publishing. Here are a couple I know of that come to mind:

Would love to hear of any others that might fit in that category, I'm also interested in small publishers that have a focus on limiting their carbon footprint. Let me know and I will update this list.
Thanks!
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2022.01.25 17:32 Boing_gaming Dirt 5 Review! Awesome Multiplayer

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2022.01.25 17:32 snodoubts Inazuma to Romance manga subreddit

If someone is looking for a subreddit about this manga, here it is InazumaToRomance
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2022.01.25 17:32 frumpygreasebizcuit I see post quite often asking, "is this arrow safe to shoot?". My LBS has a picture like this hanging on the wall, poster sized, and it says always check your arrows. It is gruesome, but has stuck with me since the day I bought my first bow. When in doubt, throw it out.

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2022.01.25 17:32 jrcl39 Thoughts on Alice Crary?

I recently watched this conference against Efective Altruism and I started reading "Inside Ethics". She supports a view of morality where we empirically discover moral values through sensitivity. She draws from Wittgenstein, John McDowell, Anscombe and Philippa Foot and, apparently, she supports virtue ethics.
I think she has a epistemologically and metaphysically loaded view and her normative ethics doesn't seem to solve much. But I also think I may be missing the point.
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2022.01.25 17:32 mymonet Cute littles things. Nano backpack, EBB, 1/2 zip, lulu hats, restless long sleeve olive

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2022.01.25 17:32 DovahkiinX88 How do i access the e learning needed for the stars?

Is it on fred or campus?
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2022.01.25 17:32 reinnes13 Economics and Retirements of Existing Nuclear Power Reactors

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2022.01.25 17:32 toyhb Wintery Trail

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2022.01.25 17:32 Psychological_Tie257 Save A Teen's Future🙏

Hi everyone, please help me!
Background: I'm originally form the indian subcontinent but grew up here in Italy (I'm 15yo M). My parents are very religious and closed minded who hate the "western" culture
Situation: first, let me tell you about my dad, he's a very hard working person, works 10-11 hrs/day as a cook and almost always very tired. He never physically hurt me, loved me and my sister a lot, bought us good/costly clothes and toys but he used the same jacket, shoes, shirts for years. My dream is to finish my studies and finally give him some rest. I'm very grateful to them but after they noticed that I'm getting aways from Islam and "embracing" the western culture they told me we'll go back to my original country. I can just walk away after 3 yrs but this is not something i want to do, my love for them is unconditonal. I wish I had less money but understanding parents, normal parents. I'm constantly stressed and anxious abt my future, I can't have friends and go out with them and I'll have to go to the mosque and waste hours behind the quran and hadiths to learn.
Sorry it's very long...I really need your help, your advice will be very helpful.
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2022.01.25 17:32 kamihaze ELI5: Why was Shakespeare such a big deal for English literature?

Didnt he just wrote poems or sonnets or stories? What precisely makes him such an important figure in our culture?
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2022.01.25 17:32 dirrtyremixes Kelvin Cedeno - Sanctum [LJR438]

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2022.01.25 17:32 Krabonszcz I drew how it feels like to have a panic attack. Based on my individual situation.

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2022.01.25 17:32 Comedymemecenter WWI American soldier 80% done

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