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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?

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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

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2022.01.25 17:32 babyodathefirst Thoughts on Young Sheldon?

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?

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2022.01.25 17:32 Vast_Garage7334 List of independent board game publishers

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

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.

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