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December 12, 2017, 09:12:08 pm

Author Topic: First Year University Mathematics Questions  (Read 410 times)  Share 

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Shadowxo

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First Year University Mathematics Questions
« on: September 10, 2017, 11:04:09 pm »
+7
Here's a place to ask all your first year uni mathematics-related questions!
I couldn't find any others so I thought it would be a good idea to make one :)
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Shadowxo

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Re: First Year University Mathematics Questions
« Reply #1 on: September 10, 2017, 11:06:41 pm »
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I'll start it off with a question of my own:
Do the vectors in the basis of the row space of a matrix + the vectors in the basis of the nullspace / solution space of a matrix make up a basis of Rn, where n is the number of columns in the matrix?
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Sine

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Re: First Year University Mathematics Questions
« Reply #2 on: September 10, 2017, 11:12:26 pm »
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I'm first year uni but doing a 2nd year maths unit LOL, i'll probably be using this thread soon. Great idea Shadowxo

RuiAce

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Re: First Year University Mathematics Questions
« Reply #3 on: September 10, 2017, 11:36:19 pm »
+8
I'll start it off with a question of my own:
Do the vectors in the basis of the row space of a matrix + the vectors in the basis of the nullspace / solution space of a matrix make up a basis of Rn, where n is the number of columns in the matrix?

This can be achieved by simply changing the dimensions of the matrix.


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Handwavy - Some results are assumed trivial and left as an exercise. Also potentially poorly explained with my 11:36PM dead brain.
« Last Edit: September 11, 2017, 04:25:45 am by RuiAce »
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AngelWings

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Re: First Year University Mathematics Questions
« Reply #4 on: December 07, 2017, 09:40:42 pm »
0
Started elsewhere:
Spoiler
Still need help with Taylor series/ approximations here. Learnt it back in first year maths (MTH1030) and need to revise this. Forgotten most of it. Mostly I just need a proof and how it works again. I've also forgotten mostly about limits, so yeah... that'd be great if you could help. :)
Which parts were you expected to prove? The existence of the \(k\)-th order Taylor expansion, or that if remainder -> 0 then f is represented by the Taylor series?
To be honest, Ive forgotten some of the basics. The most common one in these books, after a bit of dissecting, incorporates Taylor series on e^x, giving approximately 1 + x + x^2 + x^3 +... Not so sure how we got from Point A to B and would just like to see how to do it again, how we can prove this and so forth.

Did you mix some of them up?
That is entirely possible considering the only line of working I have been given is: e^x is approximately 1 + x where any term of order 2+ is ignored because itd be ridiculously small and thus negligible. (X is meant to be tiny e.g. 10^- 8, hence why itd be negligible. At least in the context where I got these from - a genetics book. See below.)
Intending on a theoretical genetics project for Honours, which involves some first year math, parts of which my memory stalls on. After previous experience, my intended supervisor advised that during this break, I should go through two genetics books. Both of them indirectly expect you to use Taylor approximations, which I can't remember how it works or how to do them. Hence the revision.   

I got that answer because my notes are old and written. I must've been haphazardly copying them down quickly during the lectures. Must've missed the factorials.
Still not quite so sure how we would get the one your wrote for e^x above though, but maybe it's because I've forgotten large chunks of content.
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RuiAce

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Re: First Year University Mathematics Questions
« Reply #5 on: December 07, 2017, 10:00:25 pm »
+3
Started elsewhere:
Spoiler
That is entirely possible considering the only line of working I have been given is: e^x is approximately 1 + x where any term of order 2+ is ignored because itd be ridiculously small and thus negligible. (X is meant to be tiny e.g. 10^- 8, hence why itd be negligible. At least in the context where I got these from - a genetics book. See below.)
I got that answer because my notes are old and written. I must've been haphazardly copying them down quickly during the lectures. Must've missed the factorials.
Still not quite so sure how we would get the one your wrote for e^x above though, but maybe it's because I've forgotten large chunks of content.


Optionally you may also memorise the one for \(\ln(1-x)\), but that can be obtained by integrating \( \frac1{1-x}\).
Remark on first order approximations
This means, that for small \(x\), the following are reasonable:
\begin{align*}e^x&\approx 1+x\\ \sin x &\approx x\\ \cos x&\approx 1\end{align*}

These four are generally regarded as the most useful. The first one really isn't anything fancy as it's literally just the geometric series, but exponentials and sinusoids appear quite common naturally. All the other stuff either stem out of these (e.g. \( \sinh\)), are inverses of these (e.g. \(\tan^{-1}\)) or are just random shit mathematicians use for convenience.


We omit justification as to why that is the case for the sake of mere computations.

So all we really need to compute is \( f(0), f^\prime(0), f^{\prime\prime}(0), f^{\prime\prime\prime}(0), f^{(4)}(0) \) and pretty much, all of the derivatives of \(f\), evaluated at 0.
The actual computation starts here.


Expanding this sum out gives \(1 + \frac{x}{1!}+\frac{x^2}{2!}+\dots\) as required.

Small note - The factorials basically appear because they're actually a part of the Taylor series formula.
« Last Edit: December 07, 2017, 10:14:44 pm by RuiAce »
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