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November 21, 2018, 09:15:16 pm

Author Topic: First Year University Mathematics Questions  (Read 2378 times)  Share 

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Shadowxo

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First Year University Mathematics Questions
« on: September 10, 2017, 11:04:09 pm »
+9
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 »
0
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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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
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RuiAce

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Re: First Year University Mathematics Questions
« Reply #3 on: September 10, 2017, 11:36:19 pm »
+11
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 »

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 »
+4
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 »

legorgo18

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Re: First Year University Mathematics Questions
« Reply #6 on: March 10, 2018, 02:15:10 pm »
0
Hello rui, im really trying my best but these questions just arent clicking for me, ie 32 b,c (not even gonna try d or e with those threatening stars), 33 b, 35 for now!
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RuiAce

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Re: First Year University Mathematics Questions
« Reply #7 on: March 10, 2018, 02:32:10 pm »
+5
Hello rui, im really trying my best but these questions just arent clicking for me, ie 32 b,c (not even gonna try d or e with those threatening stars), 33 b, 35 for now!


Note that I took the positive square root here I'll take the negative square root for 32c. The intuition behind this is that for 32c, I actually want \( y \in (-\infty, x)\), i.e. \( y < x\), so I'd prefer the negative square root for that one. But here I want \( y > x\), and hence i prefer the positive square root here.



And if you look closely, there will always be a bit of the red region overlapping with the green region.

____________________________________________________________________





I'll let you try doing part c now whilst I look at the other questions.

RuiAce

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Re: First Year University Mathematics Questions
« Reply #8 on: March 10, 2018, 03:03:06 pm »
+5
I'll get back to 33. That one is probably the hardest of the bundle you mentioned.


Now, by inspection this will just be 1. But let's argue this properly.



_________________________________________________________________________



(Note that I don't know what happens as \(x \to \infty\), nor do I know what happens as \(x \to 0^+\). GeoGebra does, but I don't.)







i.e. since the max of the function does not occur at an integer, the max of the sequence must occur at one of the two integers, bordering that original number.

(Side note: If the max exists, then the max must equal the sup. Notice how in the last problem the max didn't exist, because we asymptotically approached 1. Here, \( f(x) \) didn't just approach \( 3e^{-1}\), but it actually hit it.)
« Last Edit: March 10, 2018, 03:08:54 pm by RuiAce »

RuiAce

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Re: First Year University Mathematics Questions
« Reply #9 on: March 10, 2018, 06:53:06 pm »
+6
Hello rui, im really trying my best but these questions just arent clicking for me, ie 32 b,c (not even gonna try d or e with those threatening stars), 33 b, 35 for now!


This may come as a surprise. In high school, you were taught that \(f\) is a monotonic increasing function, if for any \( x \in (a,b) \) it satisfied \( f^\prime (x) \ge 0\), where \( f^\prime (x) \) is the derivative. And similarly, \( \le \) for decreasing. This is actually not a definition, but a theorem. The proof of this theorem requires a neat-ass formula known as the "mean value theorem", which you will learn later on.

But the idea is, we will borrow the theorem to "prove" something is monotonic decreasing, but then relate back to the actual definition.
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which, coincidentally, means that \(x\) is NOT in our set. Because our set is the set of all \(p\) such that \( (p+1) \ln x < p \ln (p+1) \). Note that I'm just using \(p\) to not mix up three different \(x\)'s in the same question.


Remark: \(e\) may not necessarily be the least upper bound, i.e. the supremum. In fact, the supremum is actually approximately at 2.293, which is certainly smaller than 2.718

Also - the lower bound for this set is pretty much 0, because you can't have negative elements in that set to begin with. (Reason - the domain restriction of the log function)
« Last Edit: March 10, 2018, 08:56:04 pm by RuiAce »

Mackenzie2000

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Re: First Year University Mathematics Questions
« Reply #10 on: April 05, 2018, 08:07:21 am »
+1
When working with matrices, and determining their geometric description, how do we know whether they are parallel, the same, or have the planes intersect in either a line or at a single point?
I am referring to both questions with three equations in three variable, and questions with two equations, but three variables.

Thank you in advance!!

RuiAce

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Re: First Year University Mathematics Questions
« Reply #11 on: April 05, 2018, 08:45:43 am »
+5
When working with matrices, and determining their geometric description, how do we know whether they are parallel, the same, or have the planes intersect in either a line or at a single point?
I am referring to both questions with three equations in three variable, and questions with two equations, but three variables.

Thank you in advance!!
Essentially, the geometric interpretation of the solutions (and hence the original planes) depends on the nature of the solutions we had found.

In the 3 equation, 3 variable case, we are considering 3 planes in \( \mathbb{R}^3\). So we have the following:
- There is no solution. When this is the case, we're saying that the 3 planes have no common point of intersection. This could be because they appear to form a triangular prism between them, or because at least 2 out of the 3 planes are parallel to one another.
- There is one unique solution. When this is the case, the planes are essentially intersecting at a common point. An easy example of this is just the x-y, y-z and x-z planes, which only intersect at (0,0,0).
- There are infinite solutions. Infinite solutions are characterised by the number of parameters there are in our solution.
   - If there is only one parameter, say \(t\), then the solutions are of the form \( \textbf{x} = a + t\textbf{v} \). This represents a line in \(\mathbb{R}^3\), passing through the point \(A\) and in the direction of the vector \( \textbf{v} \). In general, the intersection forms a lie when the planes are just rotations of one another, but all 3 of them aren't the same plane as in the case below.
   - If there are two parameters, say \(\lambda\) and \(\mu\), then the solutions are of the form \( \textbf{x} = \textbf{a}+ \lambda \textbf{u} + \mu \textbf{v} \). This now represents a plane in \(\mathbb{R}^3\), which passes through \(A\) and is spanned by \( \textbf{u} \) and \( \textbf{v} \). However, this is only possible if the three planes at the start actually all coincided with one another.

I find that figure 1.1.2 of this article gives a good visual representation of them.

On the other hand, when we have 2 equations, we're simply saying we now have only 2 planes in \( \mathbb{R}^3\). Because there's only 2 planes, our possibilities are severely narrowed
- The only way there can be no solutions is if the planes are parallel to one another.
- There can never be a unique solution. (Why?) If two planes in \( \mathbb{R}^3\) are not parallel then they must have infinite solutions.
   - If there are two parameters, then once again the planes coincided.
   - If there is only one parameter, again the intersection is a line. This is essentially every case when the planes aren't parallel with one another.

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Re: First Year University Mathematics Questions
« Reply #12 on: April 05, 2018, 11:19:33 am »
0
Hi,
I was wondering if you are able to solve these questions for me? It would be greatly appreciated!

1) In the triangle ABC, AB = sqrt(2), AC = 1/sqrt(2) and the angle at A is 60◦. Find the length of BC and the size of the angle at C.
2) If sin A = 3/5 where 90◦ < A < 180◦ and cos B = 5/13 where 0◦ < B < 90◦, find sin(A − B)
3) In the triangle ABC, AB = 2, BC = 1 + sqrt(3) and the angle at B is 30◦, Find the length of AC and the size of the angle at C.
4) If sin A = 5/13 where 90◦ < A < 180◦ and cos B = 3/5 where 0◦ < B < 90◦, find sin(A − B).

Thanks!

TrueTears

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Re: First Year University Mathematics Questions
« Reply #13 on: April 05, 2018, 02:38:50 pm »
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1) is just a straightforward application of the cosine rule. 2) and 4) are applications of the sine difference rule. Have you had a go at applying those?

RuiAce

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Re: First Year University Mathematics Questions
« Reply #14 on: April 05, 2018, 02:47:37 pm »
+3
Hi,
I was wondering if you are able to solve these questions for me? It would be greatly appreciated!

1) In the triangle ABC, AB = sqrt(2), AC = 1/sqrt(2) and the angle at A is 60◦. Find the length of BC and the size of the angle at C.
2) If sin A = 3/5 where 90◦ < A < 180◦ and cos B = 5/13 where 0◦ < B < 90◦, find sin(A − B)
3) In the triangle ABC, AB = 2, BC = 1 + sqrt(3) and the angle at B is 30◦, Find the length of AC and the size of the angle at C.
4) If sin A = 5/13 where 90◦ < A < 180◦ and cos B = 3/5 where 0◦ < B < 90◦, find sin(A − B).

Thanks!

Essentially Q1 and Q3 are just high school trigonometry - the main thing you require is the cosine rule (and possibly the sine rule.) Since Q4 follows the exact same as Q2, I will only do Q2.