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?
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, I’ve 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?
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.
Started elsewhere: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 it’d be ridiculously small and thus negligible. (X is meant to be tiny e.g. 10^- 8, hence why it’d be negligible. At least in the context where I got these from - a genetics book. See below.)Spoiler
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.
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!
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!
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?Essentially, the geometric interpretation of the solutions (and hence the original planes) depends on the nature of the solutions we had found.
I am referring to both questions with three equations in three variable, and questions with two equations, but three variables.
Thank you in advance!!
Hi,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.
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!
Question: Find maximal domain and range for f(x)= root(1-2sinx).
I got for range f > or equal 0, then domain x < or equal pi/6 or x> or equal 5pi/6 for 0< or equal x< or equal pi. But when i graph it out using graphing calculator, graph looks really weird, so how are you meant to do this one?
Hi can someone please help me. How can I find a bijection between [0,1] and [0,1)?Is this a first year question? What subject and uni is it?
Hi,
Could I please get some help on this question?
Thanks
Hey,The one thing that you might not fully understand due to limitations of the HSC is what a "position vector" is yet. For the sake of these proofs we can assume that the space we're working in is some Euclidean space, and to simplify things further we can assume it's just \( \mathbb{R}^2\) or \(\mathbb{R}^3\). (You can interpret these as the usual 2-dimensional Cartesian plane, and the 3-dimensional Cartesian space. The 3D version basically also has a \(z\)-axis.)
It took me a while to find this question page but I just have this question and was wondering how you do it via vectors. I didn't really understand any of the online working out!
Prove using vector methods that the midpoints of the sides of a convex quadrilateral form a parallelogram.
Hey, I just have two questions, one of which had been posted on the forum previously. I'm a bit confused about the difference between the two solutions of 32b and 32c if done algebraically. I read and understood the solution for 32b but I'm not sure what the difference the new domain would createFirstly note that your rearranged condition is off. You're right about that \(y^2 < x^2 + 1\), but this quadratic inequality solves to give \( \boxed{-\sqrt{x^2-1} < y < \sqrt{x^2+1}} \). On Desmos, you can just type the original condition \(x^2-y^2 < 1\) and the correct plot will still show.
for c) x will have to be smaller than y, right, so if we compute it out, wouldn't it still give up the same algebraic solution as b) did? or would the signs be flipped as we are considering it from the negative side?
I also tried to graphically solve it and have attached it. Could you see if I'm on the right track?
My second question is also attached. I'm just a bit unsure of how it works though I get why theta=0 is a solution ofc.
It took me a while but I actually fully understand that so thank you for your extremely in-depth explanation :)(P.S. This is a generic thread and not just for UNSW students. Not everyone will know about the MapleTA quizzes.)
I was trying to figure out how to do this question on the matlab quiz, but I'm confused on what we do regarding the exponential/logarithm form.
Express the rational function (27x^3+1)/x^3/2 in terms of a hyperbolic function, logarithms and other functions as needed, for x>0.
(P.S. This is a generic thread and not just for UNSW students. Not everyone will know about the MapleTA quizzes.)
Assuming that there's no typo. Also assuming that the 3/2 is in the bottom power - please make this a bit clearer in the future.
\[ \text{Let }\boxed{e^t = \sqrt{27}x^{3/2}}\implies \boxed{t = \ln \left(\sqrt{27}x^{3/2} \right)}.\text{ Then,}\\ \begin{align*}\frac{27x^3+1}{x^{3/2}} &= \tag{clever factorising}\frac{27^{1/2} x^{3/2} \left(27^{1/2} x^{3/2} + 27^{-1/2} x^{-3/2} \right)}{x^{3/2}}\\ &= \sqrt{27} (e^t + e^{-t})\\ &= 2\sqrt{27} \cosh t\\ &= 2\sqrt{27} \cosh \left( \ln \left(\sqrt{27}x^{3/2}\right) \right) \end{align*} \]
Optionally, we may use log laws to re-express what's on the inside as \( \frac32 \ln (3x) \).
Note that the factorisation was the critical step. In general, because \( \cosh t = \frac{e^t+e^{-t}}{2}\), we need to have some \( f(x) + \frac1{f(x)}\) pattern appearing, before we can sub \(e^t = f(x)\). Same goes for \(\sinh t\) except we have a minus instead.