Post by: TheAspiringDoc on March 16, 2015, 08:42:18 pm
I'm trying to get an intuitive grasp of the derivatives and antiderivatives of trigonometric functions.. Any idea's how?
Also, Find the length of the largest square that will fit inside a regular hexagon of edge length 30cm?
Post by: keltingmeith on March 19, 2015, 10:52:36 pm
Ok, thought my math's posts were a bit out of place in the mainstream question threads so I thought I'd make my own :)
I'm trying to get an intuitive grasp of the derivatives and antiderivatives of trigonometric functions.. Any idea's how?
Try it by sight - sketch the graph of sin(x), and use the major points on it to see if you can figure out what its gradient is at each point, and then sketch it out.
Another thought - sin(x) is cyclic (or periodic), so its derivative should be cyclic, no?
Post by: kinslayer on March 19, 2015, 11:42:37 pm
I'm trying to get an intuitive grasp of the derivatives and antiderivatives of trigonometric functions.. Any idea's how?
Also, Find the length of the largest square that will fit inside a regular hexagon of edge length 30cm?
Picture the unit circle and suppose there is a particle travelling along it. Its y-position is sin(t) and its x-position is cos(t), where t is the particle's angular distance from the positive x-axis.
Now think about what happens to the particle's velocity as it travels around. In particular, when the particle starts at (1, 0), its x-position is hardly changing at all (actually a maximum), while its y-position is changing fast. As it reaches the top, pi/2 radians later, the x-position is now changing fast and the y-position is hardly moving at all.
If you continue around the circle you can see that maxima/minima of the sine and cosine functions should occur at each others' zeroes, so you shouldn't be surprised that they are each other's derivatives (up to multiplication by -1).
For the hexagon thing, see this page:
http://www.drking.org.uk/hexagons/misc/deriv3.html
Post by: TheAspiringDoc on March 28, 2015, 07:18:03 pm
Also, am I right in thinking that standard deviation can only be applied to a normal distribution?
and one more thing, does the VCE at anytime use the n-1 approach for samples of the population when calculating standard deviation?
Thanks ;)
Post by: 99.90 pls on March 28, 2015, 08:01:13 pm
Post by: keltingmeith on March 28, 2015, 08:07:06 pm
Can someone please explain to me in simple terms how you would go about integrating the volume of a torus (doughnut)?
Define its cross section as an ellipse/circle, centred by the distance of the difference by the major and minor lengths. Then, the volume of the torus is a simple volume of revolution around the x/y axis (depending on how you've centred it).
Also, am I right in thinking that standard deviation can only be applied to a normal distribution?
No - you are very, VERY, wrong. Standard deviation is only the second centred moment, and many different distributions have defined moments. (in fact, it's often harder to come up with an counter example).
In fact, the exponential distribution with parameter
and one more thing, does the VCE at anytime use the n-1 approach for samples of the population when calculating standard deviation?
What do you mean by the "n-1" approach...?
Post by: TheAspiringDoc on March 28, 2015, 08:15:44 pm
Quote
A random sample drawn from some larger parent population (for example, they were 8 marks randomly chosen from a class of from a class of 20), then we would have divided by 7 (which is n−1) instead of 8 (which is n) in the denominator of the last formula, and then the quantity thus obtained would be called the sample standard deviation. Dividing by n−1 gives a better estimate of the population standard deviation than dividing by n. This is known as Bessel's correction.
Post by: kinslayer on March 28, 2015, 08:26:54 pm
Wikipedia:
That's talking about the sample variance, which is an estimator of the population variance. The correction is there to ensure that the estimator is unbiased. It is also called a "degrees of freedom" correction, because the use of another estimator (i.e. the sample mean) reduces the degrees of freedom by 1.
http://en.wikipedia.org/wiki/Degrees_of_freedom_%28statistics%29
And as Eulerfan101 pointed out, the standard deviation is a general concept, not limited to the Normal distribution. However, the Normal distribution has the characteristic that one of its parameters is its standard deviation/variance (the other being the mean) which is quite nice and not true of other common probability distributions.
The volume of the torus can be found in several ways, probably the easiest is by rotating a solid of revolution about the y-axis:
http://dean.serenevy.net/teaching/classes/Summer2007/M216/VolumeOfTorus.pdf
Post by: EspoirTron on March 28, 2015, 08:29:00 pm
I think that makes sense? I would imagine in VCE, yes you would divide my n-1. I do however doubt you'll be expected to provide rigorous meaning behind your choice of choosing n-1 over n.
Post by: keltingmeith on March 28, 2015, 08:34:15 pm
I would imagine in VCE, yes you would divide my n-1. I do however doubt you'll be expected to provide rigorous meaning behind your choice of choosing n-1 over n.
The proof of this actually isn't very difficult, and is definitely something I would expect a VCE student capable of (with a little guidance, of course). However, they don't go over what makes a good estimator (and hence, how to unbias a biased estimator), so I don't think it will pop up even in the new study design.
But yes - in further, methods AND specialist you should be diving by n-1 to find the sample standard deviation. There are some specific cases where you divide by n, but they will not appear in VCE (unless they suddenly change the study designs to include estimators - which would not be a bad move, IMHO).
Post by: TheAspiringDoc on March 30, 2015, 12:11:25 pm
Thanks :)
Post by: TheAspiringDoc on April 13, 2015, 05:33:29 pm
umm, LaTeX is confusing?
Post by: keltingmeith on April 13, 2015, 07:32:43 pm
is merely coincidence the 2 Pi r is the derivative of Pi r^2 and that Pi r^2 is the Antiderivative (integral) of 2 Pi r?
Nope - I have forgotten the logic behind it, though. Same happens for TSA and the volume of a sphere.
Also, how do we find the volume of the hollow centre of the torus, i.e. The part you would cut out of the pastry to make the doughnut?
Thanks :)
Welp, it depends on how you model the "hollow" part. Where does in start/end? One method could be to model a similar ellipsoid and take the volume of the torus from it.
Hey, just wondering, is sin(1)+sin(2)+sin(3)+...sin(358)+sin(359)+sin(360) equal to sin(0)+sin(45)+sin(90)+sin(135)+sin(180)+sin(225)+sin(270)+sin(315)+sin(360)? and are either of them equal to [tex]\int_{0}^{360} sin(x) dx\[int][tex]? It's just that my calculator gave different answers for each ones and also a different answer to what I'd expected.. Thanks :)They're definitely not all equal. Why would they be equal?
umm, LaTeX is confusing?
Post by: TheAspiringDoc on April 13, 2015, 08:26:05 pm
Nope - I have forgotten the logic behind it, though. Same happens for TSA and the volume of a sphere.Inside of the torus - I am referring to the 'apple core' shaped part.
Welp, it depends on how you model the "hollow" part. Where does in start/end? One method could be to model a similar ellipsoid and take the volume of the torus from it. They're definitely not all equal. Why would they be equal?
Well, why should they (the sine sums in my previous post) all be equal? isn't it like finding the mean of 5 + 10 + 15 or 5 + 7.5 + 10 + 12.5 + 15 or everything between 5 and 15 inclusive?
Post by: keltingmeith on April 13, 2015, 08:28:28 pm
Well, why should they (the sine sums in my previous post) all be equal? isn't it like finding the mean of 5 + 10 + 15 or 5 + 7.5 + 10 + 12.5 + 15 or everything between 5 and 15 inclusive?
No - because a mean has a normalisation factor. You've just listed sums without any factor.
Post by: ldunn on April 13, 2015, 09:24:04 pm
Maybe I'm off my head, though
Post by: kinslayer on April 14, 2015, 02:05:38 am
To be clear, those sine arguments are in degrees, right? I make all of those expressions to be zero, which makes sense to me... not for the reasons mentioned earlier, though. It's just because of the symmetry involved. Each term has a corresponding term which is just the negative: sin(359) = -sin(1), sin(358) = -sin(2), etc.
Maybe I'm off my head, though
Yep -- provided those are degrees, each expression is definitely equal to zero.
TheAspiringDoc, switch your calculator to degrees, or multiply the arguments by
Post by: TheAspiringDoc on April 14, 2015, 07:04:00 am
Yep -- provided those are degrees, each expression is definitely equal to zero.I'd always been talking degrees?
TheAspiringDoc, switch your calculator to degrees, or multiply the arguments by.
And in that case they're all equal?
Post by: kinslayer on April 14, 2015, 04:31:57 pm
I'd always been talking degrees?
And in that case they're all equal?
Yes, they are equal to zero. This is because (in radians):
which is essentially what ldunn noted.
Post by: TheAspiringDoc on April 25, 2015, 07:19:51 am
Also, does Fermat's Last Theorem apply to complex numbers?
Thanks :)
P.S. Did you know the (5 or more)! always has a units digit of 0? This is beacause it contains 2 and 5, and when they are multiplied you get 10 and 10 times anything = 0 in the units column.
Post by: TheAspiringDoc on April 29, 2015, 05:13:58 pm
Post by: cosine on April 29, 2015, 05:19:28 pm
Express x in terms of a if
Post by: TheAspiringDoc on April 29, 2015, 05:27:22 pm
But my textbook says
Post by: cosine on April 29, 2015, 05:36:03 pm
But my textbook saysthat's not the same is it?
It's the same thing, but just different lay out to it :D
Post by: keltingmeith on April 29, 2015, 07:04:12 pm
})
It's the same thing, but just different lay out to it :D
Conjecture:
We evaluate:
And so, we see that they are only the same thing if a=0,2.
Post by: cosine on April 29, 2015, 07:25:16 pm
Conjecture:
We evaluate:
And so, we see that they are only the same thing if a=0,2.
Not really relevant to me, AspiringDoc asked why the book had a different lay out, I showed him why. I don't think he asked about what a can, and cannot equal?
Post by: TheAspiringDoc on April 29, 2015, 07:37:57 pm
Not really relevant to me, AspiringDoc asked why the book had a different lay out, I showed him why. I don't think he asked about what a can, and cannot equal?Still, it's from an extension maths book so the more maths the better.
Post by: keltingmeith on April 29, 2015, 07:39:40 pm
Not really relevant to me, AspiringDoc asked why the book had a different lay out, I showed him why. I don't think he asked about what a can, and cannot equal?No, you misunderstand. You said that:
=
But, I just showed that this is only true for a=0,2. However, it never said that a had to be restricted to those values, so what you've written is false.
Post by: cosine on April 29, 2015, 08:06:45 pm
No, you misunderstand. You said that:
But, I just showed that this is only true for a=0,2. However, it never said that a had to be restricted to those values, so what you've written is false.
What values are you referring to? I don't recall restricting anything to certain values, i left it as it was until you started to complicate things about the restrictions on a..
Besides it said express x in terms of a, hence no restrictions are involved as the only thing we are required to do is find an expression of x that is in expressed in a. When you write log(x) by itself, you don't usually write log(x), x cannot equal zero do you?
Post by: keltingmeith on April 29, 2015, 08:12:42 pm
What values are you referring to? I don't recall restricting anything to certain values, i left it as it was until you started to complicate things about the restrictions on a..That's my point - what you've written is only true for a=0,2. For you to state that
Besides it said express x in terms of a, hence no restrictions are involved as the only thing we are required to do is find an expression of x that is in expressed in a. When you write log(x) by itself, you don't usually write log(x), x cannot equal zero do you?
Post by: cosine on April 29, 2015, 08:15:56 pm
That's my point - what you've written is only true for a=0,2. For you to state that(which you did), then it HAS to be true for all a. It's not. What you've written is wrong.
The book stated that too, and aspiringdoc wanted to see how to get it into that form. My original answer is at it is, but because I was being generous and decided to enlighten aspirigdoc in how to get it into the book's form, I decided too. Not sure where you are trying to get at man?
Post by: IntelxD on April 29, 2015, 08:30:47 pm
The book stated that too, and aspiringdoc wanted to see how to get it into that form. My original answer is at it is, but because I was being generous and decided to enlighten aspirigdoc in how to get it into the book's form, I decided too. Not sure where you are trying to get at man?
Can I please just interject so we can put an end to this nonsense? Eulerfan is trying to point out that
Post by: cosine on April 29, 2015, 08:36:25 pm
Can I please just interject so we can put an end to this nonsense? Eulerfan is trying to point out thatisn't equal to
. Saying they are equal is akin to saying 1/(3*6) is equal to (1/3)*6. However, as even a broken clock is right twice a day, there are certain values of a which hold true for the proposed equality. Therefore, if you chose simplify in this manner, you must restrict the values of a so that your simplification is valid.
I would usually ignore banter like this, but I am not seeing the point of your unnecessary needs to invade..
Simplify: log(x^2)
= 2log(x)
Pretty sure you would not include the x cannot equal 0 after it, as we are not worried about the values that can go into it in this certain context, but rather more worried about how we can simplify it. So, are you saying that the above expression is wrong?
Post by: cosine on April 29, 2015, 08:41:42 pm
Simplify: log(x^2) does not equal
2log(x)
log(x)^2
= 2log(x)
Incorrect.
(log(x))^2 = log(x) * log(x)
Post by: keltingmeith on April 29, 2015, 09:22:55 pm
I would usually ignore banter like this, but I am not seeing the point of your unnecessary needs to invade..
Simplify: log(x^2)
= 2log(x)
Pretty sure you would not include the x cannot equal 0 after it, as we are not worried about the values that can go into it in this certain context, but rather more worried about how we can simplify it. So, are you saying that the above expression is wrong?
Yes, actually. That simplification only makes sense if x is positive, and you have to note that, or you will run into problems later. Try solving log_10(x^2)=20 using that method and see what happens, then compare your answer to a graph.
Post by: TheAspiringDoc on May 04, 2015, 05:09:06 pm
Thanks :)
Post by: Callum@1373 on May 04, 2015, 05:21:30 pm
Solve the following equation, expressing the value of x with a rational denominator
Thanks :)
Then just multiply by
It's just using the basic algebra rules ;)
Post by: TheAspiringDoc on May 04, 2015, 05:49:03 pm
but then we end up with
Then just multiply byto get a rational denominator
It's just using the basic algebra rules ;)
Post by: StupidProdigy on May 04, 2015, 05:55:06 pm
but then we end up withmultiplying by root 3 is just rationalising, which doesn't actually change the expression value, meaning you're basically just writing the answer in a different way even though it's the same value. Have a look into surd rationalisation a bit more is my suggestion :)rather than
?
Post by: cosine on May 04, 2015, 05:59:23 pm
Solve the following equation, expressing the value of x with a rational denominator
Thanks :)
AspiringDoc, between the last three lines, when I multiplied by sqrt3, it was because I was rationalising the denominator, i.e. expressing it in a more appropriate way, it is
Edit: Beaten, but ill leave the working out for ya.
Post by: TheAspiringDoc on May 04, 2015, 06:03:09 pm
Post by: TheAspiringDoc on May 09, 2015, 02:19:32 pm
my book says 2A-ah/h this is wrong, right?
Cheers :)
Post by: LoadedWithPotatoes on May 09, 2015, 02:39:22 pm
->
2A = (a+b)h
2A= ah+bh
(2A-ah)/h = b
If you did 2A/h - a = b instead that would be the same thing as the book's answer
Post by: TheAspiringDoc on June 25, 2015, 08:20:24 am
So I thought since this is my math thread, I can also add worked solutions to problems I thought were cool so other people can learn from them and if I got it wrong then someone might pick me up on it (if anyone still ever reads this thread!) :P
So here's the first one:
Three students attempt to solve a mathematical puzzle independently of each other. The first student has a chance of 0.5 of solving the puzzle, the second student has a chance of 0.6 of solving the puzzle, while the third student has a chance of 0.7 of solving the puzzle.
(a) What is the probability that all three solve the puzzle?
Pr(all solve puzzle) = 0.5x0.6x0.7
= 21/100 or 0.21 or 21%
(b) What is the probability none of the students solve the puzzle?
Pr(none solve puzzle) = (1-prob. studentA solves puzzle) x (1- prob student B solves puzzle) x (1-prob. student C solves puzzle)
=(1-0.5)x(1-0.6)x(1-0.7)
=0.5x0.4x0.3
=3/50 or 0.06 or 6%
Thanks!!
Post by: TheAspiringDoc on June 25, 2015, 08:51:24 am
Another worked solution (I hope!!)
Find the units digit of
Okay, so whilst this may initially look like it needs a supercomputer and simply isn't solvable with a handheld calculator, the fact is that it actually is.
So lets establish one thing:
we only care about the units digits of
So, to find the units digit for
6^1 = 6
6^2 = 36
6^3 = 216
6^4 = 1296
6^5 = 7776
etc.
So we now have:
find the units digit of
For the other two, we are going to have to use a somewhat trickier method, as I will now demonstrate on
When we raise 7 to any power, we see a pattern, that is:
7^0 = 1
7^1 = 7
7^2 =49
7^3 = 343
7^4 = 2401
7^5 = 16807
7^6 = 117649
7^7 = 823543
7^8 = 5764801
etc.
the pattern becomes clear when we look at the units digit of each one:
7^0 = 1
7^1 = 7
7^2 =9
7^3 = 3
7^4 = 1
7^5 = 7
7^6 = 9
7^7 = 3
7^8 = 1
etc.
So they are cycling through the numbers 1,7,9 and 3.
and there are four numbers in this cycle, and the 52 in 7^52 is a multiple of four, so it will have the first number of the cycle: 1.
So we now have:
find the units digit of
Doing the same for
Units digits:
8^0 = 1
8^1 = 8
8^2 = 4
8^3 = 2
8^4 = 6
8^5 = 8
8^6 = 4
etc.
So our cycle is:
8,4,2,6.
Which also happens to have four components.
if the 22 in
(units digit of)
so now we just look at our cycle list (8,4,2,6) and look two places before 6, which is 4.
therefore (units digit of)
So we now have:
find the units digit of
which is 11, the units digit of which is 1.
Thus, the units digit of
Hope you guys enjoyed B)
GAHHHH I HATE LATEX!!!!! (they're factorials btw)
Post by: keltingmeith on June 25, 2015, 10:05:11 am
You should try quoting people's posts to see how they've acheived writing certain things, and then use their method of typesetting to do it yourself. Also, a couple of things you might be interested in:
1. The Bernoulli and Binomial distributions. These can be used to model certain problems similar to the one in your first question.
2. Modular arithmetic. Believe it or not, you actually did this in your second question - it is an actual mathematical thing used in algebra and number theory, you might enjoy a read about it.
Post by: TheAspiringDoc on July 02, 2015, 06:58:10 pm
A hundred people board a plane (tha has 100 seats) with assigned seating. For some reason the first person on board takes a random seat. The second person looks around and if their seat is taken, takes a random seat and otherwise sits in their assigned seat. The third person does the same and this goes on for all remaining passengers. What is the probability that the 100th person sits in their assigned seat?
Help?
Thunks :)
Post by: keltingmeith on July 02, 2015, 08:07:36 pm
Wow, the more you look into this problem, the harder it is.Try doing the problem for 2 or 3 people, first. THEN try putting it up to 100.
A hundred people board a plane (tha has 100 seats) with assigned seating. For some reason the first person on board takes a random seat. The second person looks around and if their seat is taken, takes a random seat and otherwise sits in their assigned seat. The third person does the same and this goes on for all remaining passengers. What is the probability that the 100th person sits in their assigned seat?
Help?
Thunks :)
The problem itself isn't terribly difficult, but the moment you use a big number, is the moment it becomes incredibly daunting.
Post by: kinslayer on July 03, 2015, 09:33:14 am
Another way of looking at it: the probability is equal to the probability that the first 99 people boarding the plane choose a seat that is NOT the 100th person's seat.
EDITED:
If there are n seats on the plane (n>1), the probability that the (n-1)'th person will take a seat that is not theirs is 1/2, the probability that the (n-2)'th person chooses a seat that is not theirs 2/3, etc. The probability that each of the n-1 people choose a seat that is not their own is equal to
So the probability that 99 people do not pick the 100th person's seat (the case n = 100) is equal to 1/100.
Post by: TheAspiringDoc on July 03, 2015, 10:56:29 am
If all seat configurations are equally probable then the probability should be 1/100.I'm pretty sure that answer is incorrect.
Another way of looking at it: the probability is equal to the probability that the first 99 people boarding the plane choose a seat that is NOT the 100th person's seat.
If there are n seats on the plane (n>1), the probability that the (n-1)'th person will take a seat that is not theirs is
So the probability that the 99th person takes takes a seat that is not theirs is 1/100, which is the probability you are after.
Think of it in terms of how the last person could not get their seat.
The probability that the wild person at the start chooses the final person's seat straight up is 1/100. Then we must also add the probabilities that the wild person at the start chooses some other person's seat, who in turn takes the final persons seat.
These two alone add up to more than 1/100. And there are plenty of others too.
Post by: lzxnl on July 03, 2015, 11:26:12 am
I'm pretty sure that answer is incorrect.
Think of it in terms of how the last person could not get their seat.
The probability that the wild person at the start chooses the final person's seat straight up is 1/100. Then we must also add the probabilities that the wild person at the start chooses some other person's seat, who in turn takes the final persons seat.
These two alone add up to more than 1/100. And there are plenty of others too.
You misinterpreted what kinslayer wrote. In your defence he didn't spell everything out
He worked out the chance that the 100th person wouldn't sit in their assigned seat. So the answer is 0.99
Post by: TheAspiringDoc on July 03, 2015, 11:29:24 am
You misinterpreted what kinslayer wrote. In your defence he didn't spell everything outWait, you're sating that the chance that the 100th person sits in their seat is 0.99?
He worked out the chance that the 100th person wouldn't sit in their assigned seat. So the answer is 0.99
That doesn't seem right..
Post by: keltingmeith on July 03, 2015, 12:44:14 pm
Basically, it is very common to come up with mathematical results that don't seem correct, which is why we have a bunch of results collectively known as "paradox"es. It basically means that just because your intuition tells you one thing, don't expect it to be true unless the maths tells you it is. In converse, just because something seems wrong, if the mathematics is correct, that means it must be right.
Post by: kinslayer on July 03, 2015, 02:57:26 pm
I'm pretty sure that answer is incorrect.
Think of it in terms of how the last person could not get their seat.
The probability that the wild person at the start chooses the final person's seat straight up is 1/100. Then we must also add the probabilities that the wild person at the start chooses some other person's seat, who in turn takes the final persons seat.
These two alone add up to more than 1/100. And there are plenty of others too.
You're right, I misinterpreted the question. My solution assumes everyone chooses a set randomly, when in fact they only choose randomly if their seat is already taken. The probability should be much higher than 1/100. It wouldn't be as high as 0.99, but perhaps not far off. I will revisit when I have more time.
Post by: lzxnl on July 03, 2015, 09:36:19 pm
Repeating question for my own convenience.
A hundred people board a plane (that has 100 seats) with assigned seating. For some reason the first person on board takes a random seat. The second person looks around and if their seat is taken, takes a random seat and otherwise sits in their assigned seat. The third person does the same and this goes on for all remaining passengers. What is the probability that the 100th person sits in their assigned seat?
Let's break this problem down a bit. Let P(n) be the chance the nth passenger gets his seat.
With n seats, there are a few cases.
The first person can, with 1/n probability, choose his own seat. This then forces all of the other passengers to get their own seat.
The first person can, with 1/n probability, choose the nth person's seat. This leads to a null result.
They also can, with 1/n probability, choose any other person's seat. This results in the same dilemma with fewer passengers. If the person chooses the seat x from the end, then all of the people from 2 to n-x will have to choose their own seat. The (n-x+1)th person then has to pick a random seat, beginning the dilemma again.
So the chance that the nth passenger will get his own seat is the probability the first event occurs * 1 (as it's then guaranteed that person n gets his seat) + probability third event occurs * (sum of all the different dilemma probabilities)
In other words, P(n) = 1/n + (sum (P(i))/n) where the sum is taken from i = 2 to n
P(n-1) = 1/(n-1) + 1/(n-1)* sum(P(i)) where the sum is taken from i=2 to n-1
n*P(n) = 1 + sum(P(i)), 2<i<n
(n-1)*P(n-1) = 1 + sum(P(i)), 2<i<n-1
n*P(n) - (n-1)*P(n-1) = P(n)
(n-1)*P(n) = (n-1)*P(n-1)
P(n) = P(n-1)
By induction, as P(2) = 1/2, P(n) = 1/2 for all n
Post by: keltingmeith on July 03, 2015, 09:53:44 pm
In other words, P(n) = 1/n + (sum (P(i))/n) where the sum is taken from i = 2 to n
I'm really not sure how you've managed to arrive at this conclusion, care to walk me through it some more?
Post by: lzxnl on July 03, 2015, 10:35:05 pm
Person n gets his seat if the first person to choose gets any seat that's not n. The first person has chance 1/n to pick seat 1, which necessarily means person 2 gets seat 2 and so forth, so this option contributes 1/n * 1 = 1/n
Now person 1 has a 1/n chance of choosing any one of the seats from 2 to n-1. Let the person choose seat i from the end (end being seat 1). Then, all of the people from person 2 to person n-i must choose their own seat. For the i people from n-i+1 to n, they have a similar selection dilemma as the original situation, so the probability in the latter case of the nth person choosing his own seat is P(i). Each choice contributes 1/n * P(i). Hence the summation.
Although I realised that the sum should be taken from 2 to n-1 here. It doesn't actually make much of a difference though:
P(n) = 1/n + (sum (P(i))/n) where the sum is taken from i = 2 to n-1
P(n-1) = 1/(n-1) + 1/(n-1)* sum(P(i)) where the sum is taken from i=2 to n-2
n*P(n) = 1 + sum(P(i)), 2<i<n-1
(n-1)*P(n-1) = 1 + sum(P(i)), 2<i<n-2
n*P(n) - (n-1)*P(n-1) = P(n-1)
(n)*P(n) = (n)*P(n-1)
P(n) = P(n-1)
By induction, as P(2) = 1/2, P(n) = 1/2 for all n
As required anyway
Post by: TheAspiringDoc on August 10, 2015, 02:42:17 pm
How many consecutive zeros are in that consecutive string?
Hand-held calculator allowed.
Thanks
Post by: lzxnl on August 10, 2015, 10:35:11 pm
The expression 15^80 x 28^60 x 55^70 gives a number that ends with a string of zeros.
How many consecutive zeros are in that consecutive string?
Hand-held calculator allowed.
Thanks
Prime-factorise everything.
15 = 3*5
28 = 2^2 * 7
55 = 5*11
Taking the necessary powers gives (3^80*5^80) * (2^120 * 7^60) * (5^70*11^70) = 3^80 * 5^150 * 2^120 * 7^60 * 11^70
The logic is that zeros only come from powers of 10 in the prime factorisation.
I can clearly take out a maximum of 10^120 here so there are 120 zeroes.
Post by: MathsGuru on August 11, 2015, 10:43:42 pm
Ok, thought my math's posts were a bit out of place in the mainstream question threads so I thought I'd make my own :)
I'm trying to get an intuitive grasp of the derivatives and antiderivatives of trigonometric functions.. Any idea's how?
Also, Find the length of the largest square that will fit inside a regular hexagon of edge length 30cm?
Another way to see why the derivative of sin(x) is cos(x), and why that of cos(x) is -sin(x), is to look at their power series, if you're familiar with what they are (sort of "infinite degree polynomials", roughly speaking).
This is more algebraic than intuitive though.
Post by: lzxnl on August 11, 2015, 10:51:24 pm
Another way to see why the derivative of sin(x) is cos(x), and why that of cos(x) is -sin(x), is to look at their power series, if you're familiar with what they are (sort of "infinite degree polynomials", roughly speaking).
This is more algebraic than intuitive though.
Few questions.
Firstly, how do you get to these power series? Either you construct these as the sine and cosine functions and derive all of the other known properties later, or you use the Generalised Mean Value Theorem, show convergence for the power series everywhere, show that the remainder term vanishes etc...which all requires knowing what the derivatives are.
For an intuitive grasp, you're better off looking at the graphs. Derivative of sine is cosine can be reasoned as follows.
At x=0, y = sin x is at its steepest -> derivative should be at its greatest at x=0. Function is periodic -> derivative should also be periodic. Function has extrema at pi/2, 3pi/2, 5pi/2 etc -> derivative should be zero here. Etc.
Post by: MathsGuru on August 12, 2015, 10:21:29 am
Few questions.
Firstly, how do you get to these power series? Either you construct these as the sine and cosine functions and derive all of the other known properties later, or you use the Generalised Mean Value Theorem, show convergence for the power series everywhere, show that the remainder term vanishes etc...which all requires knowing what the derivatives are.
For an intuitive grasp, you're better off looking at the graphs. Derivative of sine is cosine can be reasoned as follows.
At x=0, y = sin x is at its steepest -> derivative should be at its greatest at x=0. Function is periodic -> derivative should also be periodic. Function has extrema at pi/2, 3pi/2, 5pi/2 etc -> derivative should be zero here. Etc.
Absolutely - the trig graphs or the unit circle are much better for intuitive understanding.
So I'm not trying to put forward a rigorous proof, only a plausibility argument (which, by the way, is about what the graphs provide anyway, since it takes further reasoning to justify why the slope of sine at the origin coincides with the maximum value of cosine - as long as angles are measured in radians!)
For VCE level, I think it's more than enough just to be aware that there is such a thing as a power series, and to do a bit of differentiantion practice with it, to get an alternative algebraic view of the sin-cos relationship to complement the geometric view.
Power series are also a nice way to think about what sort of form a function would have if it equals its own derivative :-)
Post by: TheAspiringDoc on August 28, 2015, 01:11:23 pm
Thank you!
Post by: MathsGuru on September 09, 2015, 12:58:58 pm
Which three digit number has the greatest number of different factors?
Thank you!
I think it's 840, with 32 distinct factors
Post by: TheAspiringDoc on September 09, 2015, 01:42:40 pm
I think it's 840, with 32 distinct factorsOkay, thanks :D
But.. Why?
Post by: lzxnl on September 09, 2015, 05:58:18 pm
Okay, thanks :D
But.. Why?
Let's have a look at what constitutes 'distinct factors'.
32 = 2^5. This number has factors 2, 4, 8, 16, 16. So a number a^n, where a is prime, has n-1 factors.
Let's now look at 2000 = 5 * 400 = 5 * 16 * 25 = 2^4 * 5^3
How do we generate the number of factors? Hopefully you can see that the number of factors it has is 18. We get to this by considering the five powers of 2 from 2^0 to 2^4 and the four powers of 5 from 5^0 to 5^3. For each power of 2, there are four possible powers of 5 you can combine with to generate new factors. Then, we subtract the two cases 1 and 2000. That gives 5 x 4 - 2 = 18.
Generalising for a1^(x1) a2^(x2) a3^(x3)...an^(xn), the number of factors would be (x1+1)(x2+2)(x3+1)(x4+1).....(xn+1) - 2.
As an example, 324 = 18^2 = (2*3^2)^2 = 2^2 * 3^4
Here n = 2, number of factors should be 3*5 - 2 = 13
Indeed, the factors of 324 are:
2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, which totals 13 factors.
So to get the largest number of factors, you would ideally want to find a number that has a large number of primes as factors, but not too many as the primes get big. You need a mix of lots of primes and large powers of primes. This suggests that you raise the smaller primes to higher powers first.
Then, you're trying to maximise the product (x1+1)(x2+1)(x3+1)(x4+1)....(xn+1) subject to the constraint 2^(x1) 3^(x2) 4^(x3)....(nth prime)^(xn) < 1000
You can get a feel for how this would work by looking at the primes. Intuitively, you don't want primes past about 13 because then your powers aren't going to be too big; 13 is around 2^4 already. So you'd then play around with numbers. That's all I can think of atm.
Post by: MathsGuru on September 10, 2015, 10:09:54 am
Okay, thanks :D
But.. Why?
lzxnl's post sums it up pretty well, with a couple of minor comments:
- 4^(x3) near the end should of course be 5^(x3) (the bases are the primes in increasing order)
- and normally 1 and 2000 would be counted as factors of 2000, which eliminates the need for the messy -2
So here's how it works for 840:
So building up a number as a product of distinct primes (i.e. a 'squarefree' number), the number of factors doubles each time another prime is included.
The furthest we can go using as small primes as possible while not exceeding 999 is
We then increase the powers of some of the primes we already have.
We can't put in another 5 or 7 without exceeding 999.
If we increase the power of 3, we can make
However, we can fit in another two 2s, making
These factors can be enumerated simply as all the integers of the form
Post by: Orson on September 11, 2015, 12:18:15 am
Anyway...have fun buddy!
Post by: TheAspiringDoc on September 13, 2015, 02:01:10 pm
Thanks also Orson ;)
Could anyone offer some advice on how I could go about learning calculus over the upcoming holidays?
Post by: keltingmeith on September 13, 2015, 03:48:59 pm
Thanks heaps for your explanations MathsGuru and lzxnl. :DPaul's Online Notes are pretty good, do recommend.
Thanks also Orson ;)
Could anyone offer some advice on how I could go about learning calculus over the upcoming holidays?
Post by: lzxnl on September 13, 2015, 06:46:05 pm
Paul's Online Notes are pretty good, do recommend.
^
Learnt more than half of the second year UoM subject MAST20009 in a day using that site.
Post by: Orson on September 14, 2015, 11:29:32 am
Thanks also Orson ;)
Anytime.
Could anyone offer some advice on how I could go about learning calculus over the upcoming holidays?
I'd have to say Khan Academy again. Just go through his series on Calculus from the beginning...get an old textbook (I'd recommend this years Essentials, you will be able to pick them up cheap because they are changing the study design) and go for it!
Post by: TheAspiringDoc on September 14, 2015, 05:32:28 pm
I've found that other useful resources are Math Is Fun and Maths Methods Podcasts by Justin Vincent if anyone was interested ;)
On a side note, if I have 18 scores on a stem and leaf plot, all of which are different, what should I write if I'm asked to find the mode?
Thanks :D
Post by: keltingmeith on September 14, 2015, 06:09:10 pm
Paul's notes are indeed very good :)There is no mode.
I've found that other useful resources are Math Is Fun and Maths Methods Podcasts by Justin Vincent if anyone was interested ;)
On a side note, if I have 18 scores on a stem and leaf plot, all of which are different, what should I write if I'm asked to find the mode?
Thanks :D
Post by: TheAspiringDoc on September 14, 2015, 08:28:16 pm
Thank you :)
Post by: keltingmeith on September 14, 2015, 09:31:56 pm
I'm aware that the derivative of trig functions like sin(x) is only cos(x) if x is in radians (right?), but does that also apply for inverse trig [ I.e. Does the derivative of cos^-1 (x) only end up being -1/sqrt(1-x^2) if x is in radians?]
Thank you :)
First point - correct.
Second point - not quite. It has something to do with the output instead. What is the relationship between sin(x) and arcsin(x)?
Post by: MathsGuru on September 15, 2015, 09:47:14 am
I'm aware that the derivative of trig functions like sin(x) is only cos(x) if x is in radians (right?), but does that also apply for inverse trig [ I.e. Does the derivative of cos^-1 (x) only end up being -1/sqrt(1-x^2) if x is in radians?]
Thank you :)
As EulerFan101 pointed out, you're almost right - what you want to say is that the expression you gave for the derivative of y = cos^-1(x) depends on y, not x, being measured in radians.
Here's a nice way to think about the radian dependence geometrically.
As the Methods textbooks will tell you (around early Yr12 material), the graph of an inverse function can be obtained by reflecting the original graph in the line y = x, so that x- and y-coords are swapped.
So, while changing the angle measurement unit dilates the graph horizontally for a trig function, it dilates it vertically for an inverse trig fn. In either case, this dilation is going to stretch the rise or run of any triangle you use to measure or to approximate the gradient, by the same factor, so the numerical value of the derivative must also change accordingly.
Post by: TheAspiringDoc on October 05, 2015, 06:56:19 pm
Quote
The product rule is as follows: d/dx (f/g) = (f’ g − g’ f )/g^2In the product rule as shown above, do 'f' and 'g' represent two (different?) functions of x?
So it'd then be better to write as:
thnx ;D ;D
Post by: lzxnl on October 06, 2015, 01:36:51 pm
In the product rule as shown above, do 'f' and 'g' represent two (different?) functions of x?
So it'd then be better to write as:?
thnx ;D ;D
That's the quotient rule.
Post by: TheAspiringDoc on October 07, 2015, 06:36:20 pm
So once you've learnt the product, quotient and chain rules, as well as all of the minor rules (e.g. d/dx [ln(x)] = 1/x or d/dx [tan^-1] = 1/(1+x^2)), what's the next thing to learn? Surely there's more to it, I just can't find 'what's next' I guess.. ?
:D
Post by: _fruitcake_ on October 07, 2015, 06:39:34 pm
Hey,
So once you've learnt the product, quotient and chain rules, as well as all of the minor rules (e.g. d/dx [ln(x)] = 1/x or d/dx [tan^-1] = 1/(1+x^2)), what's the next thing to learn? Surely there's more to it, I just can't find 'what's next' I guess.. ?
:D
U can learn some further maths equations :')
hehehe
Post by: lzxnl on October 07, 2015, 07:00:35 pm
Hey,
So once you've learnt the product, quotient and chain rules, as well as all of the minor rules (e.g. d/dx [ln(x)] = 1/x or d/dx [tan^-1] = 1/(1+x^2)), what's the next thing to learn? Surely there's more to it, I just can't find 'what's next' I guess.. ?
:D
Do you know what a derivative means? Can you apply it to other contexts? Are you comfortable with every possible application of differentiation, from finding tangents and normals to relating rates of change?
If you get bored with that, differentiate (sin x)^(sin x) and get back to me :P
If you really want to learn more calculus, integration is generally far harder than differentiation. Proving that the integral of sin x / x = pi/2 if integrated from 0 to infinity takes some level of extreme ingenuity in at least one step. Then there's the host of substitutions to learn, like trig subs, Weierstrauss subs, integrating by parts, integrals of quadratic radicals etc
If you get done with that, and you're done with the VCE maths courses, look up some other area of maths that interests you. For instance, linear algebra, group theory, multivariate calculus, differential equations, real analysis...there's so much out there to learn.
(For the interested people, you can do the above integral via a complex integral, differentiation under the integral sign or even interchanging the order of integration).
Post by: TheAspiringDoc on October 07, 2015, 08:07:01 pm
Do you know what a derivative means? Can you apply it to other contexts? Are you comfortable with every possible application of differentiation, from finding tangents and normals to relating rates of change?Thanks lzxnl - you never cease to inspire me :D
If you get bored with that, differentiate (sin x)^(sin x) and get back to me :P
If you really want to learn more calculus, integration is generally far harder than differentiation. Proving that the integral of sin x / x = pi/2 if integrated from 0 to infinity takes some level of extreme ingenuity in at least one step. Then there's the host of substitutions to learn, like trig subs, Weierstrauss subs, integrating by parts, integrals of quadratic radicals etc
If you get done with that, and you're done with the VCE maths courses, look up some other area of maths that interests you. For instance, linear algebra, group theory, multivariate calculus, differential equations, real analysis...there's so much out there to learn.
(For the interested people, you can do the above integral via a complex integral, differentiation under the integral sign or even interchanging the order of integration).
I guess I think of the derivative as the slope of a curve at point x, produced by the a particular function of x.
Finding tangents and normals hey?
So to find a tangent I believe you just plug in your x value and your good, right?
As for finding the normal, if it has the same meaning as it does in physics, then I guess you want to find the tangent, and then somehow perform a 90 degree rotation (how?).
And yeah, integration is a good idea. Since I learnt quite a bit of my first calculus from my dad, I actually ended up learning integration first as he thought it is "more intuitive". The reason I've gone back to differentiation is that in pretty much all of the integration courses, they're like "as we know from our study of derivatives" - nope.
And sorry, this probably isn't the best way of going about it, but I legit am struggling to find any info anywhere on what a differential equation is..?
Thanks again!
Post by: MightyBeh on October 07, 2015, 08:28:39 pm
So to find a tangent I believe you just plug in your x value and your good, right?To find a tangent, you need a point from your original function or an x value; take, for example, f(x) = x2+1:
We know that the derivative is 2x,
The tangent is different at different points (mostly ::)), so we need to specify a point or an x value. Let's say we have to find the tangent at x=2 - first, we have to plug this into f(x).
Next, we put this same x-value into the derivative:
tl;dr you also need to take the + c into account.
As for finding the normal, if it has the same meaning as it does in physics, then I guess you want to find the tangent, and then somehow perform a 90 degree rotation (how?).Not sure what it means in Physics, but the normal is a 90 degree rotation. Not sure if it applies in 100% of cases because it's been taken off the study design for methods and I haven't covered it in class, but the slope is
And sorry, this probably isn't the best way of going about it, but I legit am struggling to find any info anywhere on what a differential equation is..?
Paul's notes can probably explain better than I can
Post by: zsteve on October 07, 2015, 08:45:24 pm
Thanks lzxnl - you never cease to inspire me :D
I guess I think of the derivative as the slope of a curve at point x, produced by the a particular function of x.
Finding tangents and normals hey?
So to find a tangent I believe you just plug in your x value and your good, right?
As for finding the normal, if it has the same meaning as it does in physics, then I guess you want to find the tangent, and then somehow perform a 90 degree rotation (how?).
And yeah, integration is a good idea. Since I learnt quite a bit of my first calculus from my dad, I actually ended up learning integration first as he thought it is "more intuitive". The reason I've gone back to differentiation is that in pretty much all of the integration courses, they're like "as we know from our study of derivatives" - nope.
And sorry, this probably isn't the best way of going about it, but I legit am struggling to find any info anywhere on what a differential equation is..?
Thanks again!
A differential equation simply put is an equation that involves one or more derivatives. Examples include:
Other (more interesting) equations include separable, homogeneous, linear (all these are first order), second order linear homogenous/inhomogeneous equations - google for this, and Paul's notes were a great resource when I learned them. If you're as crazy as me, you would learn these in Year 12 too. (Or maybe before, seeing you're so keen. I only wish I was as diligent a learner at your age :P)
Post by: zsteve on October 08, 2015, 10:16:57 pm
Let
and the rest follows. Sub in y when you're done.
Right |zxn|? :)
Post by: lzxnl on October 09, 2015, 03:41:49 pm
Another way is to write sin x ^ (sin x) as e^(sin x ln sin x) but that's pretty much what you've done.
Post by: TheAspiringDoc on October 11, 2015, 11:23:48 am
So lzxnl called finding the normal of a curve an "application" of differentiation. But how called such an application ever be applied/of use in real life?
Thanks!
Post by: lzxnl on October 11, 2015, 05:01:41 pm
Hi,
So lzxnl called finding the normal of a curve an "application" of differentiation. But how called such an application ever be applied/of use in real life?
Thanks!
If you want applications in real maths, VCE maths isn't one of the best places to find it.
Normals to surfaces, not curves, come up all the frigging time in physics. In fact, the surface normal vector is the only real way of defining the direction of a surface (think about it).
Normals to curves do come up in the two-dimensional divergence theorem which can be useful when computing double integrals (these have countless applications but again, you won't see it in VCE).
tl;dr, VCE maths sucks and if you want applications, do more maths
Post by: GeniDoi on October 11, 2015, 06:27:45 pm
If you want applications in real maths, VCE maths isn't one of the best places to find it.
Normals to surfaces, not curves, come up all the frigging time in physics. In fact, the surface normal vector is the only real way of defining the direction of a surface (think about it).
Normals to curves do come up in the two-dimensional divergence theorem which can be useful when computing double integrals (these have countless applications but again, you won't see it in VCE).
tl;dr, VCE maths sucks and if you want applications, do more maths
The probability unit of methods 3/4 is pretty applicable, and spesh is getting some probability from next year.
Post by: keltingmeith on October 11, 2015, 07:14:19 pm
The probability unit of methods 3/4 is pretty applicable, and spesh is getting some probability from next year.Hah, nice joke.
Post by: GeniDoi on October 11, 2015, 11:58:31 pm
Hah, nice joke.
Cmon now, all chickens secretly strategize to lay eggs whose diameter conforms to a normal distribution curve :P
Post by: MathsGuru on October 13, 2015, 10:12:15 am
...
If you get done with that, and you're done with the VCE maths courses, look up some other area of maths that interests you. For instance, linear algebra, group theory, multivariate calculus, differential equations, real analysis...there's so much out there to learn.
...
If you want a nice bird's eye view of a whole swathe of maths, so you can get a feel for what areas inspire you the most, Norman Wildberger has a really cool series of videos (including university lectures) on the history of maths, a lot of which should be reasonably accessible to a sharp high school student :-)
https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Post by: Ssuper19 on October 13, 2015, 12:13:20 pm
If you want a nice bird's eye view of a whole swathe of maths, so you can get a feel for what areas inspire you the most, Norman Wildberger has a really cool series of videos (including university lectures) on the history of maths, a lot of which should be reasonably accessible to a sharp high school student :-)
https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Great YouTube playlist. Really goes into the depth of the history of maths. Thanks 8) 8) 8)
Post by: TheAspiringDoc on October 15, 2015, 05:26:44 pm
How do you find the equation of a line (a parabola) that passes through the x-axis at -2 and 4, and the y-axis at 24?
Thanks :)
Post by: TheMereCat on October 15, 2015, 05:52:34 pm
Where a is some constant, b, c and are the x-intercepts. So you have the equation, y= a(X+2)(X-4)
Now plug in (0,24) into the equation:
24= a(0+2)(0-4)
24=a(2)(-4)
24=-8a
a=-3
You can now express the equation as Y= -3(X-4)(X+2) or
-3(X^2+2x-4x-8)
y=-3x^2+2x+24
Post by: TheAspiringDoc on October 25, 2015, 01:21:59 pm
Things like pH involves logarithms and then I believe there's some differentiation involved in chem as well?
Thanks! :)
Post by: GeniDoi on October 25, 2015, 04:42:18 pm
Does anyone know of some nice ways to tie chemistry and maths in together?
Things like pH involves logarithms and then I believe there's some differentiation involved in chem as well?
Thanks! :)
Yup, there's a pretty tricky differential equation question in essential specialist maths 3/4:
Two chemicals, A and B, are put together in a solution where they react to form a compound, X. The rate of increase of the mass, x kg, of X is proportional to the product of the masses of unreacted A and B present at time t minutes. It takes 1 kg of A and 3 kg ofB to form 4 kg of X. Initially 2 kg of A and 3 kg ofB are put together in solution. One kg of X forms in one minute.
a) Set up the appropriate differential equation expressing dx/dt as a function of x.
b) Solve the differential equation.
c) Find the time taken to form 2 kg of X. d Find the mass of X formed after two minutes.
Our teacher did it with us in class and sent us a copy of the working, I've attached it here. According to him, definitely something that wouldn't come up in a vcaa specialist exam as it's too reliant on chemistry knowledge, but it's a good question that involves many elements to solve it like partial fractions and integrating logs. The working out ends where it does because from there it's a simple integration problem.
Post by: lzxnl on October 25, 2015, 09:50:50 pm
Does anyone know of some nice ways to tie chemistry and maths in together?
Things like pH involves logarithms and then I believe there's some differentiation involved in chem as well?
Thanks! :)
Here are many of the ways I can imagine tying chemistry and maths together.
1. Quantum chemistry. AKA quantum physics applied to chemistry. Maths absolutely everywhere, from Huckel theory to particle-in-the-box/ring approximations to density functional theory to solutions to Schrodinger's equation (particle in box, harmonic oscillator, hydrogen atom, time-dependent), perturbation theory (plus more)...I don't know when that's taught in Australia but on exchange it's a graduate level chemistry course. The maths involved here would be mainly linear algebra (tonnes of matrices and inner products), differential equations and multiple integrals.
2. Kinetics. Rate laws generally relate the rate of change of the concentration of some reactant with the instantaneous concentrations of the reactants, aka a differential equation
3. Thermodynamics. So things like thermodynamic potentials (Gibbs, enthalpy etc), energies of mixing, Raoult's law, equilibrium constants, electrochemistry (Nernst equation etc), ideal/non-ideal gas calculations...these would involve integrals and differentials everywhere.
Is that enough for you? This is all at a first/second year uni level at a minimum so it's ok if it doesn't currently make sense.
Post by: keltingmeith on October 25, 2015, 10:11:30 pm
1. Crystallography. Not only is this insanely physicsy (it's one of those things were it was basically invented by a physicist and chemists just use it to their advantage) in the way that it works, but a lot of the principles behind crystallography are very closely related to group theory (symmetry operations and the like.)
2. Molecular geometries. This stuff is pure geometry at its finest, all the way down to using things like pythagoras to help us find bond lengths or the dimensions of a particular molecule.
3. Supramolecular topologies. It's in the name - this stuff builds on basic ideas of topologies to classify particular molecules. In fact, to distinguish the difference between a polyrotaxane and a polycatenane, you will need a working level knowledge of basic topology (essentially just that lengths don't matter, coffee cup lid = doughnut type stuff)
Post by: GeniDoi on October 27, 2015, 08:26:29 am
For something a bit more obscure, I recently discovered that a lot of supramolecular chemistry is quite mathsy. Note that everything I'm about to talk about is third year chemistry/honours level, so if lzxnl's stuff is a bit confusing, this will DEFINITELY make you go "... lolwut". (and some of the maths is also third year/honours level, too)
1. Crystallography. Not only is this insanely physicsy (it's one of those things were it was basically invented by a physicist and chemists just use it to their advantage) in the way that it works, but a lot of the principles behind crystallography are very closely related to group theory (symmetry operations and the like.)
2. Molecular geometries. This stuff is pure geometry at its finest, all the way down to using things like pythagoras to help us find bond lengths or the dimensions of a particular molecule.
3. Supramolecular topologies. It's in the name - this stuff builds on basic ideas of topologies to classify particular molecules. In fact, to distinguish the difference between a polyrotaxane and a polycatenane, you will need a working level knowledge of basic topology (essentially just that lengths don't matter, coffee cup lid = doughnut type stuff)
1 and 3 both sound like applications of pure mathematics, especially 3.
Post by: keltingmeith on October 27, 2015, 09:38:24 am
1 and 3 both sound like applications of pure mathematics, especially 3.That would be because they are.
Post by: Floatzel98 on November 02, 2015, 10:13:42 pm
Post by: TheAspiringDoc on November 03, 2015, 08:35:28 pm
Probably most appropriate here. Is someone able to explain how the Mandelbrot Set works? I've watched videos on it and read up about it but it is still very confusing. I don't really know what kind of questions I have about it because though because it seems so confusing. Can anyone explain in simple terms how it works? How you can keep on going deeper into it, does it ever stop?Hi!
I haven't really got my head around it yet as exams are approaching, but this appears to be very informative: https://m.youtube.com/watch?v=0YaYmyfy9Z4 . I like it's explaination, "Mandelbrot Set = set of numbers that display certain properties, displayed on complex plain." Also, it mainly works on this function (right?):
f(x)=x^2+c
Where c is an arbitrary original number that you chose (e.g. 4+3i [I think])
And x is the result from the previous computation.
For example, let's do the first few terms for c=1, and x starts as 0.
f(0)=(0)^2+1=1
f(1)=(1)^2+1=2
f(2)=(2)^2+1=5
f(5)=(5)^2+1=26
etc.
This above series produces exponential growth (not sre how it is relevant; but it does).
Also, the result of each function seen above is then 'plugged' into the next function - this is the 'iteration' concept that is continually mentioned.
That's pretty much as far as my understanding has progressed, so I'm not sure how different values on the complex plane are then assigned colours and all, or even how the shape (i.e. the madelbrot set) is produced, but yeah, at least I've bumped it and maybe someone will pick up if I've made any mistakes.
EDIT: removed LaTeX as it doesn't seem to be working *sigh*
Post by: Floatzel98 on November 03, 2015, 08:47:42 pm
Hi!I'll definitely watch the video, thanks! I'm pretty sure I watched the numberphile video on it. But yeah, I understand the basic iteration part, I just don't really understand how all of that translates onto the graph, how the colours work etc.
I haven't really got my head around it yet as exams are approaching, but this appears to be very informative: https://m.youtube.com/watch?v=0YaYmyfy9Z4 . I like it's explaination, "Mandelbrot Set = set of numbers that display certain properties, displayed on complex plain." Also, it mainly works on this function (right?):
f(x)=x^2+c
Where c is an arbitrary original number that you chose (e.g. 4+3i [I think])
And x is the result from the previous computation.
For example, let's do the first few terms for c=1, and x starts as 0.
f(0)=(0)^2+1=1
f(1)=(1)^2+1=2
f(2)=(2)^2+1=5
f(5)=(5)^2+1=26
etc.
This above series produces exponential growth (not sre how it is relevant; but it does).
Also, the result of each function seen above is then 'plugged' into the next function - this is the 'iteration' concept that is continually mentioned.
That's pretty much as far as my understanding has progressed, so I'm not sure how different values on the complex plane are then assigned colours and all, or even how the shape (i.e. the madelbrot set) is produced, but yeah, at least I've bumped it and maybe someone will pick up if I've made any mistakes.
EDIT: removed LaTeX as it doesn't seem to be working *sigh*
Thanks Doc :)
Post by: TheAspiringDoc on November 03, 2015, 09:29:43 pm
I'll definitely watch the video, thanks! I'm pretty sure I watched the numberphile video on it. But yeah, I understand the basic iteration part, I just don't really understand how all of that translates onto the graph, how the colours work etc.Perhaps a development?
Thanks Doc :)
So I've looked into it a bit more, and it seems as though the colours are not actually part of the set, simply something that is added, which makes it look wayyyy cooler..
-Colours are assigned based on how long it takes until a sequence eventually diverges (grows indefinately). Because with complex numbers, it seems as though the value appears to orbit around the origin of the complex number plane for a while?
Post by: MathsGuru on November 04, 2015, 08:40:17 pm
Probably most appropriate here. Is someone able to explain how the Mandelbrot Set works? I've watched videos on it and read up about it but it is still very confusing. I don't really know what kind of questions I have about it because though because it seems so confusing. Can anyone explain in simple terms how it works? How you can keep on going deeper into it, does it ever stop?
Another useful documentary on the Mandelbrot set:
https://www.youtube.com/watch?v=Lk6QU94xAb8
This is a fascinating area of maths... to learn more, you can look up some of the key terms (fractals, chaos, strange attractors) online. It may keep you entertained for years ;)
When I was a student at Monash I had the privilege of doing a vacation project in this area, at Sydney Uni with Mike Field, who is one of the authors of this book:
http://www.amazon.com/Symmetry-Chaos-Pattern-Mathematics-Edition/dp/0898716721
Post by: TheAspiringDoc on November 08, 2015, 09:34:45 am
I've got my school math's exam tomorrow and I've done a past exam and I've found that I barely managed to finish on time..
This is odd because last semester when we had exams, I had about 20 minutes to check over my answers to each section before submitting it.
I'm very concerned as yesterday when I did the past paper I didn't get time to check my answers to sections BC (rule is that you hand in the non-calculator section A in the first 45 minutes), which led to multiple mistakes.. What can I do about this?
I'm worried. :'(
Thanks
Post by: MightyBeh on November 08, 2015, 10:26:03 am
Hi :)
I've got my school math's exam tomorrow and I've done a past exam and I've found that I barely managed to finish on time..
This is odd because last semester when we had exams, I had about 20 minutes to check over my answers to each section before submitting it.
I'm very concerned as yesterday when I did the past paper I didn't get time to check my answers to sections BC (rule is that you hand in the non-calculator section A in the first 45 minutes), which led to multiple mistakes.. What can I do about this?
I'm worried. :'(
Thanks
What content are you covering? It's hard to give you good advice without knowing. :)
Post by: TheAspiringDoc on November 08, 2015, 11:01:26 am
What content are you covering? It's hard to give you good advice without knowing. :)Trig, shapes and angles, Linear graphs, statistics, and quadratic graphs.
Thank you :-\
Post by: 99.90 pls on November 08, 2015, 11:23:38 am
Another useful documentary on the Mandelbrot set:
https://www.youtube.com/watch?v=Lk6QU94xAb8
This is a fascinating area of maths... to learn more, you can look up some of the key terms (fractals, chaos, strange attractors) online. It may keep you entertained for years ;)
When I was a student at Monash I had the privilege of doing a vacation project in this area, at Sydney Uni with Mike Field, who is one of the authors of this book:
http://www.amazon.com/Symmetry-Chaos-Pattern-Mathematics-Edition/dp/0898716721
That was an awesome doco, but I do wish they'd delve into more of how it relates to complex numbers and stuff, there was a scene where they just scrolled over the psychedelic M-set for like 2 minutes straight with trippy music hahaha
@AspiringDoc, it's pretty much all practice for quadratics, linear and trig. For instance, the more you'll practice, the quicker you'll be able to find delta of a quadratic, x and y-intercepts of linear graphs and trig exact values. If you do it enough, you'll be able to do it in your head, saving precious time.
Another thing I'm guilty of is being perfectionist with my handwriting, but after I ditched that on maths exams, my speed went up quite a bit.
Post by: MightyBeh on November 08, 2015, 11:30:24 am
Trig, shapes and angles, Linear graphs, statistics, and quadratic graphs.
Thank you :-\
Diagrams are your friends for everything but statistics (and maybe even statistics). Write more working out than you think you need. Context check your answers (-2.8 is an odd answer for the length of wall), highlight, the list goes on.
Because it's time that seems to be the issue, maybe try pumping out some more questions from your textbook or asking for more practice questions? (although given that it's tomorrow that's probably not that viable) I could probably hook you up with some decent practice material if you're desperate :)
@AspiringDoc, it's pretty much all practice for quadratics, linear and trig. For instance, the more you'll practice, the quicker you'll be able to find delta of a quadratic, x and y-intercepts of linear graphs and trig exact values. If you do it enough, you'll be able to do it in your head, saving precious time.this ^^
Another thing I'm guilty of is being perfectionist with my handwriting, but after I ditched that on maths exams, my speed went up quite a bit.
Post by: TheAspiringDoc on November 08, 2015, 11:39:36 am
Diagrams are your friends for everything but statistics (and maybe even statistics). Write more working out than you think you need. Context check your answers (-2.8 is an odd answer for the length of wall), highlight, the list goes on.Thank you both very much.
Because it's time that seems to be the issue, maybe try pumping out some more questions from your textbook or asking for more practice questions? (although given that it's tomorrow that's probably not that viable) I could probably hook you up with some decent practice material if you're desperate :)
this ^^
It's as though I might be showing to many workings perhaps?
Post by: MightyBeh on November 08, 2015, 11:44:41 am
Thank you both very much.
It's as though I might be showing to many workings perhaps?
For me personally:
more working = less mistakes, but less checking time
less working = more checking time, but more mistakes
There's a sweet spot between both of those that you can have concise working without spending too much time on it, and consequently you'll have more time to check but not many mistakes to fix, if that makes sense? It really does just come down to practice :)
Post by: TheAspiringDoc on November 08, 2015, 12:00:53 pm
For me personally:But also like aren't marks given for workings?
more working = less mistakes, but less checking time
less working = more checking time, but more mistakes
There's a sweet spot between both of those that you can have concise working without spending too much time on it, and consequently you'll have more time to check but not many mistakes to fix, if that makes sense? It really does just come down to practice :)
I have a tendency to do like twenty lines for a four mark question - probably not good?
Post by: MightyBeh on November 08, 2015, 12:21:48 pm
But also like aren't marks given for workings?
I have a tendency to do like twenty lines for a four mark question - probably not good?
Not sure, depends on your school. If they're emulating VCAA exams, full marks would be given the the correct answer regardless of working. It might be that your teachers decide that there's certain 'signposts' for each question; for example you might have to show a substitution when finding the value of a constant or they'll deduct a mark for poor methodology.
You should use as many lines as you feel comfortable with - there's no good reason to skip working that would be helpful to you because someone else doesn't write it. 20 does seem a little excessive, though ::)
Post by: GeniDoi on November 08, 2015, 12:42:31 pm
If they're emulating VCAA exams, full marks would be given the the correct answer regardless of working.
This is not true at all. It used to be for physics but they changed it last year, where you need to show valid working for a problem to get full marks.
For mathematics (especially specialist) the strictness is ridiculous, if you have so much as a single line with an error (eg, you meant -(5+2) = -7 but you actually wrote -5+2 (forgot the brackets), but still said it = -7) you will be penalized marks.
Post by: TheMereCat on November 09, 2015, 04:23:54 pm
But also like aren't marks given for workings?
I have a tendency to do like twenty lines for a four mark question - probably not good?
I'd say it depends on the question you're attempting, but try to do just enough working out for you to be able to get the answer. Don't do the working out for the sake of just doing it, do it for the sake of getting the answer and make sure teachers are able to follow through, that's really all you need to score full marks on a question.
Post by: Peanut Butter on November 09, 2015, 04:56:54 pm
Post by: TheAspiringDoc on November 17, 2015, 09:14:41 pm
My exam went alright - I ended up getting 94 - not quite as good as I'd hoped, but close enough for now I guess.
For the fraction
For example, it is well known that
Thanks!
Post by: pi on November 17, 2015, 09:19:31 pm
Not sure if that's true.
The way you have written it, I'd say it = a/(bcd)
Post by: keltingmeith on November 17, 2015, 09:22:20 pm
Hey :D
My exam went alright - I ended up getting 94 - not quite as good as I'd hoped, but close enough for now I guess.
For the fractionwhat would be a generalised formula for the result?
For example, it is well known that/(bc))
Thanks!
First, see pi's comment for consistent writing in mathematics. :P
Second, try figuring it out for yourself! Ever heard of the stay, change, flip rule?