BEC'S methods questions

[bec]

On the study design, a couple of the key knowledge dotpoints are:

1. Derivatives of xⁿ, for n ∈ Q, e^x, log_e(x), sin(x) and cos(x) and tan(x) (formal derivation is not required)

What do they mean by "formal derivation is not required"? Does this mean we need to know how to plug x^cos(x) into our calculator to find its derivative, but don't need to show any working?

2. Bernoulli trials and two state markov chains, including the length of run in a sequence, steady
values for a markov chain (familiarity with the use of transition matrices to compute values of
a markov chain will be assumed)

I'm sure I've learnt about them before, maybe just with a different name...but what are "bernoulli trials"?

Thanks

[cara.mel]

1. Do you remember (probably in unit 1/2) doing differentiation by first principles, with the limits and such?
It means you don't have to do that, and can just go from d/dx (x^3) = 3x^2. Aka what you normally do so don't worry about it

2. Yeah you would have learnt then, stuff like "I do something 5 times whats the probability I win 3 times"

[excal]

Bernoulli trails = Binomial (two) events - success or failure.

(i.e., binomial distribution)

[bec]

thanks

[bec]

Does anyone know if you use the 2nd derivative in methods? Sign diagrams are so... messy...and I make mistakes more often in them than I do finding the 2nd deriv...

Also, if the equation is y = bla, and you need to show the derivative when x=a, is it alright to just write out ? I noticed kilbaha likes _x=a, but I'd never seen that before...

[shinny]

I've been told we're not allowed to. But then people on this forum keep saying you're allowed to use anything thats mathematically correct (so we're even allowed to use uni maths theorems? hmph). I'll let someone who's got a better source to give a definitive answer to that...

For the second thing, yeh, I tend to just write when x=a, dy/dx=etcetc. Never seen Kilbaha's notation before...Examiners aren't ridiculously fussy on notation at this level since even a lot of the top students won't have been taught notation perfectly due to a lack of good teachers+sources+a curriculum that doesn't really care about it...

[bec]

Thanks for that, I think I'll ask my teacher what she thinks about the 2nd derivative thing..
Also, this probably doesn't matter either, but when you define variables in probability, is it like with curly brackets or not?
eg. "Let X = {Bob catches the train}" or "let X = Bob catches the train"?

[Glockmeister]

I've personally never used the curly brackets myself.

Does anyone know if you use the 2nd derivative in methods? Sign diagrams are so... messy...and I make mistakes more often in them than I do finding the 2nd deriv...

Also, if the equation is y = bla, and you need to show the derivative when x=a, is it alright to just write out \frac {dy}{dx} \_ when \_ x=a? I noticed kilbaha likes \frac {dy}{dx}|x=a, but I'd never seen that before...

If the question asks you to use a particular method, then you need to use that method. My guess otherwise would be that it's ok to use non-methods techniques but perhaps someone should email VCAA to make sure.

[shinny]

From what I've been told, we're not allowed to because spesh students aren't meant to have any advantage in terms of techniques over methods students, otherwise it would be just outright unfair. While on the topic anyhow, do we ALWAYS have to verify min/max? Even if its a 1 or 2 mark question? o_O I've seen some past papers skip it while others don't. Very confusing.

[bec]

I always specify min/max, just in case - it doesn't take long if you've got your calculator, and if not it's still normally quite straightforward.
I just sent VCAA an email about using methods outside the scope of the course, I'll post their reply here when I hear from them!

[Collin Li]

From what I've been told, we're not allowed to because spesh students aren't meant to have any advantage in terms of techniques over methods students, otherwise it would be just outright unfair.

I side with this point of view. Also, on a practical point, a Methods examiner is not necessarily aware of the technique, so it may just be marked as a 'fudge.' That said, however, you are rarely asked to justify the nature of a stationary point on a Methods exam. I would stick to the sign diagram. It's actually easier than differentiating again, in a lot of cases.

While on the topic anyhow, do we ALWAYS have to verify min/max? Even if its a 1 or 2 mark question? o_O I've seen some past papers skip it while others don't. Very confusing.

Nope. You can just assume it. It's not part of the marks to justify it unless they ask you to. Part of this is because the functions you study which have minima and maxima are functions that you should be able to visualise, and hence graphically justify.

[Collin Li]

Thanks for that, I think I'll ask my teacher what she thinks about the 2nd derivative thing..
Also, this probably doesn't matter either, but when you define variables in probability, is it like with curly brackets or not?
eg. "Let X = {Bob catches the train}" or "let X = Bob catches the train"?

I'd just write "Let X be the event that Bob catches the train"

If I wanted to define X as a space of events, I'd go: X = {A, B} (where A and B are previously defined), so that's where it comes from really. I guess your event spaces should really be comprised of individual events.

Read more here: http://en.wikipedia.org/wiki/Probability_space

It's too formal for them to care about really.

[bec]

Thanks for the answers coblin, I saw them but forgot to reply
Can anyone give me a hand with this question from the thread about "possible questions" here: http://vcenotes.com/forum/index.php/topic,5600.0.html?

The weight of a colony of penguins follows a normal distribution. 80% weigh more than 100kg, 10% weight less than 70kg. Find .

penguins, penguins


Use binomial distribution in conjunction with normal distribution. How many penguins would need to be weighed to ensure that the probability of between 30 and 50 penguins weighing over 120kg is greater than 0.65?

Let X be the weight of a penguin (is this definition right?)







I have no idea how to do this! The only thing I can think to do is ridiculously tedious (and possibly wouldn't even work):
Solve for Pr(30<X<40)=0.65
₃₀ .65 + ₃₁ .65 ....... ₄₀ .65 = 0.65
Then whatever the solution for "n", state that the population must be LESS THAN "n"

What's the best way to work it out though, because I'm sure it's not that...

[ReVeL]

Ok for the first question.
The weight of a colony of penguins follows a nomal distribution. 80% weigh more than 100kg, 10% weight less than 70kg. Find .

Ok so and

To find and we must standardise the data and apply it to the standard normal distribution(mean=0, sd=1)

Standardising the first part:

To find this we use . It is 0.2 because invNorm gives the upper limit, so 0.8 would give us the top 20% not top 80%.


Therfore, as
We can say that

Standardising the second part:
To do this we use . This time it is 0.1 as it is Pr that X is LESS than 70.


Therfore, as
We can say that

Now we have 2 equations, and two unknowns. We can solve them simultaneously to find and .

Equation 1: and Equation 2:

Rearrange each to make the subject.

Equation 1: and Equation 2:

Let them equal each other, then solve for .





Then sub this back into either equation to find which is equal to .


Sorry for the awful setting out you probably can't understand what I mean... just get coblin to do it.

[bec]

Thanks Revel, that's what I got too and I worked it out the same way.
Anyone have any ideas for the second question?