VCE Methods Question Thread!

[samuelbeattie76]

Let's say we solve for x and the solutions are 1 and 2.

Which is more correct to say?:
1) x= 1 and 2
2) x= 1 or 2.

[FallingStar]

Let's say we solve for x and the solutions are 1 and 2.

Which is more correct to say?:
1) x= 1 and 2
2) x= 1 or 2.

Hmm... I don't really know since I actually answer these questions as x=1,2.  As in I use a comma to separate the two possibilities. Can a mod say which is the correct method to write, or are all these methods equally valid?

[sweetiepi]

Let's say we solve for x and the solutions are 1 and 2.

Which is more correct to say?:
1) x= 1 and 2
2) x= 1 or 2.

Just my personal preference, I would say:
x=1 and x=2. However, most assessors do allow x=1 and 2, from memory.

[RuiAce]

The comma is most mathematically correct, surprisingly. Not "or" or "and"

In my opinion, if they're going to fuss about this in the VCE then they're being harsh.

What comes next will overcomplicate a lot of things for the average VCE (AND average HSC) student. Technicality issues with grammar (and in particular, logic) are really annoying and it really forces someone to think.

Overcomplicated analysis

Firstly, the comma has preference here due to the fact that it demonstrates 'listing'. Fact is, the solutions to the quadratic (or logarithmic or etc.) equation do not impact each other. It is better to just list the solutions here, rather than try to use words unnecessarily.

x=1,2

Some pedantic punctuation freaks such as me will ask why the comma is favoured over the semi-colon due to the grammatical issues. Short answer: Mathematicians are lazy. Alternate answer: I'm not sure if the semi-colon even appears in maths. But if it does, I either forgot, or just haven't seen it yet.

Now, the deal with the words "and" and "or" come to exclusivity/inclusivity.

"And" is a bit more flawed. In the context of logic, "and" usually means that both conditions must be satisfied simultaneously. This immediately leads to x=1 and 2 being a problem, because we're saying that x must be two things at once.

x=1 and x=2 sort of avoids it depending on what angle you view it at. But from another angle it doesn't really work.

Hence we turn to "or". In the context of logic, "or" without a description means the "inclusive or". The inclusive or caters for the "and" case as well (assuming there is one). That is to say, it can be one, or the other, or both at the same time.

The "exclusive or", on the other hand, states that x has to be one or the other. Depending on the scenario this also causes problems.

And that will break people's minds a bit. Because when someone says "or", we're used to interpreting the exclusive or. Not the inclusive or.

So what about x=1 or 2 v.s x=1 or x=2? This is a bit harder and both look roughly equally valid to me. Here though, I would favour the former in a way because the latter might imply x=1 and 2

[solution]

I calculated a 98% confidence interval, but the interval goes over 1. (0.8340, 1.0091) Did I make an error?

[blacksanta62]

because it is a rectangle, f(a) to f(b) must be parallel to the side on the X axis (a,0) to (b,0)
Therefore f(a)=f(b)
and because you know the domains of a,b
you can sub both values in to f(x)
to get 2a= f(a)        and      30-3b=f(b)
equate both and you get
2a=30-3b
solve for b and you get:

Thank you for the explanation
Would you (or anyone) know what part of the course this question relates to?
I'm thinking it relates heavily to min and max problems. If so, will doing enough practice enable me to do these questions more fluently? See attachment for part b)

Thanks

[MB_]

Thank you for the explanation
Would you (or anyone) know what part of the course this question relates to?
I'm thinking it relates heavily to min and max problems. If so, will doing enough practice enable me to do these questions more fluently? See attachment for part b)

Thanks

I'm not exactly sure what part of the course it relates to but from what I've seen usually those questions relate to max/min problems but I've seen a few questions similar to that in the later and last questions of the exam 1s (see attached). Also, I'm not sure if you needed help with that question but I'll help anyway.

[samuelbeattie76]

Would it be fair to say the function g(x) = 8x³ - 9x² -14x +12 which is of the 3rd degree imitates the function f(x) = 2x⁴ - 3x³ - 7x² +12x -4 which is of the 4th degree?

I know this is an important aspect of calculus and finding the turning points of polynomials but I’m not sure what this is called and how I would describe what is happening.

[MightyBeh]

Would it be fair to say the function g(x) = 8x³ - 9x² -14x +12 which is of the 3rd degree imitates the function f(x) = 2x⁴ - 3x³ - 7x² +12x -4 which is of the 4th degree?

I know this is an important aspect of calculus and finding the turning points of polynomials but I’m not sure what this is called and how I would describe what is happening.

Looks like g(x) is just the derivative of f(x). From what I can tell that's all there really is to say about them, unless I've misunderstood the question.

[RuiAce]

Would it be fair to say the function g(x) = 8x³ - 9x² -14x +12 which is of the 3rd degree imitates the function f(x) = 2x⁴ - 3x³ - 7x² +12x -4 which is of the 4th degree?

I know this is an important aspect of calculus and finding the turning points of polynomials but I’m not sure what this is called and how I would describe what is happening.

Yeah like MightyBeh said it's just the derivative. I've never heard of "imitation" in the world of mathematics.

The method of finding the turning points is called the method of finding the turning points. It doesn't have a special name because it's a tool that we use,

Why does the tool work? Because by DEFINITION, the derivative is 0 when the tangent is horizontal. And by definition, the tangent being horizontal means that we are at a stationary point

[blacksanta62]

I'm not exactly sure what part of the course it relates to but from what I've seen usually those questions relate to max/min problems but I've seen a few questions similar to that in the later and last questions of the exam 1s (see attached). Also, I'm not sure if you needed help with that question but I'll help anyway.

Why b - a? I'm not getting that at all
Sorry, really struggling with this part of the course
Thank you for the help though

Also, could someone explain the attachment to me? Why do we replace it with a t?

Edit:
And CDF in general. The textbooks explanation isn't very helpful right now. Trying to self learn this stuff is quite a challenge
Thank you

[RuiAce]

Why b - a? I'm not getting that at all
Sorry, really struggling with this part of the course
Thank you for the help though

Also, could someone explain the attachment to me? Why do we replace it with a t?

Edit:
And CDF in general. The textbooks explanation isn't very helpful right now. Trying to self learn this stuff is quite a challenge
Thank you

____________________




They used c because they assumed that the lower bound wasn't negative infinity, but an actual number. I've seen questions in here that make it go to negative infinity though.

[MB_]

Why b - a? I'm not getting that at all
Sorry, really struggling with this part of the course
Thank you for the help though

Also, could someone explain the attachment to me? Why do we replace it with a t?

Edit:
And CDF in general. The textbooks explanation isn't very helpful right now. Trying to self learn this stuff is quite a challenge
Thank you

Its b-a because that is the length of the rectangle. Say for example b=10 and a=2, we want to find the length from b to a so we subtract a from b giving us 8. Hopefully that helps but if it doesn't I hope someone else can explain it better. Also, I've attached the question with a few arrows, hopefully it helps visualize it.

[lzxnl]

____________________




They used c because they assumed that the lower bound wasn't negative infinity, but an actual number. I've seen questions in here that make it go to negative infinity though.

The usual definition in higher level probability has the lower bound as negative infinity, with the density function set to zero at all unwanted values.

[RuiAce]

The usual definition in higher level probability has the lower bound as negative infinity, with the density function set to zero at all unwanted values.

I know that lol