VCE Methods Question Thread!

[knightrider]

This is just a poorly written question. In ANY VCAA exam, they will be more obvious as to if they want the line or the distance. In a SAC, just ask your teacher - if they don't want questions, they'll write the SAC better.

Thanks Eulerfan101  How would VCAA show you what you wanted like if they wanted you to use distance or find the line.

I have not yet covered this topic buddy, but from personal knowledge, if it was asking for the distance (magnitude) it would most likely be denoted as |AB|. This means that we want to find the length of the line segment from the point A (-2, 5) to point B (1, 3).

So, without confirmation, I would say they are asking you to find the line AB.

Thanks Cosine

[Spxtcs]

How come when writing notation in maths.

Whenever infinity is mentioned it is always in closed brackets. eg

why cant it be in square brackets eg

Is there a reason behind this?

someone else, maybe Eulerfan could clarify better than I, but infinity cannot be included because it isn't a finite (heh) value that can be pinpointed on a graph. Square brackets denote that the interval is up to and including that number, e.g. [2,9] includes the numbers 2 and 9, but (2,9] does not include the number 2 on the axis. Infinity is such a big thing that can't be included because it just keeps going.

EDIT: cosine beat me to it xD

[keltingmeith]

Thanks Eulerfan101  How would VCAA show you what you wanted like if they wanted you to use distance or find the line.

They would say something along the lines of "find the equation of the line connecting AB" or "find the distance AB". They'd just put in another word to make it a bit more obvious. It's true that often in high stress situations, you'll interpret something wrong, and then do the question wrong - but VCAA are nice enough that out of the stress, you can see what they would've wanted. Namely because if you couldn't, the teachers tell them off (hell - my old methods teacher said that he was going to call up about last year's exam, just because ONE multiple choice question required you to assume a probability distribution was continuous).

How come when writing notation in maths.

Whenever infinity is mentioned it is always in closed brackets. eg

why cant it be in square brackets eg

Is there a reason behind this?

It's because infinity is not a number. Infinity is a concept - what happens when you keep counting forever and ever? Well, numbers don't end, because if you find the final number, I can just add one to it. And if you THEN get the highest number, I'll just add one again, and find a HIGHER number. Infinity is the idea of what happens at the end - when you get so big that multiplying it by itself 37419872802970923870298350 times doesn't even make a difference at just how big it is.

THAT is why we use open brackets - infinity isn't a number we can just treat like a number. If f(x)=1/x, then f(infinity) doesn't exist. BUT , because whilst we can't jump to infinity, we can slowly approach it until we get to a number so high that it may as well be infinity.

Understanding infinity, and why we do things such as use open brackets with it or why we can only use limits with it, is an abstract concept. It's DEFINITELY a concept worth pursuing and trying to understand (for fun! in fact, a good recreational maths class will at least discuss infinity), but don't expect it to be easy or as concrete as things such as the quadratic formula.

EDIT: Already two comments, but I felt this offered up something new about the innate beauty of infinity, so I'll keep it~

[Spxtcs]

Doesn't matter, your explanation is better than mine!

And still EulerFan smashes our explanations combined, lol -.-

that's what I was thinking lmao.....I'll leave this thread to him

[knightrider]

This is because infinite is endless, so how can we include infinite, if it never really ends, that is, how can we include a number that really goes on forever?

Just think of it as the graph goes on forever, it never ends, so it has the open brackets for infinite (,) which means it is NOT included. But, when we have closed brackets [,] this means that the points are included, and we know we cannot include infinity because the graph doesn't stop at infinite because it doesnt stop at all!

Thanks Cosine 

someone else, maybe Eulerfan could clarify better than I, but infinity cannot be included because it isn't a finite (heh) value that can be pinpointed on a graph. Square brackets denote that the interval is up to and including that number, e.g. [2,9] includes the numbers 2 and 9, but (2,9] does not include the number 2 on the axis. Infinity is such a big thing that can't be included because it just keeps going.

EDIT: cosine beat me to it xD

Thanks Spxtcs 

They would say something along the lines of "find the equation of the line connecting AB" or "find the distance AB". They'd just put in another word to make it a bit more obvious. It's true that often in high stress situations, you'll interpret something wrong, and then do the question wrong - but VCAA are nice enough that out of the stress, you can see what they would've wanted. Namely because if you couldn't, the teachers tell them off (hell - my old methods teacher said that he was going to call up about last year's exam, just because ONE multiple choice question required you to assume a probability distribution was continuous).

It's because infinity is not a number. Infinity is a concept - what happens when you keep counting forever and ever? Well, numbers don't end, because if you find the final number, I can just add one to it. And if you THEN get the highest number, I'll just add one again, and find a HIGHER number. Infinity is the idea of what happens at the end - when you get so big that multiplying it by itself 37419872802970923870298350 times doesn't even make a difference at just how big it is.

THAT is why we use open brackets - infinity isn't a number we can just treat like a number. If f(x)=1/x, then f(infinity) doesn't exist. BUT , because whilst we can't jump to infinity, we can slowly approach it until we get to a number so high that it may as well be infinity.

Understanding infinity, and why we do things such as use open brackets with it or why we can only use limits with it, is an abstract concept. It's DEFINITELY a concept worth pursuing and trying to understand (for fun! in fact, a good recreational maths class will at least discuss infinity), but don't expect it to be easy or as concrete as things such as the quadratic formula.

EDIT: Already two comments, but I felt this offered up something new about the innate beauty of infinity, so I'll keep it~

Thanks eulerfan101  everything makes sense now.

[keltingmeith]

Doesn't matter, your explanation is better than mine!

And still EulerFan smashes our explanations combined, lol -.-

Nah, I've just done units of maths where infinity as a concept has been discussed.

In fact, it's really quite amazing just how infinity works. If you take the two number sets, the natural numbers and the integers, one would assume that the integers are twice as big as the natural numbers. HOWEVER, it is possible to make a construction where the two sets have a one-to-one correspondence - that is to say, they have the same "size". Even the rational numbers isn't any bigger, and this defines a level of infinity we call "countable".

Even more impressive is an "uncountable" infinity. The real numbers ARE bigger than the natural numbers, but they're SO big that even the interval [0, 1] is shown to have the same "size" as ALL the real numbers (as someone mentioned earlier - Conic, I think?)

Even the quotient isn't well defined, and to try and figure out what the answer is we often talk about "rates", where we consider the numerator and denominator as functions, and see how "fast" each function moves off to infinity. For example, is ALWAYS infinity, just because the exponential function is so much faster than any polynomial. However, is one, because both functions move off at the same rate.

And I could ramble about this forever and ever, moving from topic to topic about things that don't seem to really relate at all, buuuuut there are much better ways to waste your time, hahah.

[knightrider]

what would be the domain for this graph and why?

[brightsky]

Domain = {4}
Range = R

[knightrider]

Domain = {4}
Range = R

thanks brightsky

that would mean 4 is the only x value for which the function is defined.

[Zues]

Regarding the first screen shot, I would agree that the function would also have an inverse with the domain restricted to (-∞,1]. Regarding the second screen shot, they want S to be the restriction of the implied domain of the function so that it's inverse is also a function. In this case S = [1,∞), as the function is one-to-one over its implied domain, and hence its inverse is a function.

Somewhat off topic

The phrase "largest domain" is used a lot in methods. The intervals [1,∞) and [2,∞) contain the same number of elements, but in methods [2,∞) is considered "smaller".

From wiki: "Between any two real numbers a<b, no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the open interval (a,b) is equinumerous with ."

Cardinality of the Continuum

i know how to solve the second screenshot, but they didnt mention anything like the largest set of S to produce an inverse function. how they have said it can mislead me to to put its domain, rather then restricting it?

also with the graph of y = loge(x^2)  and (logex)^2, why is graphed the way it is? i.e. with the cusp at 0,0 (if thats what you call it)

[brightsky]

thanks brightsky

that would mean 4 is the only x value for which the function is defined.

Bear in mind that the graph that you've provided is not the graph of a function, as it does not pass the vertical line test. It is a one-to-many relation.

[knightrider]

Domain is considered as the set of all x-values in the graph. As we can clearly see, the only x-value that is possible with this graph is 4. So obviously domain: {4}

Range isn't 4, because there are heaps and heaps of y-values! So the range: R (which is basically every single number, positive, negative and zero).


You said this graph is a function, to be specific it isnt a function, in order for a graph to be classified as a function, the first ordered pair (the x-values) must only occur once, but as we can see x=4 occurs infinitely number of times. For example, at y=-1, the x value is still 4, and at y=1 the x value is 4.


To make it easier for yourself, draw a line down at any point on the graph, if the line intercepts the graph more than once, then the graph is not a function.

Hope this helped!

yep thanks

Bear in mind that the graph that you've provided is not the graph of a function, as it does not pass the vertical line test. It is a one-to-many relation.

Thanks

[knightrider]

i just wanted to know if there is anything in terms of geometry and shapes from junior years that can pop up on VCAA exams like things that are unexpected.

[pi]

i just wanted to know if there is anything in terms of geometry and shapes from junior years that can pop up on VCAA exams like things that are unexpected.

Knowing basic things like areas, volumes, basic angles (in relation to polygons, or directions on maps and so forth), and the concept of "similar shapes" are probably it.

[knightrider]

Knowing basic things like areas, volumes, basic angles (in relation to polygons, or directions on maps and so forth), and the concept of "similar shapes" are probably it.

Thanks pi