Ho Ho... No

[blacksanta62]

Thank you Crypt

[blacksanta62]

Today was packed with maths! Here s one of the questions I need help with:

2) 'party poppers' are plastic devices which use gun powder to launch streamers into the air. As the streamers unfurl, they get more wind resistance. The height of the centre of the streamers (y meters, measured from the solid ground) is modelled t seconds after pulling the cord, and has the following equation:

                                    y(t) = 1/6 (12t - t^2)e^2/5(2-t)

a) State a sensible domain for this equation (0<x<2 ??)
b) Find the highest point the streamers reach
c) The velocity at time t is given by y'(t). What is the maximum velocity, and when is it reached? (Would I differentiate to find the gradient function and the set it equal to 0?)

Thanks. If some working out or guidance is provided I thank you

Not doing the VCESS program anymore but if anyone could explain this question It would be sweet.

[blacksanta62]

A bit of an application question....
In specialist maths, if told to "prove" something what should I do first? This is the first time I've encountered a "prove that" question in my textbook questions which is under the preliminary topics section... lol, this is the first time I'm doing modulus functions

The question is: Prove that |x-y| is greater than/equal to |x| + |y|
This is not the only "prove that" question, there's 2 more after it and then I've done the whole chapter...
I have spesh tomorrow so I'll ask any questions I have to my a-maze-zinga teacher but wanted a bit of insight into how to even attempt these questions

On a more "newbie" question: for each of the following, illustrate the set on the number line and represent the set using interval notation: C) {x : |x-2|</equal to 1}
I was okay with questions like this: {x : |x| < 3} but I'm thrown off. What should I do?

Thank you

[wyzard]

A bit of an application question....
In specialist maths, if told to "prove" something what should I do first? This is the first time I've encountered a "prove that" question in my textbook questions which is under the preliminary topics section... lol, this is the first time I'm doing modulus functions

The question is: Prove that |x-y| is greater than/equal to |x| + |y|
This is not the only "prove that" question, there's 2 more after it and then I've done the whole chapter...
I have spesh tomorrow so I'll ask any questions I have to my a-maze-zinga teacher but wanted a bit of insight into how to even attempt these questions

On a more "newbie" question: for each of the following, illustrate the set on the number line and represent the set using interval notation: C) {x : |x-2|</equal to 1}
I was okay with questions like this: {x : |x| < 3} but I'm thrown off. What should I do?

Thank you

Ah proofs, proofs are the real deal in mathematics and make them a lot more fun How do we prove a theorem is true? Unfortunately there are usually no direct method when it comes proving theorems, such as applying a formula, and involves a bit of trial-and-error.

Usually for proofs, you'll be given a statement and asked to prove a certain property of that statement, like |x+y| being less than or equal to |x| + |y| (You've got it the other way round by the way, this is called the triangular inequality). For specialist, these two methods of proofs will be sufficient:
1) Direct Proof
2) Proof by cases

Generally for proving questions, start off by listing down what you know is true, like for |x+y|, we know both x and y are real numbers (or possibly complex), then we also know the properties of real numbers, such as a+b=b+a etc. With this, try to see if you apply what is already known to be true, and show that |x+y| is indeed less than or equal to |x|+|y|. To do so you can apply the method of proofs I've listed out earlier. For absolute values, it's a good idea to do by cases.

Other method of proofs, but not limited to, include:
- Contrapositive
- Proof by contradiction
- Mathematical Induction

To understand proofs better and how it can be used, I've listed 4 different proofs for a different theorem where the sum of the first n odd number equals n squared here in my blog: http://www.nerdofpassion.com/blog/fun-with-maths-4-ways-to-prove-that-the-sum-of-the-first-n-odd-numbers-equals-to-n-squared for you to check out!

Hope this helps!

[blacksanta62]

The only proofs that I saw in year 11 was the geometric proof for vectors. It was the only time I was told to prove something and I didn't understand how to do it. I would be given a triangle with vectors a and b etc then a midpoint, p, for example and told to prove that b is 2a or something (not accurate ). The fact that I couldn't use an equation or a shortcut was what, I feel, confused me the most and even got me to the point of just hating it.... lol, I've built a bridge and gotten over it but I'm going to have to face it this year.

Thanks Wyzard, I'll read the blog and probably bookmark it too!

[wyzard]

The only proofs that I saw in year 11 was the geometric proof for vectors. It was the only time I was told to prove something and I didn't understand how to do it. I would be given a triangle with vectors a and b etc then a midpoint, p, for example and told to prove that b is 2a or something (not accurate ). The fact that I couldn't use an equation or a shortcut was what, I feel, confused me the most and even got me to the point of just hating it.... lol, I've built a bridge and gotten over it but I'm going to have to face it this year.

Thanks Wyzard, I'll read the blog and probably bookmark it too!

Yeah I get that feeling, mathematical proofs are daunting at first. When I first come across proofs I was stumped. But once you start to get the hang of them, you'll start to see the beauty of maths, how one thing is built upon another, and you'll truly understand maths. If formulas are machines of mathematics, then proofs are the gears.

Don't rush through the proofs, make sure you examine every step properly, examine the details and practice coming out with some yourself. It's a really good practice to start examining proofs early in high school, as uni maths is filled with mathematical proofs.

[blacksanta62]

Hello, I just need a bit of explanation on this question:
Consider the circles with the following equations:
4x^2 + 4y^2 - 60x -76y +563 = 0 and x^2 + y^2 - 10x -14y +49 = 0

a) Find the radius and the coordinates of the centre of each circle

b) Find the coordinates of the points of intersection of the two circles

I'm thrown off by the first equation and can't seem to get it into the circle equation form
Also, with the second equation, I graphed these two circles and found that the second equation has a much larger radius which allows it to intersect the other circle. How would I prove it? Or am I just allowed to write down the coordinates by sight e.g. it intersects at the coordinates (5,6) and (7,8)

Thank you

[zsteve]

Hello, I just need a bit of explanation on this question:
Consider the circles with the following equations:
4x^2 + 4y^2 - 60x -76y +563 = 0 and x^2 + y^2 - 10x -14y +49 = 0

a) Find the radius and the coordinates of the centre of each circle

b) Find the coordinates of the points of intersection of the two circles

I'm thrown off by the first equation and can't seem to get it into the circle equation form
Also, with the second equation, I graphed these two circles and found that the second equation has a much larger radius which allows it to intersect the other circle. How would I prove it? Or am I just allowed to write down the coordinates by sight e.g. it intersects at the coordinates (5,6) and (7,8)

Thank you

(a) I would complete the square here. I'm guessing you will be armed with CAS so use completeSquare(4x^2 + 4y^2 - 60x -76y +563 = 0, x, y) on the TI, which returns:
.
Do the same for the other circle.

The rest follows - find radius and coords from completed-square equations.

To find the intersections, I'd say use a CAS . There are probably ways to find the intersection, though, but my mind is currently blank. You can try to solve simultaneously, but you'd only end up with another linear equation, so further investigation is required.

[blacksanta62]

Thanks zsteve. The textbook were using for spesh, atm, is Cambridge. There wasn't a calculator icon like there was in the maths quest book we had last year. My question is, is this still doable by hand? I started and then got lost in a sea of pencil lead, rubber shavings and smudges on paper 

For both that is. Thank you for answering

[wyzard]

Thanks zsteve. The textbook were using for spesh, atm, is Cambridge. There wasn't a calculator icon like there was in the maths quest book we had last year. My question is, is this still doable by hand? I started and then got lost in a sea of pencil lead, rubber shavings and smudges on paper 

For both that is. Thank you for answering

It is doable by hand, but you'll have to be very careful dealing with the fractions, it is a very intricate manipulation. Just practice a couple of times and you'll get the hang of it.

[blacksanta62]

My teacher went through it today, by hand. You're right, it was easy for us to make mistakes, especially with part b of the question. Is it true that with the tech free exams that we're expected to be able to manipulate 'hard' numbers like 429 for example.

[blacksanta62]

That wasn't a very good example but I hope anyone reading gets what I'm trying to say: Like this polynomial: 34x^5+76x^5+25x^4-87y^2+|-87xy|.....  lol

[blacksanta62]

Throwing a hypothetical scenario at the reader: say you were in a SAC or exam situation. The next question to appear on the paper was "sketch the graph of: y= 4 cos(2x) + 3 and asked to show all relevant intercepts and end points. The domain is [-2Pi/3, 2Pi]. This is a tech enabled question and worth 4 marks.

How many marks are allocated to the shape and look of the graph? My methods teacher since last year consistently tells me off for having "bad graphs" and says he won't even look at them. How does one get better at sketching any graph since I get dogged out for them too   

Show relevant working out but a sketch isn't needed, just some tips on shape and flow which will help me.

P.s. I have done this question and many others but he probably won't mark them off as done  . And how much should that question be worth? There's no horizontal shift or - to worry about

Thank you

[lzxnl]

That wasn't a very good example but I hope anyone reading gets what I'm trying to say: Like this polynomial: 34x^5+76x^5+25x^4-87y^2+|-87xy|.....  lol

That's not a polynomial because of the mod sign

Throwing a hypothetical scenario at the reader: say you were in a SAC or exam situation. The next question to appear on the paper was "sketch the graph of: y= 4 cos(2x) + 3 and asked to show all relevant intercepts and end points. The domain is [-2Pi/3, 2Pi]. This is a tech enabled question and worth 4 marks.

How many marks are allocated to the shape and look of the graph? My methods teacher since last year consistently tells me off for having "bad graphs" and says he won't even look at them. How does one get better at sketching any graph since I get dogged out for them too   

Show relevant working out but a sketch isn't needed, just some tips on shape and flow which will help me.

P.s. I have done this question and many others but he probably won't mark them off as done  . And how much should that question be worth? There's no horizontal shift or - to worry about

Thank you

How to sketch a sine curve? This is what I do, and it ALWAYS gives me a decent-looking sine curve.
Let's just take y = sin x for simplicity; you can reflect and translate for cos curves or other curves.
Anyway, the trick is to start at a minimum, so (-pi/2, -1) works fine.
Start off flat and draw a curve whose slope is gradually increasing and passes through the origin.
As soon as you hit the origin, make sure the slope of your curve stops increasing and starts decreasing. Make this curve have a turning point at (pi/2, 1). Do a similar thing on the way down.

Try this a few times and see if it helps. It's what I do all the time.

I'd say surely one mark at least for the shape of the graph, although I don't VCAA will be too strict on these.

[blacksanta62]

That's not a polynomial because of the mod sign. 

I was adding anything and everything on my keyboard to make and example and get an answer for the question above it  What I wanted to know was does VCAA want students to be able to manipulate what's given to them or hand them with nice turning out numbers. Wanted to know this for tech free tests

I just wanted tips for trig graphs in general  not only sin. My example just happened to be a cos graph doe