Would it be a good idea to know of any 4u techniques? (1 Viewer)

[rsingh]

Hey guys,

I was just wondering, for the trials and HSC itself, would it be a good idea to know any particular 4u techiques that would make particular 3u questions easier and shorter to do. I mean, my friend is doing 4u and he has taught me some techniques that make a few integration and polynomial divison easier.

But the thing is, even if we knew it, are we allowed to use 4u techniques on a 3u paper?

Any comments would be great.

 

[who_loves_maths]

^ yes you are allowed to use 4u techniques in 3u in a maths exam. ppl who do 4u must do the 3u exam too, so they can use whatever technique(s) they want from the three maths courses.

i can't remember specific examples of where 4u can come in useful for 3u questions, but i am sure that bits of the 4u topics like Graphing, Polynomials, etc... can be seen as supplements for their 3u counterparts. so if you have time, then it certainly wouldn't hurt to go through some 4u techniques/topics i suppose.

Edit: there are four maths courses, sorry, not three in the HSC.

 

[FinalFantasy]

some of the graph adding and subtracting techniques in 4u would turn out useful in 3u, and having a good understanding of 4u polynomials would allow u to do 3u ones easily

 

[thunderdax]

4u polynomials makes a lot of difference but graphing isn't that important. The best thing to know in 4u polynomials for 3u is how to find the equation with roots a², b², etc given equation with roots a, b and c and stuff like that. It takes so little time to do it the 4u way then using 3u methods. There are a few other examples but I can't really think of them at the moment.

 

[withoutaface]

I wouldn't bother personally, as the 3u method is almost without exception the most elegant one, and to fill your head with other methods that leave you trying decide which is the best to use probably isn't the best of ideas.

 

[ishq]

Hey guys,

I was just wondering, for the trials and HSC itself, would it be a good idea to know any particular 4u techiques that would make particular 3u questions easier and shorter to do. I mean, my friend is doing 4u and he has taught me some techniques that make a few integration and polynomial divison easier.

But the thing is, even if we knew it, are we allowed to use 4u techniques on a 3u paper?

Any comments would be great.

Whatever you use, make sure you're comfortable with it. ^ withoutaface is correct about sitting there trying to decide...
I always feel uneasy about using short-cuts (with my history of silly mistakes) - so I prefer to stay away from them...

But yeah, 4U stuff is accepted in 3U.

 

[who_loves_maths]

can someone plz correct me (or my teacher) here if what i say is wrong:
my teacher told me that in any HSC maths exam, you are allowed to use ANY technique(s) to solve a problem, incl. those not pertinent to the syllabus courses, as long as it's mathematical. is that true?

 

[Slidey]

Yes, I've also heard that. I use L'Hopital's rule whenever possible on trig limit questions.

 

[haboozin]

can someone plz correct me (or my teacher) here if what i say is wrong:
my teacher told me that in any HSC maths exam, you are allowed to use ANY technique(s) to solve a problem, incl. those not pertinent to the syllabus courses, as long as it's mathematical. is that true?

ofcourse...

unless they tell u what method to use.
eg find the integral by parts.
but sometimes they say something like
by using similar triangles, or otherwize prove this..

so we can prove it whatever way we want

even with physics formula's about projectile, they are the same as maths ones just simplified. we can still use them if we prove them by deriving.
U just cant quote them.

but mostly they set the questions in such a way for you to follow the way they want you to do it..

eg in 3u. for a 5 mark graph scetching question they will ask like

i: find stationary points, and their nature.
ii. points of inflextion

so using 4u graphing in that question is pretty useless since u'll miss out on most of the marks.

 

[who_loves_maths]

^ okay, thanks.
just making sure that you can use extraneous theory.

Originally Posted by Slide_Rule
Yes, I've also heard that. I use L'Hopital's rule whenever possible on trig limit questions.

lol, we already use L'Hopital's rule implicitly with trig limits as it is taught in the 4u course don't we? there's no need to explicitly quote it whenever you apply the rule (unless it's not trig limits) is there?

 

[withoutaface]

Yes, I've also heard that. I use L'Hopital's rule whenever possible on trig limit questions.

<3 L'Hopitals

 

[Slidey]

lol, we already use L'Hopital's rule implicitly with trig limits as it is taught in the 4u course don't we? there's no need to explicitly quote it whenever you apply the rule (unless it's not trig limits) is there?

I thought we implicitly used the Squeeze? That's the method I used to prove the sin/x limit.

As for explicitly quoting it, hell no. Though you should be sure to ALWAYS mention what you are doing. If there's ever any possibility for ambiguity, explain your steps.

Suppose they say: "Find Lim [x-->0] 3x/sin(5x)"
Then you'd go:
Lim [x-->0] 3x/sin(5x) = Lim [x-->0] 3/{5cos(5x)} = 3/5 (by L'Hopital's rule)

 

[FinalFantasy]

is dere n e other quick tools like L Hopitals rule dat we can use and not in 4u syllabus?

 

[Slidey]

Let's see. This is not a proof of what we do in class, but it's still interesting:

-1 <= sin(x) <= 1
-1/x <= sin(x)/x <= 1/x

Lim [x-->inf] -1/x <= Lim [x-->inf] sin(x)/x <= Lim [x-->inf] 1/x
0 <= Lim [x-->inf] sin(x)/x <= 0

.'. Lim [x-->inf] sin(x)/x =0

Which is intuitively obvious since sin(x) will, at the most be |1|.

FF: Uh. I suppose. You've learnt synthetic division, right?

 

[who_loves_maths]

Originally Posted by Slide_Rule
I thought we implicitly used the Squeeze? That's the method I used to prove the sin/x limit.

Originally Posted by Slide_Rule
Let's see. This is not a proof of what we do in class, but it's still interesting:

-1 <= sin(x) <= 1
-1/x <= sin(x)/x <= 1/x

Lim [x-->inf] -1/x <= Lim [x-->inf] sin(x)/x <= Lim [x-->inf] 1/x
0 <= Lim [x-->inf] sin(x)/x <= 0

.'. Lim [x-->inf] sin(x)/x =0

Which is intuitively obvious since sin(x) will, at the most be |1|...

well, both The Squeeze Theorem and L'Hopitals Rule can be applied to prove the trig limits, whichever is more convenient. so implicitly, we're using the truth of both theorems in the 4u course.

 

[FinalFantasy]

FF: Uh. I suppose. You've learnt synthetic division, right?

if i have i don't know about it LoL

 

[who_loves_maths]

Originally Posted by FinalFantasy
if i have i don't know about it LoL

synthetic division is just Ruffini's Rule. it's hard to demonstrate here, so click on this:

http://en.wikipedia.org/wiki/Ruffini's_rule

for further examples:

http://www.themathpage.com/aPreCalc/synthetic-division.htm

 

[Slidey]

I like it because it is both long division and the remainder test combined, and quicker than both.

Here's the page I learnt it from: http://www.purplemath.com/modules/synthdiv.htm

Great site overall, actually. It covers a lot of different things.

*goes and checks out Ruffini's rule*

EDIT: Oh, and a note on L'Hopital's rule: NEVER use it if the denominator and numerator don't both evaluate to 0 or + infinity. And it can't be a combination, they must either both go to zero, or both go to infinity. And of course they must be differentiable at c if x->c.

 

[who_loves_maths]

Originally Posted by Slide_Rule
I like it [synthetic division] because it is both long division and the remainder test combined, and quicker than both...

yea in most cases it certainly is. but i suppose you have to becareful when you are met with a polynomial with only complex conjugate roots, in which case it'll be hard to know what linear factor to divide by.
ie. (just to remind FF) synthetic division (in the sites we posted above) doesn't work if you have/are looking for a quadratic factor {or in fact, even higher order factors) to divide - in this case, long division is still faster and more useful :uhhuh:

also, the division transformation and the remainder theorem combine to offer a much more effective mechanism with which one can find a remainder whose degree is equal to or larger than 1.

P.S. synthetic division is possible with quadratic factors, but is more complicated - check out the Lin-Bairstow Method:
http://math.fullerton.edu/mathews/n2003/BairstowMethodMod.html

 

[acmilan]

FF: If youre interested, heres some stuff you could take a look at. Theres a proof for one form of l'Hopital's rule in the 2nd one. There may be some general spelling errors because I havent had time to look over them after I initially converted them to electronic form, but most would be ok.

http://www.geocities.com/tehac86/2Functions.pdf
http://www.geocities.com/tehac86/4FoundationsofOneVariableCalculus.pdf
http://www.geocities.com/tehac86/6Functionsof2Variables.pdf

I loved the third topic, although I havent added the pictures so it may be hard to visualise. I'm not sure how much it would help in 4u since I didnt take it up and hence dont know the entire course