Having some trouble with some of last year's exam questions (CAS)

[Lulu]

This question from last year's VCAA exam 2 paper is driving me mad!! I have the answers, but don't exactly follow what they're doing...

Question 4:
d.

For f(x) = 1-e^(-x)

show that f(u)f(v) = f(u) + f(v) - f(u+v) where u and v are real numbers.

If anyone could help me out I'd greatly appreciate it.

Another question is regarding question 3, part d ii. It's a pretty long question so if anyone's completed it, could you please explain it to me?

Is it just me, or was last year's exam paper very difficult? I'm usually pretty confident with my methods, but this paper...has just made me so insecure about the damn subject.

By the way, nice to meet you all!

[Collin Li]

As required

[Collin Li]

If anyone could help me out I'd greatly appreciate it.

Another question is regarding question 3, part d ii. It's a pretty long question so if anyone's completed it, could you please explain it to me?

Is it just me, or was last year's exam paper very difficult? I'm usually pretty confident with my methods, but this paper...has just made me so insecure about the damn subject.

I can't find these questions on 2007 MM Exam 2.

[Lulu]

Hey coblin,

Thanks for the quick reply. I understand most of it except the factorisation in this line:

By the way, the exam is the CAS version  Sorry, should have made that a bit clearer.

[Collin Li]

OH! Okay. I'll have a look. That line wasn't factorisation. I've put some things in brackets to make it clearer, but all I've done is, from the line before:



(add 1 and subtract 1)

(just rearranging the order of them, nothing has changed)



I put brackets around some terms for clarity, once again, nothing has changed, although I have factorised out a negative in one of the terms.

[cara.mel]

Welcome Lulu
Yeah, 2007 exam was hard, in comparison 2006 was incredibly easy so hopefully this year they will hit somewhere between the two extremes.

coblin, CAS exam: http://vcaa.vic.edu.au/vce/studies/mathematics/cas/pastexams/2007/2007mmCAS2.pdf


Part d ii is easier in my opinion ignoring the hence part.
By thinking about it/plotting the cos graph, get the intercepts over the domain [-3,1] (-3,-1,1)
You know that the maximum and minimum are the max/min of g is +6 and -6
so your equation is in the form y = A(x+3)(x+1)(x-1)
and then fiddle with A until you get the maxmim required
probably a better way

[ReVeL]

Wow I've gotta touch up on that step where you add and subtract 1 to factorise.

[Collin Li]

Wow I've gotta touch up on that step where you add and subtract 1 to factorise.

It's not really factorising, but it is a clever trick to show what was needed

All the terms are still separated by pluses or minuses. The only factorisation I've done is taken out a negative 1, so that it is clear that:



Here's what you might find useful though (similar technique):



Say goodbye to long division if you're doing a linear-linear division (can also work with other degree polynomials if you do it properly)

[Lulu]

OH! Okay. I'll have a look. That line wasn't factorisation. I've put some things in brackets to make it clearer, but all I've done is, from the line before:



(add 1 and subtract 1)

(just rearranging the order of them, nothing has changed)



I put brackets around some terms for clarity, once again, nothing has changed, although I have factorised out a negative in one of the terms.

Awesome! I get it now.

Thanks for the tips caramel, I've gotten up to what you have y= A(x+3)(x+1)(x-1)

But I'm not sure which point to sub-in in order to find the A value, which the answer has as 2 or -2.

[ReVeL]

Thanks thats lots of help.

[Collin Li]

Thanks thats lots of help.

No problem. It gets slightly trickier if you're doing something like:



Hint:

Then it's clear that you get:

The difference is that you've got to create a factor (a half, in this case), to compensate for the different co-efficient in the denominator. The key is you want your numerator to contain your denominator.

[Collin Li]

Thanks for the tips caramel, I've gotten up to what you have y= A(x+3)(x+1)(x-1)

But I'm not sure which point to sub-in in order to find the A value, which the answer has as 2 or -2.

Hmm... I was looking at that question for a while, but if you work from the start, it should be clear that this cubic you're finding to 'fit' is going to be a lot like , in that slightly fits (and is a transformed ).

If you look at , and , it should be clear that they differ by translation, and since we want their maxima/minima to be the same, we want the same level of dilation, so we should set: (no dilation factor in between and ).

[Lulu]

Thanks coblin, I understand it now

I would pep you if I could, but I don't think I have the rights yet