Evaluation of HSC 2015 Maths Extension 1 Exam - Prelim versus HSC Content

I took a look at the HSC 2015 Math Extension 1 paper and concluded that about 1/3 is Prelim, and 2/3 is HSC.

The Prelim questions are easier in general than those with HSC content.
Strategy wise, I think it would be good to get the Prelim questions 100% correct! so it's a good idea to ensure proficiency in all the preliminary topics.
Holidays are a good time to do that sort of thing.
Here are the BOSTES 3U syllabus topics. 
We have essentially finished the preliminary content.

Here is my break down of Prelim/HSC content for those interested :-)

Merry Christmas, Happy New Year, and Fantastic Festive Season to everyone. 

Regards, :-) JH

EXTRA-CURRICULAR EXTENSION - Combining Binomial Theorem, Differentiation and Induction !! Differentiation of the product of two functions $n$ times - A Formula

This is outside of syllabus but many of you should extend yourself beyond the syllabus a bit. Is there a formula for differentiating the product of two functions $$y=uv$$  many times over (say, $n$ times)

To get an idea of what the answer to this might be, actually perform the differentiation several times over to see what you get. So find the first derivative, the second derivative, the third derivative, and the fourth derivative. Is there a pattern? Here are the answers.

First derivative,
${d \over dx } (uv) = u'v+uv'$

Second derivative,
$\begin{array}{rl}{d^2 \over dx^2 } (uv) & = u''v+u'v'+u'v'+uv''\\ &= u''v+2u'v'+uv''\end{array}$

Third derivative,
$\begin{array}{rl}{d^3 \over dx^3 } (uv)
& = u'''v+u''v'+2u''v'+2u'v''+u'v''+uv'''\\
& = u'''v+3u''v'+3u'v''+uv'''\end{array}$

Fourth derivative,
$\begin{array}{rl}{d^4 \over dx^4 } (uv)
& = u''''v+u'''v'+3u'''v'+3u''v''+3u''v''+3u'v'''+u'v'''+v''''\\
& = u''''v+4u'''v'+6u''v''+4u'v'''+v''''\end{array}$ 

Can you see that the coefficients (numbers in front) follow the pattern of Pascal's Triangle below? 

1

1   1

1  2  1

1  3   3  1

1  4  6   4   1

Here is the general case. It uses the binomial theorem and sigma notation, so hopefully you have seen these before.

THEOREM:
If $f(x) = uv$ where $u$ and $v$ are functions of $x$ then the
nth derivative of this product is given by
$$f^{(n)}(x)={d^n \over dx^n } (uv)=\sum^n_{r=0} {n\choose r} u^{(r)}v^{(n-r)}$$
where we use the notation for the $r^{th}$ derivative

$\displaystyle u^{(r)}={ d^r\over dx^r} u$

(When it's a derivative there are brackets around the 'index')

Using this notation differentiating one more time looks like this
$$\displaystyle{ d\over dx}\left( u^{(r)}\right) = { d\over dx} { d^r\over dx^r} u={ d^{r+1}\over dx^{r+1}}=u^{(r+1)}$$
PROOF: 
By induction on $n$.
For $n=0$, $f^{(0)}(x)=f(x)=uv$ true. and

$ {0\choose 0} u^{(0)}v^{(0)}=uv$ and so it is true for $n=0$.

[Remark, $n=0$ corresponds to not differentiating at all, so it should be the original product $uv$.]

For $n=1$,

$ {1\choose 0} u^{(0)}v^{(1)}+{1\choose 1} u^{(1)}v^{(0)}=uv'+u'v$ and so it is true for $n=1$.


[Remark, $n=1$ corresponds to differentiating once, the result is nothing more than the usual product rule for two functions.]

Assume the statement is true for $n=k$, prove it is true for $n=k+1$.


$\begin{array}{rcl}f^{(k+1)}(x)&=&{ d\over dx} f^{(k)}(x)\\
{ \;}&=&\displaystyle{ d\over dx} \sum^k_{r=0} {k\choose r} u^{(r)}v^{(k-r)}\\
&=&\displaystyle
\sum^k_{r=0} {k\choose r} u^{(r+1)}v^{(k-r)}+
\sum^k_{r=0} {k\choose r} u^{(r)}v^{(k+1-r)}\\
&=&\displaystyle
\sum^{k+1}_{s=1} {k\choose s-1} u^{(s)}v^{(k+1-s)}+

\sum^k_{r=0} {k\choose r} u^{(r)}v^{(k+1-r)}\\\end{array}$

where we changed variables to $s=r+1$ in the first summation.

$\begin{array}{lcl}
f^{(k+1)}(x) &= &\displaystyle{k\choose k} u^{(k+1)}v^{(0)}+{k\choose 0} u^{(0)}v^{(k+1)}\\&&\displaystyle +\sum^{k}_{s=1} {k\choose s-1} u^{(s)}v^{(k+1-s)}\\&&\displaystyle +
\sum^k_{r=1} {k\choose r} u^{(r)}v^{(k+1-r)}\\&= &\displaystyle u^{(k+1)} + v^{(k+1)}\\&&\displaystyle +\sum^{k}_{r=1} \left({k\choose r-1} +{k\choose r}\right) u^{(r)}v^{(k+1-r)} \\
&= &\displaystyle \sum^{k+1}_{r=0} \left({k\choose r-1} +{k\choose r}\right) u^{(r)}v^{(k+1-r)}
\\&= &\displaystyle \sum^{k+1}_{r=0} \left({k\choose r-1} +{k\choose r}\right) u^{(r)}v^{(k+1-r)}
\\&= &\displaystyle \sum^{k+1}_{r=0} {k+1\choose r}u^{(r)}v^{(k+1-r)}\end{array}$
and so it is proven true for $n=k+1$. Therefore by induction the statement is true for all integers $n\ge 0$.

SUGGESTED EXTRA PROBLEM.

Can you apply this formula (or otherwise) and find an expression for the nth derivative of the function $\displaystyle y={u\over v}$?