**Question**

**Introduction**

This question
was most probably taken from an Integrated
Programme (IP) school in Singapore. For your information, students in Integrated
Programme schools do not take the GCE ‘O’ Levels, and each IP school is free to
design its own individual curriculum. In
practice, they incorporate the mainstream GCE ‘O’ Level topics, as well as
additional topics, and/or teach topics in advance, and may call their
syllabuses by different names. They tend
to set more challenging questions than the mainstream schools, which are
already targetting their internal examination standards above the ‘O’ Levels. In other words, they tend to cram in more, but
never less. Part (d) tests approximate
change, which had been taken out of the mainstream syllabus at this time of
writing.
Note that the
IP schools tend to be schools that traditionally attract the academically best
students from each cohort. Even before the
IP programme was introduced, these schools were already setting harder
questions. Anyway, this blog welcomes
everybody from all around the world who is willing to learn, regardless of the
type of school they are from, even home-schoolers and independent learners! Let us see how we can employ re-usable tactics
to tackle this question.

**Get to the root of the matter, fast!**

The
question is on the applications of differential calculus on a quadratic curve
(a parabola). Observe that the equation
of the curve is given in completed square form.
From the equation, can you spot the line of symmetry (centre line) and
the *y*-intercept
immediately (like within 5 seconds)? [You
need to *know all the basic facts at your
finger tips* and *make observations*.]

The line of symmetry always passes through the maximum
or (in this case) minimum point. We know
that the minimum is when the squared term (*x* –
3)^{2} is zero. So

the line of symmetry is *x* =
3.

To get the *y*-intercept, put *x* =
0. This gives *y* =
(-3)^{2} = 9. So

*C* = (0, 9) and equation of *CD*
is *y*
= 9

Since *PQ* = 2*k*, distance from *P* to the centre line = *k*. All the above are basic
observations that should be carried out mentally within one minute and you
should be able to mark the diagram with pencil notes (shown above in blue). Once this is done, let us get on to the real
business.

**Solution**

**Final Remarks**

For part (c), they have already told you it’s
maximum, so you do not need to prove that it is maximum. However, intuitively it is obvious there is a
maximum: imagine if *k* = 0 or *k* =
3, then we get very thin rectangles with area zero. As *k* increases from
0, the area gets bigger and after that shrinks towards zero again.

Actually, was this problem really so difficult? What did we do to solve it? Let’s review

· Know all
basic facts and skills thoroughly at immediate recall (e.g. extremum of
parabola lies on line of symmetry, how to spot that from completed square form,
how to find *y*-intercept)

· Use
heuristics: e.g. make observations. Use simple
facts you already know (e.g. subtract lengths, substitute values of *x* to find
*y* which is “height” have *x*-axis etc,). Practice
positive psychology: instead of worrying, write down everything you can deduce. Then try to find connections.

· Apply the
formulas

· Use your
intuition to see whether your answer makes sense.

· For part (d), if it is not in your syllabus, do not
worry about it. But if you are curious
or feel the itch to learn more, it is also not too difficult. See the boxed formulas in the solution above. Just remember that the ratio of small changes
^{DA}/_{Dk} is
approximately equal to the derivative ^{dA}/_{dk}. You can detach
the D*k*, bring it to
the other side of the equation, and that allows you to approximate D*A*.

H02. Use a diagram / model

H04. Look for pattern(s)

H09. Restate the problem in
another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a
Mathematical Sentence

**Suitable Levels**

**·**** **GCE ‘O’ Level Additional Mathematics

**·** other syllabuses that involve applications of differentiation

**·** anyone who is interested in calculus!