a) Use the Product Rule to differentiate the function:

*p*(*q*) = (*q* - 4)(*q *- 1)

b) Multiply out the brackets in *p*(*q*). Then show
that the differential of *p*(*q*) in this form is the
same as that obtained in part (a) above.

c) Determine the co-ordinates of the turning point of *p*(*q*),
and prove whether it is a maximum or a minimum.

d) Sketch the curve *p*(*q*).

e) What is the area beneath this curve between *q *= 0 and
*q *= 1 ?

1) Product Rule: *dp/dq* = *u*.(*dv/dq*) + *v*.(*du/dq*),
where *u* = (q - 4) and *v* = (*q *- 1).

So, *du/dq* = 1, and *dv/dq* = 1

Therefore the Product Rule gives:

*dp/dq* = (*q *- 1).1 + (*q-* 4).1

= __2 q - 5
__

b) Multiplying out the brackets gives: *p*(*q*) = *q*^{2}
- 5*q* +4

Differentiating this gives: __2 q -5__, as we found
before.

c) At the turning point, *dp/dq *= 0, so 2*q* - 5 =
0, so *q *= 2.5. Putting this value back into *p*(*q*),
we find that *p *= -2.25. So the turning point is at (2.5,
-2.25).

*d*^{2}*p*/*dq*^{2} = +2, which
is __positive__. This means the turning point is a __minimum__.

- We know the curve crosses the
*p*-axis at +4. We can also see from part (a), that if*q*= +1 or*q*= +4, then*p*(*q*) = 0,*i.e.*it crosses the*q*-axis at +1 and +4. Since the curve is a function of*q*^{2}, it will be parabola. So the curve looks like:

Need to label every point where the curve cross an axis, plus the t.p, plus label the axes correctly.

e) Area is given by: = =

__ Area = 1.833 sq. units__ (= sq.units)