# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | May-Jun | (P2-9709/02) | Q#6

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Question

The equation of a curve is

i.       Show, by differentiation, that the gradient of the curve is always negative.

ii.       Use the trapezium rule with 2 intervals to estimate the value of

iii.

The diagram shows a sketch of the curve for . State, with a reason, whether the  trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part  (ii).

Solution

i.

We are required to show that the gradient of the curve with following equation is always negative.

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to  is:

Therefore, we need to differentiate the given equation of the curve for the gradient.

We apply chain rule to solve this derivative.

If we define , then derivative of  is;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is;

Another way of applying chain rule is as follows.

If we have  and then derivative of  is;

We have;

Let , therefore;

First we find ;

Rule for differentiation of  is:

Next we find .

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is;

Now we can find .

Since , therefore;

It is evident that and  both
being squares will always be positive. There is a negative sign in
, therefore, derivative will always  be negative and since the derivative of a curve is its gradient, the gradient of the curve will  be always negative.

ii.

We are required to apply Trapezium Rule to evaluate;

The trapezium rule with  intervals states that;

We are given that there are two intervals, .

We are also given that and .

Hence;

 0 1 2

Therefore;

iii.

If the graph is bending upwards over the whole interval from  to , then trapezium rule will give an overestimate of the true area (as shown in the diagram below).

It is evident that for the given graph trapezium rule will give an overestimate.