# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | Oct-Nov | (P2-9709/02) | Q#6

Question

The diagram shows the part of the curve  for , and its minimum point M.

i.       Find the coordinates of M.

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

iii.       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 find the coordinates of point M which is minimum point of the curve;

A stationary point on the curve is the point where gradient of the curve is equal to zero;

Since point M is minimum point, therefore, it is stationary point of the curve and, hence, gradient of  the curve at point M must ZERO.

We can find expression for gradient of the curve at point M and equate it with ZERO to find the x- coordinate of point M.

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

Therefore;

If  and  are functions of , and if , then;

If , then;

Let  and ;

Rule for differentiation of natural exponential function , or ;

Rule for differentiation of  is:

Now we need expression for gradient of the curve at point M.

Gradient (slope) of the curve at a particular point can be found by substituting x- coordinates of that point in the expression for gradient of the curve;

Since point M is a maximum point, the gradient of the curve at this point must be equal to ZERO.

Hence, x-coordinate of point M on the curve is .

To find the y-coordinate of point M on the curve, we substitute value of x-coordinate in equation of  the curve.

Hence;

Hence, y-coordinate of the point M is .

Therefore, coordinates of point M

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;

 1 2 3

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.