# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/21) | Q#5

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Question

The diagram shows the curve  and its minimum point M.

i.       Show that the x-coordinate of M can be written in the form , where the value of a is to be  stated.

ii.       Find the exact value of the area of the region enclosed by the curve and the lines x = 0, x = 2 and y = 0.

Solution

i.

We are required to find the x-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;

Rule for differentiation of  is:

Rule for differentiation natural exponential function , or ;

Rule for differentiation of  is:

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 minimum point, the gradient of the curve at this point must be equal to ZERO.

Taking logarithm of both sides;

Since  for any ;

Power Rule;

Hence, x-coordinate of point M on the curve is given by;

ii.

We are required to find the exact value of the area of the region enclosed by the curve and the lines  x = 0, x = 2 and y = 0.

It is evident that desired area is area under the curve from x=0 to x=2.

Next we find area under the curve.

To find the area of region under the curve , we need to integrate the curve from point to   along x-axis.

For the given case;

Therefore;

Rule for integration of  is:

Rule for integration of , or ;

Rule for integration of  is:

Rule for integration of  is: