# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#10

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Question

The equation of a curve is

i.      Calculate the gradient of the curve at the point where .

ii.       A point with coordinates  moves along the curve in such a way that the rate of increase of  has the constant value 0.03 units per second.  Find the rate of increase of   at the instant when .

iii.       Find the area enclosed by the curve, the x-axis, the y-axis and the line .

Solution

i.

To find the gradient of the curve at the point where ;

First we need gradient of the curve;

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

In the given case:

We can rewrite the equation in the form of exponents:

Hence;

Rule for differentiation of  is:

In the given case:

Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that particular point.

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;

To find the gradient of the curve where , we substitute  in the equation of the gradient  of the curve:

ii.

Rate of change of  with respect to  is derivative of  with respect to  ;

Rate of change  of  with respect to  at a particular point  can be found by substituting x-coordinates of that point in the expression for rate of change;

As the point  moves along the curve, rate of increase of  has the constant value 0.03 units per second i.e.  is changing with respect to time;

To find rate of increase of  at ; first we need to find rate of increase of  i.e.

From (i) we have ;

Therefore:

iii.

To find the area enclosed by the curve, the x-axis, the y-axis and the line ;

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

We need to integrate the equation of the curve along the x-axis from  to .

We can rewrite the equation in the form of exponents:

Rule for integration of  is:

The given case is: