# 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:

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