# 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:               