Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#9

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Question

A curve is such that . The curve has a maximum point at (2, 12).

i. Find the equation of the curve.

A point P moves along the curve in such a way that the x-coordinate is increasing at 0.05 units per second.

ii. Find the rate at which the y-coordinate is changing when x = 3, stating whether the y-coordinate
is increasing or decreasing.

Solution


i.
 

We can find equation of the curve from its derivative through integration;

We are given that;

To find  we need to integrate .Therefore;

Rule for integration of  is:

We are given that curve has a maximum point at (2, 12), therefore at this point;

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

Hence at point (2, 12);

Therefore;

Now we find equation of the curve from this derivative of the given curve.

We can find equation of the curve from its derivative through integration;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

We are given that curve has a maximum point at (2, 12), therefore at this point we a stationary point.

Hence;


ii.
 

We are given that;

We are required to find;

We know that;

From (i), we have;

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;

Therefore;

Therefore, at  we have .

To test whether a function  is increasing or decreasing at a particular point , we take derivative of a function at that point.

If  , the function  is increasing.

If  , the function  is decreasing.

If  , the test is inconclusive.

Since  y-coordinate is decreasing.

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