Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#8

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Question

The function  is such that  for ,

     i.       Obtain an expression for  and explain why  is a decreasing function.

   ii.       Obtain an expression for .

  iii.       A curve has the equation . Find the volume obtained when the region bounded by the curve, the coordinate axes and the line is rotated through  about the x-axis.

Solution


i.
 

We have the function;

The expression for  represents derivative of .

We can write it as;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

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   is negative for all values of , hence,   is an increasing function.


ii.
 

We have;

We write it as;

To find the inverse of a given function  we need to write it in terms of  rather than in terms of .

Interchanging ‘x’ with ‘y’;


iii.
 

We are given that;

Therefore;

Expression for the volume of the solid formed when the shaded region under the curve  is rotated completely about the x-axis is;

Hence we can find the volume as;

Rule for integration of  is:

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