# 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:             