Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2009  OctNov  (P29709/21)  Q#7
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Question
The diagram shows the curve y = e^{– x}. The shaded region R is bounded by the curve and the lines y = 1 and x = p, where p is a constant.
i. Find the area of R in terms of p.
ii. Show that if the area of R is equal to 1 then p = 2 – e^{p}.
iii. Use the iterative formula
with initial value p_{1 }= 2, to calculate the value of p correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
Solution
i.
We are required to find area of shaded region R.
It is evident that area of shaded region R is given by;
First we find area of rectangle.
Expression for the area of the rectangle is;
Therefore;
Next we find area under the curve.
To find the area of region under the curve , we need to integrate the curve from point to along xaxis.
For the given case;
Therefore;
Rule for integration of , or ;
Finally, we can find area of shaded region R as;
ii.
We are given that area of shaded region R is 1.
We have demonstrated in (i) that;
Therefore;
iii.
If the sequence given by the inductive definition , with some initial value , converges to a limit , then is the root of the equation .
Therefore, if , then is a root of .
Therefore, iterative formula we use;
We use as initial value.



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It is evident that .
Hence, is a root of .
The root given correct to 2 decimal places is 1.84.
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