Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | Oct-Nov | (P2-9709/21) | 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 p1 = 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 x-axis.
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
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The root given correct to 2 decimal places is 1.84.
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