CAIE Further Paper 2 (Further Paper 2) 2022 June

4 The diagram shows the curve with equation \(\mathrm { y } = 2 ^ { \mathrm { x } }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\).
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1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } 2 ^ { x } d x < U _ { N }\), where $$\mathrm { U } _ { \mathrm { N } } = \frac { 2 ^ { \frac { 1 } { \mathrm {~N} } } } { \mathrm {~N} \left( 2 ^ { \frac { 1 } { \mathrm {~N} } } - 1 \right) }$$
2. Use a similar method to find, in terms of \(N\), a lower bound \(\mathrm { L } _ { \mathrm { N } }\) for \(\int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x\).
3. Find the least value of \(N\) such that \(\mathrm { U } _ { \mathrm { N } } - \mathrm { L } _ { \mathrm { N } } < 10 ^ { - 4 }\).

6 Use the substitution \(y = v x\) to find the solution of the differential equation $$x \frac { d y } { d x } = y + \sqrt { 9 x ^ { 2 } + y ^ { 2 } }$$ for which \(y = 0\) when \(x = 1\). Give your answer in the form \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), where \(\mathrm { f } ( x )\) is a polynomial in \(x\). [10]
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