Given that
$$y = 2 ^ { x }$$
write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
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(ii) Hence find
$$\int 2 ^ { x } \mathrm {~d} x$$
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The area, \(A\), bounded by the curve with equation \(y = 2 ^ { x }\), the \(x\)-axis, the \(y\)-axis and the line \(x = - 4\) is approximated using eight rectangles of equal width as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-23_1319_978_450_532}
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Show that the exact area of the largest rectangle is \(\frac { \sqrt { 2 } } { 4 }\)
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(ii) The areas of these rectangles form a geometric sequence with common ratio \(\frac { \sqrt { 2 } } { 2 }\)
Find the exact value of the total area of the eight rectangles.
Give your answer in the form \(k ( 1 + \sqrt { 2 } )\) where \(k\) is a rational number. [0pt]
[3 marks]
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(iii) More accurate approximations for \(A\) can be found by increasing the number, \(n\), of rectangles used.
Find the exact value of the limit of the approximations for \(A\) as \(n \rightarrow \infty\)