Composite substitution expansion

4．（a）Find the binomial series expansion for \(( 4 + y ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) ．Simplify the coefficient of each term．
（3）
（b）Hence show that the binomial series expansion for \(\left( 4 + 5 x + x ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\) is $$2 + \frac { 5 x } { 4 } - \frac { 9 x ^ { 2 } } { 64 } + \frac { 45 x ^ { 3 } } { 512 }$$ （c）Show that the binomial series expansion of \(\left( 4 + 5 x + x ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\) will converge for \(- \frac { 1 } { 2 } \leqslant x \leqslant \frac { 1 } { 2 }\)
（d）Use the result in part（b）to estimate $$\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \sqrt { 4 + 5 x + x ^ { 2 } } d x$$ Give your answer as a single fraction．

1

1. Expand \(( 1 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) as far as the term in \(x ^ { 2 }\).
2. Hence expand \(\left( 1 - 2 y + 4 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(y\) as far as the term in \(y ^ { 2 }\).

7 A student is trying to find the binomial expansion of \(\sqrt { 1 - x ^ { 3 } }\).
She gets the first three terms as \(1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 }\).
She draws the graphs of the curves \(y = \sqrt { 1 - x ^ { 3 } } , y = 1 - \frac { x ^ { 3 } } { 2 }\) and \(y = 1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 }\) using software.
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1. Explain why \(1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 } \geqslant 1 - \frac { x ^ { 3 } } { 2 }\) for all values of \(x\).
2. Explain why the graphs suggest that the student has made a mistake in the binomial expansion.
3. Find the first four terms in the binomial expansion of \(\sqrt { 1 - x ^ { 3 } }\).
4. State the set of values of \(x\) for which the binomial expansion in part (c) is valid.
5. Sketch the curve \(y = 2.5 \sqrt { 1 - x ^ { 3 } }\) on the grid in the Printed Answer Booklet. \section*{(f) In this question you must show detailed reasoning.} The end of a bus shelter is modelled by the area between the curve \(\mathrm { y } = 2.5 \sqrt { 1 - x ^ { 3 } }\), the lines \(x = - 0.75 , x = 0.75\) and the \(x\)-axis. Lengths are in metres. Calculate, using your answer to part (c), an approximation for the area of the end of the bus shelter as given by this model.