Product of separate expansions

8 Scientists predict the velocity ( \(v\) kilometres per minute) for the new "outer explorer" spacecraft over the first minute of its entry to the atmosphere of the planet Titan to be modelled by the equation: $$v = \frac { 5000 } { ( 1 + t ) ( 2 + t ) ^ { 2 } } , 0 \leq t \leq 1 \text { where } t \text { represents time in minutes. }$$

1. Use a binomial expansion to expand \(( 1 + t ) ^ { - 1 }\) up to and including the term in \(t ^ { 2 }\).
2. Use a binomial expansion to expand \(( 2 + t ) ^ { - 2 }\) up to and including the term in \(t ^ { 2 }\).
3. Hence, or otherwise, show that \(v \approx 1250 \left( 1 - 2 t + \frac { 11 t ^ { 2 } } { 4 } \right)\).
4. The displacement of the spacecraft can be found by calculating the area under the velocity time graph. Use the approximation found in part (iii) to estimate the displacement of the spacecraft over the first half minute.
5. Write \(\frac { 1 } { ( 1 + t ) ( 2 + t ) ^ { 2 } }\) in partial fractions.
6. The displacement of the spacecraft in the first \(T\) minutes is given by \(\int _ { 0 } ^ { T } v \mathrm {~d} t\) Calculate the exact value of the displacement of the spacecraft over the first half minute given by the model.
7. On further investigation the scientists believe the original model may be valid for up to three minutes. Explain why the approximation in (iii) will be no longer be valid for this time interval.

3

1. Expand \(( 1 + 2 x ) ^ { \frac { 1 } { 2 } }\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. Hence find the expansion of \(\frac { ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } } { ( 1 + x ) ^ { 3 } }\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
3. State the set of values of \(x\) for which the expansion in part (ii) is valid.

6

1. Find the first three terms in the expansion of \(( 8 - 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
2. State the range of values of \(x\) for which the expansion in part (a) is valid.
3. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { ( 8 - 3 x ) ^ { \frac { 1 } { 3 } } } { ( 1 + 2 x ) ^ { 2 } }\).

3

1. Find the binomial expansion of \(( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
   (2 marks)
2. 1. Find the binomial expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
   2. State the range of values of \(x\) for which the expansion in part (b)(i) is valid.
3. Find the binomial expansion of \(\sqrt { \frac { 1 + 4 x } { 4 - x } }\) up to and including the term in \(x ^ { 2 }\).
   (2 marks)

3

1. Find the binomial expansion of \(( 1 - 4 x ) ^ { \frac { 1 } { 4 } }\) up to and including the term in \(x ^ { 2 }\).
   [0pt] [2 marks]
2. Find the binomial expansion of \(( 2 + 3 x ) ^ { - 3 }\) up to and including the term in \(x ^ { 2 }\).
3. Hence find the binomial expansion of \(\frac { ( 1 - 4 x ) ^ { \frac { 1 } { 4 } } } { ( 2 + 3 x ) ^ { 3 } }\) up to and including the term in \(x ^ { 2 }\).
   [0pt] [2 marks]

2. (a) Expand \(( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(b) Hence, or otherwise, show that for small \(x\), $$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$

11

1. Expand \(( 1 + x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
2. (a) Expand \(\sqrt { 2 + 3 x ^ { 2 } }\) up to and including the term in \(x ^ { 4 }\).
   (b) For what range of values of \(x\) is this expansion valid?
3. Find the first three terms of the expansion of \(\frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x }\) in ascending powers of \(x\) and hence show that \(\int _ { 0 } ^ { 0.1 } \frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x } \mathrm {~d} x \approx 0.135\).