1.04c Extend binomial expansion: rational n, |x|<1

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. The first four terms in the series expansion of \(( 1 + a x ) ^ { n }\) in ascending powers of \(x\) are $$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where \(a , n\) and \(k\) are constants and \(| a x | < 1\).

1. Find the values of \(a\) and \(n\).
2. Show that \(k = - 160\).

5.

1. Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
2. State the set of values of \(x\) for which your expansion is valid.
3. Use your expansion with \(x = 0.01\) to find the value of \(\sqrt { 399 }\), giving your answer to 9 significant figures.

6. $$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$

1. Show that \(\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }\).
2. Expand \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
3. Use your expansion to find an approximate value for \(\sqrt { 10 }\), giving your answer to 8 significant figures.
4. Find, to 1 significant figure, the percentage error in your answer to part (c).

2.

1. Expand \(( 1 + 4 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
2. State the set of values of \(x\) for which your expansion is valid.

6. $$f ( x ) = \frac { 1 + 3 x } { ( 1 - x ) ( 1 - 3 x ) } , \quad | x | < \frac { 1 } { 3 }$$

1. Find the values of the constants \(A\) and \(B\) such that $$\mathrm { f } ( x ) = \frac { A } { 1 - x } + \frac { B } { 1 - 3 x }$$
2. Evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } } f ( x ) d x$$ giving your answer as a single logarithm.
3. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.

2.

1. Find the binomial expansion of \(( 2 - 3 x ) ^ { - 3 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
2. State the set of values of \(x\) for which your expansion is valid.

1

1. Find the first three terms of the binomial expansion of \(\frac { 1 } { \sqrt [ 3 ] { 1 - 2 x } }\). State the set of values of \(x\) for which
   the expansion is valid.
2. Hence find \(a\) and \(b\) such that \(\frac { 1 - 3 x } { \sqrt [ 3 ] { 1 - 2 x } } = 1 + a x + b x ^ { 2 } + \ldots\).

2 Find the first three terms in the binomial expansion of \(( 4 + x ) ^ { \frac { 3 } { 2 } }\). State the set of values of \(x\) for which the expansion is valid.

3

1. Express \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) }\) in partial fractions.
2. Hence use binomial expansions to show that \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) } = a x + b x ^ { 2 } + \ldots\), where \(a\) and \(b\) are
   constants to be determined. constants to be determined. State the set of values of \(x\) for which the expansion is valid.

4 Find the first four terms in the binomial expansion of \(\sqrt { 1 + 2 x }\). State the set of values of \(x\) for which the expansion is valid.

5 Find the first three terms in the binomial expansion of \(\sqrt [ 3 ] { 1 + 3 x }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.

6

1. Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x }$$ where \(A , B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\).
2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \(( 1 + x ) ^ { - 2 }\) and \(( 1 - 4 x ) ^ { - 1 }\). Hence find the first three terms of the expansion of \(\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }\).

7 Find the first 4 terms in the binomial expansion of \(\sqrt { 4 + 2 x }\). State the range of values of \(x\) for which the expansion is valid.

2 Given the binomial expansion \(( 1 + q x ) ^ { p } = 1 - x + 2 x ^ { 2 } + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid. [6]

3 Find the first three terms in the binomial expansion of \(\frac { 1 } { ( 3 - 2 x ) ^ { 3 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
[0pt] [7]

4 Find the first three terms in the binomial expansion of \(\frac { 1 + 2 x } { ( 1 - 2 x ) ^ { 2 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
[0pt] [7]

5 Show that \(( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots\), and find the next term in the expansion.
State the set of values of \(x\) for which the expansion is valid.

6

1. Find the first three terms in the binomial expansion of \(\frac { 1 } { \sqrt { 1 - 2 x } }\). State the set of values of \(x\) for which the expansion is valid.
2. Hence find the first three terms in the series expansion of \(\frac { 1 + 2 x } { \sqrt { 1 - 2 x } }\).

7

1. Find the first three non-zero terms of the binomial expansion of \(\frac { 1 } { \sqrt { 4 - x ^ { 2 } } }\) for \(| x | < 2\). [4]
2. Use this result to find an approximation for \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to
   4 significant figures.
3. Given that \(\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x = \arcsin \left( \frac { 1 } { 2 } x \right) + c\), evaluate \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to 4 significant figures.

6 Find the first three terms in the binomial expansion of \(\overline { 4 + x }\) in ascending powers of \(x\).
State the set of values of \(x\) for which the expansion is valid.

9

1. Find the first three non-zero terms of the binomial series expansion of \(\frac { 1 } { \sqrt { 1 + 4 x ^ { 2 } } }\), and state the set of values of \(x\) for which the expansion is valid.
2. Hence find the first three non-zero terms of the series expansion of \(\frac { 1 - x ^ { 2 } } { \sqrt { 1 + 4 x ^ { 2 } } }\).

10 Two students are trying to evaluate the integral \(\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x\).
Sarah uses the trapezium rule with 2 strips, and starts by constructing the following table.

1. Complete the calculation, giving your answer to 3 significant figures. Anish uses a binomial approximation for \(\sqrt { 1 + \mathrm { e } ^ { - x } }\) and then integrates this.
2. Show that, provided \(\mathrm { e } ^ { - x }\) is suitably small, \(\left( 1 + \mathrm { e } ^ { - x } \right) ^ { \frac { 1 } { 2 } } \approx 1 + \frac { 1 } { 2 } \mathrm { e } ^ { - x } \quad \frac { 1 } { 8 } \mathrm { e } ^ { - 2 x }\).
3. Use this result to evaluate \(\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x\) approximately, giving your answer to 3 significant figures.

6

1. Express \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) }\) in partial fractions.
2. Hence use binomial expansions to show that \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) } = a x + b x ^ { 2 } + \ldots\), where \(a\) and \(b\) are
   constants to be determined. State the set of values of \(x\) for which the expansion is valid.