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

3

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. 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 Given that the first three terms of the Maclaurin series for \(( 1 + \sin x ) \mathrm { e } ^ { 2 x }\) are identical to the first three terms of the binomial series for \(( 1 + a x ) ^ { n }\), find the values of the constants \(a\) and \(n\). (You may use appropriate results given in the List of Formulae (MF1).)

2 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

1. Write down the expansion of \(\mathrm { e } ^ { 3 x }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
2. Hence, or otherwise, find the term in \(x ^ { 2 }\) in the expansion, in ascending powers of \(x\), of \(\mathrm { e } ^ { 3 x } ( 1 + 2 x ) ^ { - \frac { 3 } { 2 } }\).
   (4 marks)

7

1. 1. Write down the first three terms of the binomial expansion of \(( 1 + y ) ^ { - 1 }\), in ascending powers of \(y\).
   2. By using the expansion $$\cos x = 1 - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \ldots$$ and your answer to part (a)(i), or otherwise, show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\sec x\) are $$1 + \frac { x ^ { 2 } } { 2 } + \frac { 5 x ^ { 4 } } { 24 }$$
2. By using Maclaurin's theorem, or otherwise, show that the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\tan x\) are $$x + \frac { x ^ { 3 } } { 3 }$$
3. Hence find \(\lim _ { x \rightarrow 0 } \left( \frac { x \tan 2 x } { \sec x - 1 } \right)\).

8

1. Find the first three terms in the expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\).
2. The expansion of \(\frac { a + b x } { \sqrt { 4 - x } }\) is \(16 - x \ldots\). Find the values of the constants \(a\) and \(b\).

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 \(\frac { ( 8 - 3 x ) ^ { \frac { 1 } { 3 } } } { ( 1 + 2 x ) ^ { 2 } }\).

8

1. Find the first three terms in the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\).
2. State the range of values of \(x\) for which the expansion in part (a) is valid.
3. In the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } } ( 1 + a x ) ^ { 2 }\) the coefficient of \(x ^ { 2 }\) is \(\frac { 107 } { 16 }\). Determine the possible values of the constant \(a\).

2

1. Find the coefficient of \(x ^ { 5 }\) in the expansion of \(( 3 - 2 x ) ^ { 8 }\).
2. 1. Expand \(( 1 + 3 x ) ^ { 0.5 }\) as far as the term in \(x ^ { 3 }\).
   2. State the range of values of \(x\) for which your expansion is valid. A student suggests the following check to determine whether the expansion obtained in part (b)(i) may be correct.
      "Use the expansion to find an estimate for \(\sqrt { 103 }\), correct to five decimal places, and compare this with the value of \(\sqrt { 103 }\) given by your calculator."
   3. Showing your working, carry out this check on your expansion from part (b)(i).

1. The binomial expansion of

$$( 1 + a x ) ^ { 12 }$$ up to and including the term in \(x ^ { 2 }\) is $$1 - \frac { 15 } { 2 } x + k x ^ { 2 }$$ where \(a\) and \(k\) are constants.

1. Show that \(a = - \frac { 5 } { 8 }\)
2. Hence find the value of \(k\) Using the expansion and making your method clear,
3. find an estimate for the value of \(\left( \frac { 17 } { 16 } \right) ^ { 12 }\), giving your answer to 4 decimal places.

1. a. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( \frac { 1 } { 9 } - 2 x \right) ^ { \frac { 1 } { 2 } }$$ giving each coefficient in its simplest form.
b. Explain how you could use \(x = \frac { 1 } { 36 }\) in the expansion to find an approximation for \(\sqrt { 2 }\). There is no need to carry out the calculation.

11. a. Find the binomial expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\), up to and including the term in \(x ^ { 2 }\). Given that the binomial expansion of \(\mathrm { f } ( x ) = \sqrt { \frac { 1 + 2 x } { 4 - x } } , | x | < \frac { 1 } { 4 }\), is $$\frac { 1 } { 2 } + \frac { 9 } { 16 } x - A x ^ { 2 } + \cdots$$ b. Show that the value of the constant \(A\) is \(\frac { 45 } { 256 }\) c. By substituting \(x = \frac { 1 } { 4 }\) into the answer for (b) find an approximate for \(\sqrt { 10 }\), giving your answer to 3 decimal places.

7. a. Use the binomial theorem to expand $$( 8 - 3 x ) ^ { \frac { 2 } { 3 } }$$ in ascending powers of \(x\), up to and including the term \(x ^ { 3 }\), as a fully simplifying each term. Edward, a student decides to use the expansion with \(x = \frac { 1 } { 3 }\) to find an approximation for \(( 7 ) ^ { \frac { 2 } { 3 } }\). Using the answer to part (a) and without doing any calculations, b. explain clearly whether Edward's approximation will be an overestimate, or, an underestimate.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$( 3 + x ) ^ { - 2 }$$ writing each term in simplest form.
2. Using the answer to part (a) and using algebraic integration, estimate the value of $$\int _ { 0.2 } ^ { 0.4 } \frac { 6 x } { ( 3 + x ) ^ { 2 } } d x$$ giving your answer to 4 significant figures.
3. Find, using algebraic integration, the exact value of $$\int _ { 0.2 } ^ { 0.4 } \frac { 6 x } { ( 3 + x ) ^ { 2 } } d x$$ giving your answer in the form \(a \ln b + c\), where \(a , b\) and \(c\) are constants to be found.

13 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt [ 3 ] { 27 - 8 x ^ { 3 } }\). Jenny uses her scientific calculator to create a table of values for \(\mathrm { f } ( x )\) and \(\mathrm { f } ^ { \prime } ( x )\).

1. Use calculus to find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why the calculator gives an error for \(\mathrm { f } ^ { \prime } ( 1.5 )\).
2. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\).
3. Jenny integrates the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) to estimate the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). Explain why Jenny's method is valid in this case. (You do not need to evaluate Jenny's approximation.)
4. Use the trapezium rule with 4 strips to obtain an estimate for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The calculator gives 2.92117438 for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The graph of \(y = \mathrm { f } ( x )\) is shown in Fig. 13. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-08_490_906_1505_568} \captionsetup{labelformat=empty} \caption{Fig. 13}
   \end{figure}
5. Explain why the trapezium rule gives an underestimate.

3 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = ( 1 - a x ) ^ { - 3 }\), where \(a\) is a non-zero constant. In the binomial expansion of \(\mathrm { f } ( x )\), the coefficients of \(x\) and \(x ^ { 2 }\) are equal.

1. Find the value of \(a\).
2. Using this value for \(a\),
   1. state the set of values of \(x\) for which the binomial expansion is valid,
   2. write down the quadratic function which approximates \(\mathrm { f } ( x )\) when \(x\) is small.

4 Using an appropriate expansion show that, for sufficiently small values of \(x\), \(\frac { 1 - x } { ( 2 + x ) ^ { 2 } } \approx \frac { 1 } { 4 } - \frac { 1 } { 2 } x + \frac { 7 } { 16 } x ^ { 2 }\).

10

1. Determine the first three terms in ascending powers of \(x\) of the binomial expansion of \(( 8 + 3 x ) ^ { \frac { 1 } { 3 } }\).
2. State the range of values of \(x\) for which this expansion is valid.