Maclaurin series for ln(a+bx)

4. $$y = \ln \left( \frac { 1 } { 1 - 2 x } \right) , \quad | x | < \frac { 1 } { 2 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\)
2. Hence, or otherwise, find the series expansion of \(\ln \left( \frac { 1 } { 1 - 2 x } \right)\) about \(x = 0\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient in its simplest form.
3. Use your expansion to find an approximate value for \(\ln \left( \frac { 3 } { 2 } \right)\), giving your answer
   to 3 decimal places.

3. (a) Given that \(y = \ln ( 1 + 5 x ) , | x | < 0.2\), find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
(b) Hence obtain the M aclaurin series for \(\ln ( 1 + 5 x ) , | x | < 0.2\), up to and including the term in \(x ^ { 3 }\).

1 It is given that \(\mathrm { f } ( x ) = \ln ( 3 + x )\).

1. Find the exact values of \(f ( 0 )\) and \(f ^ { \prime } ( 0 )\), and show that \(f ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 9 }\).
2. Hence write down the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\), given that \(- 3 < x \leqslant 3\).

2

1. Given that \(y = \ln ( 4 + 3 x )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Hence, by using Maclaurin's theorem, find the first three terms in the expansion, in ascending powers of \(x\), of \(\ln ( 4 + 3 x )\).
3. Write down the first three terms in the expansion, in ascending powers of \(x\), of \(\ln ( 4 - 3 x )\).
4. Show that, for small values of \(x\), $$\ln \left( \frac { 4 + 3 x } { 4 - 3 x } \right) \approx \frac { 3 } { 2 } x$$

3 You are given that \(\mathrm { f } ( x ) = \ln ( 2 + x )\).

1. Determine the exact value of \(\mathrm { f } ^ { \prime } ( 0 )\).
2. Show that \(\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 4 }\).
3. Hence write down the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\).

2 You are given that \(\mathrm { f } ( x ) = \ln ( 2 + x )\).

1. Determine the exact value of \(\mathrm { f } ^ { \prime } ( 0 )\).
2. Show that \(\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 4 }\).
3. Hence write down the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\). You are given that \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 1
   2 & 5 & 2
   3 & - 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 1 & 0 & 1
   - 8 & 4 & 0
   19 & - 8 & - 1 \end{array} \right)\).
4. Find \(\mathbf { A B }\).
5. Hence write down \(\mathbf { A } ^ { - 1 }\).
6. You are given three simultaneous equations $$\begin{array} { r } x + 2 y + z = 0
   2 x + 5 y + 2 z = 1
   3 x - 2 y - z = 4 \end{array}$$
   1. Explain how you can tell, without solving them, that there is a unique solution to these equations.
   2. Find this unique solution.