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