6 The function f is defined by \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } ( 1 + 3 x ) ^ { - \frac { 2 } { 3 } }\).
- Use the series expansion for \(\mathrm { e } ^ { x }\) to write down the first four terms in the series expansion of \(\mathrm { e } ^ { 2 x }\).
- Use the binomial series expansion of \(( 1 + 3 x ) ^ { - \frac { 2 } { 3 } }\) and your answer to part (a)(i) to show that the first three non-zero terms in the series expansion of \(\mathrm { f } ( x )\) are \(1 + 3 x ^ { 2 } - 6 x ^ { 3 }\).
- Given that \(y = \ln ( 1 + 2 \sin x )\), find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
- By using Maclaurin's theorem, show that, for small values of \(x\),
$$\ln ( 1 + 2 \sin x ) \approx 2 x - 2 x ^ { 2 }$$
- Find
$$\lim _ { x \rightarrow 0 } \frac { 1 - \mathrm { f } ( x ) } { x \ln ( 1 + 2 \sin x ) }$$