Maclaurin series of shifted function

1 Find the Maclaurin's series for $\tan \left( x + \frac { 1 } { 4 } \pi \right)$ up to and including the term in $x ^ { 2 }$.

1

1. Given that $\mathrm { f } ( t ) = \arcsin t$, write down an expression for $\mathrm { f } ^ { \prime } ( t )$ and show that $$\mathrm { f } ^ { \prime \prime } ( t ) = \frac { t } { \left( 1 - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$$
2. Show that the Maclaurin expansion of the function $\arcsin \left( x + \frac { 1 } { 2 } \right)$ begins $$\frac { \pi } { 6 } + \frac { 2 } { \sqrt { 3 } } x$$ and find the term in $x ^ { 2 }$.
Sketch the curve with polar equation $r = \frac { \pi a } { \pi + \theta }$, where $a > 0$, for $0 \leqslant \theta < 2 \pi$. Find, in terms of $a$, the area of the region bounded by the part of the curve for which $0 \leqslant \theta \leqslant \pi$ and the lines $\theta = 0$ and $\theta = \pi$.
Find the exact value of the integral $$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$

2 It is given that $\mathrm { f } ( x ) = \tan ^ { - 1 } ( 1 + x )$.

Find $\mathrm { f } ( 0 )$ and $\mathrm { f } ^ { \prime } ( 0 )$, and show that $\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 2 }$.
Hence find the Maclaurin series for $\mathrm { f } ( x )$ up to and including the term in $x ^ { 2 }$.

2 You are given that $\mathrm { f } ( x ) = \tan ^ { - 1 } ( 1 + x )$.

1. Find the value of $f ( 0 )$.
2. Determine the value of $f ^ { \prime } ( 0 )$.
3. Show that $f ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 2 }$.
Hence find the Maclaurin series for $\mathrm { f } ( x )$ up to and including the term in $x ^ { 2 }$.