Maclaurin series of shifted function

CAIE Further Paper 2 2020 November Q1

5 marks Standard +0.8

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

OCR MEI FP2 2010 June Q1

19 marks Standard +0.8

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 $$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$

CAIE Further Paper 2 2023 November Q3

6 marks Challenging +1.2

Find the first three terms in the Maclaurin's series for $\tanh^{-1}\left(\frac{1}{2}e^t\right)$ in the form $\frac{1}{2}\ln a + bx + cx^2$, giving the exact values of the constants $a$, $b$ and $c$. [6]

SPS SPS FM Pure 2025 February Q13

6 marks Moderate -0.3

Write down the Maclaurin series of $e^x$, in ascending power of $x$, up to and including the term in $x^3$ [1]
Hence, without differentiating, determine the Maclaurin series of $$e^{(x^3-1)}$$ in ascending powers of $x$, up to and including the term in $x^3$, giving each coefficient in simplest form. [5]