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

Find the Maclaurin's series for \(\mathrm{e}^{1+x^{2}}+\mathrm{e}^{1-x}\) up to and including the term in \(x^{2}\).

(a) Find the Maclaurin series for \(\sin ^{-1} x\) up to and including the term in \(x^{3}\).
(b) Deduce an approximation to \(\int_{0}^{\frac{1}{5}} \frac{1}{\sqrt{1-u^{2}}} \mathrm{~d} u\), giving your answer as a fraction.

Find the Maclaurin's series for \(\ln (x+2)+\ln \left(x^{2}+5\right)\) up to and including the term in \(x^{2}\).

Find the first three terms in the Maclaurin's series for \(\tanh ^{-1}\left(\frac{1}{2} \mathrm{e}^{x}\right)\) in the form \(\frac{1}{2} \ln a+b x+c x^{2}\), giving the exact values of the constants \(a, b\) and \(c\).

The variables \(x\) and \(y\) are such that \(y=0\) when \(x=0\) and
\((x+1) y+(x+y+1)^{3}=1 .\)
(a) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{3}{4}\) when \(x=0\).

(a) Find the coefficient of \(x^{2}\) in the Maclaurin's series for \(-\ln \cos x\).
(b) Find the length of the arc of the curve with equation \(y=-\ln \cos x\) from the point where \(x=0\) to the point where \(x=\frac{1}{4} \pi\).

Find the Maclaurin's series for \(\ln \left(1+\mathrm{e}^{x}\right)\) up to and including the term in \(x^{2}\).

(a) It is given that \(y=\operatorname{sech}^{-1}\left(x+\frac{1}{2}\right)\).
Express cosh \(y\) in terms of \(x\) and hence show that \(\sinh y \frac{\mathrm{~d} y}{\mathrm{~d} x}=-\frac{1}{\left(x+\frac{1}{2}\right)^{2}}\).
(b) Find the first three terms in the Maclaurin's series for \(\operatorname{sech}^{-1}\left(x+\frac{1}{2}\right)\) in the form
\(\ln a+b x+c x^{2}\)
where \(a\), \(b\) and \(c\) are constants to be determined.

Find the Maclaurin's series for \(\mathrm{e}^{x} \tan x\) from first principles up to and including the term in \(x^{2}\).

It is given that \(y=2^{x}\).
(a) By differentiating \(\ln y\) with respect to \(x\), show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=2^{x} \ln 2\).
(b) Write down \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\).

(c) Hence find the first three terms in the Maclaurin's series for \(2^{x}\).

(a) Starting from the definitions of tanh and sech in terms of exponentials, prove that
\[1-\tanh ^{2} \theta=\operatorname{sech}^{2} \theta\]

The variables \(x\) and \(y\) are such that \(\tanh y=\cos \left(x+\frac{1}{4} \pi\right)\), for \(-\frac{1}{4} \pi<x<\frac{3}{4} \pi\).
(b) By differentiating the equation \(\tanh y=\cos \left(x+\frac{1}{4} \pi\right)\) with respect to \(x\), show that
\[\frac{\mathrm{d} y}{\mathrm{~d} x}=-\operatorname{cosec}\left(x+\frac{1}{4} \pi\right) .\]
(c) Hence find the first three terms in the Maclaurin's series for \(\tanh ^{-1}\left(\cos \left(x+\frac{1}{4} \pi\right)\right)\) in the form \(\frac{1}{2} \ln a+b x+c x^{2}\), giving the exact values of the constants \(a, b\) and \(c\).

Find the Maclaurin's series for \(\ln \cosh x\) up to and including the term in \(x^{4}\).

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

It is given that \(y=\sinh \left(x^{2}\right)+\cosh \left(x^{2}\right)\).
(a) Use standard results from the list of formulae (MF19) to find the Maclaurin's series for \(y\) in terms of \(x\) up to and including the term in \(x^{4}\).
(b) Deduce the value of \(\frac{\mathrm{d}^{4} y}{\mathrm{~d} x^{4}}\) when \(x=0\).
(c) Use your answer to part (a) to find an approximation to \(\int_{0}^{\frac{1}{2}} y \mathrm{~d} x\), giving your answer as a rational fraction in its lowest terms.