RI/2021/JC2/Prelim/P2/Q03

It is given that $$\ln y=\sqrt{1+8 e ^{x}}$$.

(i) Show that $$(\ln y) \frac{ d y}{ d x}=4 y e ^{x}$$.

[1]

(ii) Show that the value of $$\frac{ d ^{2} y}{ d x^{2}}$$ when $$x=0$$ is $$\frac{68}{27} e ^{x}$$.

[4]

(iii) Hence find the Maclaurin series for $$e ^{\sqrt{1+8 e ^{x}}}$$ up to and including the term in $$x^{2}$$.

[2]

(iv) Denoting the answer found in part (iii) as $$g (x)$$, find the set of values of $$x$$ for which $$g (x)$$ is within $$\pm 0.5$$ of the value of $$e ^{\sqrt{1+8 e ^{2}}}$$.

[3]