2009 TPJC P1 Q11
(i)
Given that $y={{\text{e}}^{{{\tan }^{-1}}x}}$, find $\frac{\text{d}y}{\text{d}x}$ and show that $\left( 1+{{x}^{2}} \right)\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}+\left( 2x-1 \right)\frac{\text{d}y}{\text{d}x}=0$. Obtain the Maclaurin series for ${{\text{e}}^{{{\tan }^{-1}}x}}$ up to and including the term in ${{x}^{2}}$.
[5]
(ii)
Expand ${{\left( 1+3x \right)}^{-\frac{1}{2}}}$ in ascending powers of $x$ up to and including the term in ${{x}^{2}}$.
[2]
(iii)
Using the series that you obtained in (i) and (ii), deduce the equation of the tangent to the curve $y=\frac{{{\text{e}}^{{{\tan }^{-1}}x}}}{\sqrt{1+3x}}$ at $x=0$.
[2]
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