2023 NJC P1 Q3
It is given that $y={{\left( 1+{{\tan }^{-1}}x \right)}^{2}}$.
(a)
Prove that ${{\left( 1+{{x}^{2}} \right)}^{2}}\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}+2x\left( 1+{{x}^{2}} \right)\frac{\text{d}y}{\text{d}x}=2$.
[3]
(b)
Obtain an equation relating $\frac{{{\text{d}}^{3}}y}{\text{d}{{x}^{3}}}$, $\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}$ and $\frac{\text{d}y}{\text{d}x}$.
[1]
(c)
Find the Maclaurin’s series for $y$, up to and including the terms in ${{x}^{3}}$.
[2]
(d)
Deduce the equation of the normal to the curve $y={{\left( 1+{{\tan }^{-1}}x \right)}^{2}}$ at $x=0$.
[1]
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