It is given that $\ln \left( 1+y \right)=\tan x$. Show that \(\frac{{{\mathrm{d}}^{2}}y}

Question

It is given that $\ln \left( 1+y \right)=\tan x$. Show that $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\frac{\mathrm{d}y}{\mathrm{d}x}\left( 2\tan x+{{\sec }^{2}}x \right)$.[3] By further differentiation of the result in part (a), find the Maclaurin series for $y$, up to and including the term in ${{x}^{3}}$.[3] Hence find an estimate for the value of $\mathrm{e}$.[2]

Accepted Answer

(b) $y=x+\frac{1}{2}{{x}^{2}}+\frac{1}{2}{{x}^{3}}+...$ (c) $2.34$

2024 CJC P1 Q6

2024 CJC P1 Q6

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