It is given that $y = \sin\left(\mathrm{e}^x - 1\right)$. Show that \(\frac{\mathrm{d}^2

Question

It is given that $y = \sin\left(\mathrm{e}^x - 1\right)$. Show that $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = \frac{\mathrm{d}y}{\mathrm{d}x} - y\mathrm{e}^{2x}$.[3] Hence, find the Maclaurin series of $y$, up to and including the term in $x^2$.[2] Using the Maclaurin series found in part (b), determine the series expansion of $\sqrt{1 + \sin\left(\mathrm{e}^x - 1\right)}$, up to and including the term in $x^2$.[2]

Accepted Answer

(a) $\frac{\mathrm d^2y}{\mathrm dx^2}=\frac{\mathrm dy}{\mathrm dx}-ye^{2x}$ (shown). (b) $y=x+\frac12x^2+\cdots$. (c) $1+\frac12x+\frac18x^2+\cdots$.

2024 CJC Promo Q6

2024 CJC Promo Q6

Solution:

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