<p>It is given that&nbsp;\(y = \sin\left(\mathrm{e}^x - 1\right)\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = \frac{\mathrm{d}y}{\mathrm{d}x} - y\mathrm{e}^{2x}\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence, find the Maclaurin series of \(y\), up to and including the term in \(x^2\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">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 \(x^2\).<span class="inline-marks">[2]</span></div></li></ol>

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

Show that \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = \frac{\mathrm{d}y}{\mathrm{d}x} - y\mathrm{e}^{2x}\).

Hence, find the Maclaurin series of \(y\), up to and including the term in \(x^2\).

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 \(x^2\).

2024 CJC Promo Q6

2024 CJC Promo Q6