Deriving Maclaurin series

Secants to tangents: dy/dx

<p><strong>Do not use a calculator for this question</strong></p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Give that $y={{\mathrm{e}}^{\frac{2}{x+1}}}$, where $x\ne -1$, show that</span></div><div class="list-item-paragraph" style="text-align: center">${{(x+1)}^{2}}\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+2(x+2)\frac{\mathrm{d}y}{\mathrm{d}x}=0$.</div><div class="list-item-paragraph">By further differentiation of this result, or otherwise, find the Maclaurin series of $y$, up to and including the term in ${{x}^{3}}$.</div><div class="list-item-paragraph">Leave the coefficients in exact form.</div><p style="text-align: right;">[7]</p></li><li><div class="list-item-content"><span class="list-item-text">Using appropriate expansions from the List of Formulae (MF26), find the expansion of ${{(4-x)}^{-1}}\cos \,\,3x$, up to and including the term in ${{x}^{3}}$.</span></div><div class="list-item-paragraph">Give the coefficients as exact fractions in their simplest form.</div><p style="text-align: right;">[5]</p></li></ol>

**Do not use a calculator for this question**

Give that $y={{\mathrm{e}}^{\frac{2}{x+1}}}$, where $x\ne -1$, show that

${{(x+1)}^{2}}\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+2(x+2)\frac{\mathrm{d}y}{\mathrm{d}x}=0$.

By further differentiation of this result, or otherwise, find the Maclaurin series of $y$, up to and including the term in ${{x}^{3}}$.

Using appropriate expansions from the List of Formulae (MF26), find the expansion of ${{(4-x)}^{-1}}\cos \,\,3x$, up to and including the term in ${{x}^{3}}$.

2020 TMJC Promo Q6

2020 TMJC Promo Q6

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