Given that $\mathrm{f}(x) = \ln(3 - 2e^x)$, find $\mathrm{f}(0)$, $\mathrm{f}'(0)$, \(\ma

Question

Given that $\mathrm{f}(x) = \ln(3 - 2e^x)$, find $\mathrm{f}(0)$, $\mathrm{f}'(0)$, $\mathrm{f}''(0)$ and $\mathrm{f}'''(0)$. Hence write down the first three non-zero terms in the Maclaurin series for $\mathrm{f}(x)$. [7] The first two non-zero terms in the Maclaurin series for $\mathrm{f}(x)$ are equal to the first two non-zero terms in the series expansion of $(1 + ax)^5 \sin\left(\frac{x}{n}\right)$. Using appropriate expansions form the List of Formulae (MF 26), find the constants $a$ and $n$. Hence find the third non-zero term of the series expansion of $(1 + ax)^5 \sin\left(\frac{x}{n}\right)$ for these values of $a$ and $n$. [5]

Accepted Answer

(i) $f(0)=0,\ f'(0)=-2,\ f''(0)=-6,\ f'''(0)=-30$; $f(x)=-2x-3x^2-5x^3+\cdots$ (ii) $n=-\frac12,\ a=\frac3{10}$; third non-zero term $-\frac7{15}x^3$.

2018 TGM P1 Q2

2018 TGM P1 Q2

Solution:

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