Given that $\mathrm{f}\left( x \right)={{\mathrm{e}}^{\sin ax}}$, where a is a non-zero real

Question

Given that $\mathrm{f}\left( x \right)={{\mathrm{e}}^{\sin ax}}$, where a is a non-zero real constant, find $\mathrm{f}\left( 0 \right)$, $\mathrm{f{ }'}\left( 0 \right)$ and $\mathrm{f{ }''}\left( 0 \right)$. Hence write down the first three non-zero terms in the Maclaurin series of $\mathrm{f}\left( x \right)$. Give the coefficients in terms of $a$.[5] [It is given that arc length of a circular sector with radius $r$ and angle $\theta $ radians is $r\theta $.] The diagram shows a triangle $OAB$ with $OA=4$$\mathrm{cm}$, $OB=5$cm and $\angle AOB=\theta $radians. Given that $OA$ and $OC$ are radii of a circle with centre $O$ and $\theta $ is a sufficiently small angle, show that the perimeter of $ABC$ can be approximated by $a+b\theta +c{{\theta }^{2}}$ for constants $a$, $b$ and $c$ to be determined.[5]

Accepted Answer

(a) $f(0)=1$, $f'(0)=a$, $f''(0)=a^2$, and $e^{\sin(ax)}=1+ax+\frac{a^2}{2}x^2+O(x^3)$. (b) The perimeter is approximately $2+4\theta+10\theta^2$, so $a=2$, $b=4$, and $c=10$.

2015 MJC P1 Q7

2015 MJC P1 Q7

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