2015 MJC P1 Q7
(a)
Given that $\text{f}\left( x \right)={{\text{e}}^{\sin ax}}$, where a is a non-zero real constant, find $\text{f}\left( 0 \right)$, $\text{f{ }’}\left( 0 \right)$ and $\text{f{ }”}\left( 0 \right)$. Hence write down the first three non-zero terms in the Maclaurin series of $\text{f}\left( x \right)$. Give the coefficients in terms of $a$.
[5]
(b)
[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$$\text{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]
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