2017 ACJC Promo Q10
(a)
Given that $x$ is small enough for terms involving ${{x}^{4}}$ and above to be ignored, use the Maclaurin series for $\sin x$ and $\cos x$ to show that
$\tan x\approx x+\frac{1}{3}{{x}^{3}}$.
[3]
A straight pole, $AB$, leans against a wall as shown in the diagram below. $AC$ is horizontal where $C$ is the base of the wall and $AC=2$ metres. $B$ is vertically above $C$ and angle $BAC={{85}^{\circ }}$.
Use the Maclaurin series for $\tan x$ to find an approximate value for $BC$, giving your answer correct to $5$ decimal places.
[2]
Use a calculator to find the value of $BC$, giving your answer correct to $5$ decimal places, and hence show that the error in your approximation is about $0.0008\%$.
[2]
(b)
Given that $y=\tan \left( \frac{1}{x+a} \right)$, where $a$ is a positive constant, show that
${{\left( x+a \right)}^{2}}\frac{\text{d}y}{\text{d}x}+{{y}^{2}}+1=0$.
[2]
Using $a=\frac{4}{\pi }$, find the Maclaurin series for $\tan \left( \frac{1}{x+a} \right)$, up to and including the term in ${{x}^{2}}$, giving all coefficients in exact form.
[4]
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