N2022 P1 Q2

N2022 P1 Q2

Junior College 2
6 marks
Free

It is given that \(\mathrm{f}(x)=\tan^{-1}(\sqrt{2}+x)\).

  1. Find \(\mathrm{f}'(x)\) and \(\mathrm{f}''(x)\).[3]
  2. Hence find the first three terms of the Maclaurin series for \(\mathrm{f}(x)\). Give the coefficients correct to 3 significant figures.[3]

Default solution

  1. \(\mathrm{f}'(x)=\frac{1}{1+(\sqrt{2}+x)^2}\)
    \(\mathrm{f}''(x)=-\frac{2(\sqrt{2}+x)}{[1+(\sqrt{2}+x)^2]^2}\)
  2. \(\mathrm{f}(0)=\tan^{-1}\sqrt{2}=0.955316\ldots\)
    \(\mathrm{f}'(0)=\frac13,\qquad \mathrm{f}''(0)=-\frac{2\sqrt{2}}9\)
    \(\mathrm{f}(x)=\tan^{-1}\sqrt{2}+\frac13x-\frac{\sqrt{2}}9x^2+\cdots\)
    \(\mathrm{f}(x)=0.955+0.333x-0.157x^2+\cdots\quad\text{(3 s.f.)}\)
Answer:(a) \(\mathrm{f}'(x)=\frac{1}{1+(\sqrt{2}+x)^2},\ \mathrm{f}''(x)=-\frac{2(\sqrt{2}+x)}{[1+(\sqrt{2}+x)^2]^2}\) (b) \(\mathrm{f}(x)=0.955+0.333x-0.157x^2+\cdots\)

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