Find the series expansion for $\frac{\cos 2x}{{{\left( 1+\tan x \right)}^{2}}}$ up to and inc

Question

Find the series expansion for $\frac{\cos 2x}{{{\left( 1+\tan x \right)}^{2}}}$ up to and including the term in ${{x}^{2}}$, given that $x$ is a sufficiently small angle.[3] It is given that $\mathrm{f}(x)={{\left( 1-x \right)}^{\frac{1}{2}}}\ln \sqrt{1+x}$. Find the numerical value of $\mathrm{f}'(x)$ when $x=0.01$, leaving the answer in $5$ decimal places.[1] Using the standard series from the List of Formulae (MF27), expand $\mathrm{f}\left( x \right)$ as far as the term in ${{x}^{3}}$. State the range of values of $x$ for which the expansion is valid.[4] Denote the answer in part (b)(ii) as $\mathrm{g}\left( x \right)$. Hence find $\mathrm{g }'\mathrm{ }\left( x \right)$ when $x=0.01$, leaving the answer in $5$ decimal places.[2] Comparing your answer in part (b)(i) against (b)(iii), comment on the validity of this answer.[1]

Accepted Answer

(a) $1-2x+x^2+O(x^3)$ (b)(i) $0.49007$ (ii) $f(x)=\dfrac{x}{2}-\dfrac{x^2}{2}+\dfrac{11x^3}{48}+O(x^4)$, valid for $-1<x<1$ (iii) $0.49007$ (iv) The approximation is valid since $x=0.01$ lies in $(-1,1)$ and agrees with the exact derivative to $5$ d.p.

2024 MI P2 Q5

2024 MI P2 Q5

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