Practice section 5 of 7

Let $y=e^{1/(x+1)}$. Show that $yy^{\prime\prime}-(y^\prime)^2=\dfrac{2y^2}{(x+1)^3}$. Hence find its Maclaurin expansion through $x^2$, and deduce the expansion of $e^{-x/(x+1)}$ through $x^2$.

Let $f(x)=a\sec x+b\tan x$ for real distinct positive constants $a,b$.

(i) Show that $f^\prime(x)=f(x)\tan x+b$.

(ii) Hence find the first three nonzero Maclaurin terms in terms of $a,b$.

(iii) These terms agree with the first three terms of $e^{\sqrt2 x}\cos x$ for small $x$. Using MF27, find the exact values of $a,b$.

In triangle $ABC$, $AB=1$, $\angle BAC=\theta$ radians and $\angle ABC=3\pi/4$ radians (see diagram).

(i) Show that $AC=\dfrac1{\cos\theta-\sin\theta}$.

(ii) For sufficiently small $\theta$, show that $AC\approx1+a\theta+b\theta^2$, determining $a,b$.

Express $f(x)=\dfrac{3x^2-3x+4}{(2x-1)(x^2+3)}$ in partial fractions and hence obtain $f(x)$ as a series expansion in ascending powers of $x$ up to and including the term in $x^2$. [5]

Find the range of values of $x$ for which the expansion is valid. [1]

Expand $(4-x)^{-1/2}$ in ascending powers of $x$ through $x^3$, and state its range of validity. Find $a$ if the term independent of $x$ in $(1+ax^{-1})/\sqrt{4-x}$ is $5$. [6]

Expand $\sqrt{1-x}/(1+2x)$ through $x^2$. [4]

(i) A student substitutes $x=-1$ to estimate $\sqrt2$. Explain why this is wrong. [1]

(ii) Put $x=1/9$ in the expansion to show $\sqrt2\approx p/1296$, and find the positive integer $p$. [2]