Discussion questions · Section 2 of 2

Let $y=\ln(1+\sin x)$.

(i) Show that $e^y\dfrac{d^2y}{dx^2}=1-e^y[(\dfrac{dy}{dx})^2+1]$.

(ii) Find its Maclaurin series through $x^2$.

(iii) Verify it using the standard series for $\sin x$ and $\ln(1+x)$.

Given $y=\tan(\pi/4+x)$, show that $\dfrac{d^2y}{dx^2}=2y\dfrac{dy}{dx}$. Find its series through $x^4$. When $x=0.1$, $y=1.22304888$ correct to 8 decimal places. Verify that the fourth-degree estimate has error about $0.004\%$.

The curve $y=f(x)$ passes through $(0,\pi/4)$ and satisfies $y^\prime\tan y=\ln((e^x+1)/2)$. Find the first three nonzero Maclaurin terms of $y$. Deduce the expansion of $(\cot y)\ln((e^x+1)/2)$ through $x^2$.

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 $a,b$.