Review practice · Section 2 of 2

(i) For $f(x)=\ln(1+2\sin x)$, find $f(0),f^{\prime}(0),f^{\prime\prime}(0),f^{\prime\prime\prime}(0)$ and hence its first three nonzero Maclaurin terms.7 marks

(ii) Its first two nonzero terms match those of $e^{ax}\sin(nx)$. Use appropriate MF27 expansions to find $a,n$ and then the third nonzero term of $e^{ax}\sin(nx)$.5 marks

Given $\cos y=\sqrt{1-e^x/4}$ and $0<y<\pi/2$, show $\sin(2y)y^{\prime}=e^x/4$. By further differentiation find the Maclaurin series for $y$ through $x^2$ in exact form.5 marks

Deduce the tangent to $y=\cos^{-1}\!\sqrt{1-e^x/4}$ at $x=0$.2 marks

For $y=\tan^{-1}x$, prove $2xy^{\prime}+(1+x^2)y^{\prime\prime}=0$. Differentiate further to show its Maclaurin series through $x^3$ is $x+hx^3$, finding $h$.5 marks

Hence expand $\sqrt{1+\tan^{-1}x}$ through $x^3$.3 marks

Let $f(x)=e^x\sin x$.

(i) Sketch $f$ for $-3\le x\le3$.2 marks

(ii) Expand $f$ through $x^3$ and call that polynomial $g(x)$.3 marks

(iii) On the same diagram sketch and clearly label $g$.1 mark

(iv) For $-3\le x\le3$, find where $g(x)$ is within $\pm0.5$ of $f(x)$.3 marks

Given $f^{\prime}(x)=f(x)\ln(\sec(ax))$ and $f(0)=1/2$, where $a$ is real and nonzero, show

$$f^{\prime\prime}(x)f(x)=[f^{\prime}(x)]^2+a[f(x)]^2\tan(ax).$$

Hence find the Maclaurin series through $x^3$ in terms of $a$.4 marks

(a) Let $f(x)=e^{\tan^{-1}(x/2)}$.

(i) Show $(4+x^2)f^{\prime}(x)=2f(x)$.1 mark

(ii) Use repeated differentiation to expand $f$ through $x^2$. Use it to estimate $e^{\pi/6}$ and show the error is about $3.3\%$.6 marks

(b) For $g(x)=1/[\sqrt2\cos(x/a+\pi/4)]$, where $a>1$, show that for sufficiently small $x$, $g(x)\approx1+px+qx^2$, finding $p,q$ in terms of $a$.4 marks