Practice section 7 of 7

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

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

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

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

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

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]

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

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

(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]

(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]