Let $\mathrm{f}\left( x \right)={{\mathrm{e}}^{x}}\sin x$. Sketch the graph of \(y=\mathrm{f}

Question

Let $\mathrm{f}\left( x \right)={{\mathrm{e}}^{x}}\sin x$. Sketch the graph of $y=\mathrm{f}\left( x \right)$ for $-3\le x\le 3$.[2] Find the series expansion of $\mathrm{f}\left( x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.[3] Denote the answer to part (ii) by $\mathrm{g}\left( x \right)$. On the same diagram as in part (i), sketch the graph of $y=\mathrm{g}\left( x \right)$. Label the two graphs clearly.[1] Find, for $-3\le x\le 3$, the set of values of $x$ for which the value of $\mathrm{g}\left( x \right)$ is within $\pm 0.5$ of the value of $\mathrm{f}\left( x \right)$.[3]

Accepted Answer

(i) Zero $(0,0)$, minimum at $x=-\dfrac\pi4$, maximum at $x=\dfrac{3\pi}{4}$ (ii) $f(x)=x+x^2+\dfrac{x^3}{3}+O(x^4)$ (iii) $g(x)=x+x^2+\dfrac{x^3}{3}$, with stationary point of inflection $\left(-1,-\dfrac13\right)$ (iv) $-1.96<x<1.56$

N2008 P2 Q1

N2008 P2 Q1

Video Solution:

Default solution