A function $\mathrm{f}$ is defined by $\mathrm{f}\left( x \right)={{\mathrm{e}}^{x}}\cos x$, for

Question

A function $\mathrm{f}$ is defined by $\mathrm{f}\left( x \right)={{\mathrm{e}}^{x}}\cos x$, for $0\le x\le \frac{1}{2}\pi $. Using calculus, find the stationary point of $\mathrm{f}\left( x \right)$ and determine its nature.[5] Integrate by parts twice to show that $\int \mathrm{e}^{2x} \cos 2x \mathrm{d}x = \frac{1}{4} \mathrm{e}^{2x} (\sin 2x + \cos 2x) + c.$ [4] The graph of $y=\mathrm{f}\left( x \right)$ is rotated completely about the $x$-axis. Find the exact volume generated.[4]

Accepted Answer

(a) $\left(\dfrac{\pi}{4},\, \dfrac{e^{\pi/4}}{\sqrt{2}}\right)$ is a maximum point (b) $\dfrac{1}{4}e^{2x}(\sin 2x + \cos 2x) + C$ (c) $\dfrac{\pi}{8}(e^{\pi} - 3)$ units$^3$

N2021 P1 Q9

N2021 P1 Q9

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