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Find and simplify the rule of `f^{\prime}(x)`, where `f:R \rightarrow R, f(x)=\frac{\cos (x)}{e^x}`. (2 marks)
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`\frac{-(\sin x+\cos x)}{e^x}`
Using the quotient rule
| `f(x)` | `=\frac{\cos (x)}{e^x}` | |
| `f^{\prime}(x)` | `=\frac{-e^x \sin x-e^x \cos x}{e^{2 x}}` | |
| `= \frac{-e^x(\sin x+\cos x)}{e^{2 x}}` | ||
| `=\frac{-(\sin x+\cos x)}{e^x}` |
Let \(f(x)=\sin(x)e^{2x}\).
Find \(f^{'}\Big(\dfrac{\pi}{4}\Big)\). (2 marks)
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\(\dfrac{3\sqrt{2}}{2}e^{\frac{\pi}{2}}\ \text{or}\ \dfrac{3e^{\frac{\pi}{2}}}{\sqrt{2}}\)
\(\text{Using the product rule}\)
| \(f'(x)\) | \(=e^{2x}\cos(x)+2e^{2x}\sin(x)\) |
| \(=e^{2x}\Big(\cos(x)+2\ \sin(x)\Big)\) | |
| \(\therefore\ f’\Big(\dfrac{\pi}{4}\Big)\) | \(=e^{2(\frac{\pi}{4})}\Bigg(\cos(\dfrac{\pi}{4})+2\ \sin(\dfrac{\pi}{4})\Bigg)\) |
| \(=e^{\frac{\pi}{2}}\Bigg(\dfrac{1}{\sqrt{2}}+\sqrt{2}\Bigg)\) | |
| \(=e^{\frac{\pi}{2}}\Bigg(\dfrac{1+\sqrt2 \times \sqrt2}{\sqrt2} \Bigg) \) | |
| \(=\dfrac{3\sqrt{2}}{2}e^{\frac{\pi}{2}}\ \ \text{or}\ \ \dfrac{3e^{\frac{\pi}{2}}}{\sqrt{2}}\) |
Let `f(x) = (e^x)/(cos(x))`.
Evaluate `f^{prime}(pi)`. (2 marks)
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`text(See Worked Solutions)`
`f^{prime}(x) = (e^x)/(cos(x))`
| `u` | `= e^x` | `v` | `= cos(x)` |
| `u^{prime}` | `= e^x` | `v^{prime}` | `=-sin(x)` |
| `f^{prime}(x)` | `= (u^{prime}v-uv^{prime})/(v^2)` |
| `= (e^x · cos(x) + e^x sin(x))/(cos^2(x))` |
| `f^{prime}(pi)` | `= (e^pi · cospi + e^pi sinpi)/(cos^2 pi)` |
| `= (e^pi(-1) + e^pi · 0)/((-1)^2)` | |
| `= -e^pi` |