Show that \(\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}x^2}=k\left(\frac{\mathrm{d}y}{\mathrm{d}x}-kye^{-4x}\right)\), where \(k\) is a constant to be found.[3]
By differentiation of the result in part (a), find the first three non-zero terms of the Maclaurin expansion of \(\cos(e^{-2x}-1)\).[3]
If your expansion in part (b) is used to estimate \(\displaystyle \int_0^{0.7}\cos(e^{-2x}-1)\,\mathrm{d}x\), explain how you can improve the accuracy of your approximation.[1]
The first two non-zero terms of the Maclaurin expansion of \(\cos(e^{-2x}-1)\) are equal to the first two non-zero terms of the series expansion of \(\sqrt[3]{a-bx^2}\), where \(a\) and \(b\) are constants. Using standard series from the List of Formulae (MF27), find the values of \(a\) and \(b\).[2]
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