October 2021 Paper 2 Q7
7 Differentiate \(\cos x\) with respect to \(x\), from first principles. [4]
| Scheme | Marks |
|---|---|
| \(\cos(x + \delta x) - \cos x\) \(= \cos x\cos\delta x - \sin x\sin\delta x - \cos x\) | B1 |
| \(\displaystyle\lim_{\delta x \to 0} \frac{\cos x\cos\delta x - \sin x\sin\delta x - \cos x}{\delta x}\) | M1 |
| as \(\delta x \to 0\): \(\cos\delta x \to 1\) or \(1 - \dfrac{(\delta x)^2}{2}\) and \(\dfrac{\sin\delta x}{\delta x} \to 1\) or \(\sin\delta x \to \delta x\) | M1 |
| \(\left(\displaystyle\lim_{\delta x \to 0} \frac{\cos x - \sin x\,\delta x - \cos x}{\delta x}\right)\) \(= -\sin x\) | A1 |
| [4] |
Notes
Allow \(h\) or other letter for \(\delta x\) throughout
M1: or \(\displaystyle\lim_{\delta x \to 0} \frac{\cos(x + \delta x) - \cos x}{\delta x}\) or may be seen later. Must include \(\displaystyle\lim_{\delta x \to 0}\).
M1: Allow \(\cos\delta x = 1\) for small \(\delta x\) \(\left(\text{or } 1 - \dfrac{(\delta x)^2}{2}\right)\)
Allow \(\sin\delta x = \delta x\) for small \(\delta x\)
Both must be explicitly stated for M1
If not stated but implied, M0, but can still possibly gain final A1
A1: Dep on at least B1M1 gained, and approximations either seen explicitly or seen substituted.
and nothing incorrect seen
NB. \(\cos x - \sin x - \cos x = -\sin x\) is incorrect and scores A0