Derivative involving harmonic form

1. (a) Express \(2 \cos 3 x - 3 \sin 3 x\) in the form \(R \cos ( 3 x + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your answers to 3 significant figures.

$$\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } \cos 3 x$$ (b) Show that \(\mathrm { f } ^ { \prime } ( x )\) can be written in the form $$\mathrm { f } ^ { \prime } ( x ) = R \mathrm { e } ^ { 2 x } \cos ( 3 x + \alpha )$$ where \(R\) and \(\alpha\) are the constants found in part (a).
(c) Hence, or otherwise, find the smallest positive value of \(x\) for which the curve with equation \(y = \mathrm { f } ( x )\) has a turning point.

7 Two quantities, \(x\) and \(\theta\), vary with time and are related by the equation \(x = 5 \sin \theta - 4 \cos \theta\).

1. Find the value of \(x\) when \(\theta = \frac { \pi } { 2 }\).
2. When \(\theta = \frac { \pi } { 2 }\), its rate of increase (in suitable units) is given by \(\frac { \mathrm { d } \theta } { \mathrm { d } t } = 0.1\). Show that at that moment \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4\).

5 In this question you must show detailed reasoning. The function f is defined by \(\mathrm { f } ( x ) = \cos x + \sqrt { 3 } \sin x\) with domain \(0 \leqslant x \leqslant 2 \pi\).

1. Solve the following equations.
   1. \(\mathrm { f } ^ { \prime } ( x ) = 0\)
   2. \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) The diagram shows the graph of the gradient function \(y = \mathrm { f } ^ { \prime } ( x )\) for the domain \(0 \leqslant x \leqslant 2 \pi\). \includegraphics[max width=\textwidth, alt={}, center]{40d40a0b-5b33-4940-b15b-ee03e1291f61-05_583_741_781_242}
2. Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points \(A , B , C\) and \(D\).
3. 1. Explain how to use the graph of the gradient function to find the values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
   2. Using set notation, write down the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing in the domain \(0 \leqslant x \leqslant 2 \pi\).