Find stationary points - mixed transcendental products

5 The curve with equation \(y = \mathrm { e } ^ { - a x } \tan x\), where \(a\) is a positive constant, has only one point in the interval \(0 < x < \frac { 1 } { 2 } \pi\) at which the tangent is parallel to the \(x\)-axis. Find the value of \(a\) and state the exact value of the \(x\)-coordinate of this point.

4 The curve with equation \(y = \mathrm { e } ^ { - x } \sin x\) has one stationary point for which \(0 \leqslant x \leqslant \pi\).

1. Find the \(x\)-coordinate of this point.
2. Determine whether this point is a maximum or a minimum point.

4 A curve has equation \(y = \mathrm { e } ^ { - 3 x } \tan x\). Find the \(x\)-coordinates of the stationary points on the curve in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\). Give your answers correct to 3 decimal places.

4 The curve with equation \(y = \mathrm { e } ^ { 2 x } ( \sin x + 3 \cos x )\) has a stationary point in the interval \(0 \leqslant x \leqslant \pi\).

1. Find the \(x\)-coordinate of this point, giving your answer correct to 2 decimal places.
2. Determine whether the stationary point is a maximum or a minimum.

1. 1. The curve \(C\) has equation \(y = \mathrm { g } ( x )\) where

$$g ( x ) = e ^ { 3 x } \sec 2 x \quad - \frac { \pi } { 4 } < x < \frac { \pi } { 4 }$$

1. Find \(\mathrm { g } ^ { \prime } ( x )\)
2. Hence find the \(x\) coordinate of the stationary point of \(C\).
   (ii) A different curve has equation $$x = \ln ( \sin y ) \quad 0 < y < \frac { \pi } { 2 }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { x } } { \mathrm { f } ( x ) }$$ where \(\mathrm { f } ( x )\) is a function of \(\mathrm { e } ^ { x }\) that should be found.
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

5. (i) Find, using calculus, the \(x\) coordinate of the turning point of the curve with equation $$y = \mathrm { e } ^ { 3 x } \cos 4 x , \quad \frac { \pi } { 4 } \leqslant x < \frac { \pi } { 2 }$$ Give your answer to 4 decimal places.
(ii) Given \(x = \sin ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(y\). Write your answer in the form $$\frac { \mathrm { d } y } { \mathrm {~d} x } = p \operatorname { cosec } ( q y ) , \quad 0 < y < \frac { \pi } { 4 }$$ where \(p\) and \(q\) are constants to be determined. \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-09_2258_47_315_37}

7 Fig. 8 shows part of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \mathrm { e } ^ { - \frac { 1 } { 5 } x } \sin x\), for all \(x\). \begin{figure}[h] \end{figure}

1. Sketch the graphs of
   (A) \(y = \mathrm { f } ( 2 x )\),
   (B) \(y = \mathrm { f } ( x + \pi )\).
2. Show that the \(x\)-coordinate of the turning point P satisfies the equation \(\tan x = 5\). Hence find the coordinates of P .
3. Show that \(\mathrm { f } ( x + \pi ) = \mathrm { e } ^ { - \frac { 1 } { 5 } \pi } \mathrm { f } ( x )\). Hence, using the substitution \(u = x - \pi\), show that $$\int _ { \pi } ^ { 2 \pi } \mathrm { f } ( x ) \mathrm { d } x = \mathrm { e } ^ { - \frac { 1 } { 5 } \pi } \int _ { 0 } ^ { \pi } \mathrm { f } ( u ) \mathrm { d } u .$$ Interpret this result graphically. [You should not attempt to integrate \(\mathrm { f } ( x )\).]

1 Fig. 1 shows part of the curve \(y = \mathrm { e } ^ { 2 x } \cos x\). \begin{figure}[h] \end{figure} Find the coordinates of the turning point P .

3 The equation of a curve is \(y = \mathrm { e } ^ { 2 x } \cos x\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any stationary points for which \(- \pi \leqslant x \leqslant \pi\). Give your answers correct to 3 significant figures.