Find stationary points of hyperbolic curves

2. $$y = \ln ( \tanh 2 x ) \quad x > 0$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = p \operatorname { cosech } 4 x$$ where \(p\) is a constant to be determined.
2. Hence determine, in simplest form, the exact value of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\)

3. (a) Given that \(y = \operatorname { arsech } \left( \frac { x } { 2 } \right)\), where \(0 < x \leqslant 2\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p } { x \sqrt { q - x ^ { 2 } } }$$ where \(p\) and \(q\) are constants to be determined. In part (b) solutions based entirely on calculator technology are not acceptable. $$\mathrm { f } ( x ) = \operatorname { artanh } ( x ) + \operatorname { arsech } \left( \frac { x } { 2 } \right) \quad 0 < x \leqslant 1$$ (b) Determine, in simplest form, the exact value of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) = 0\)

7.

1. Show that \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{469976eb-f1a9-4bdc-8f52-64ab23856109-26_1088_691_251_676} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \arccos ( \operatorname { sech } x ) + \operatorname { coth } x \quad x > 0$$ The point \(P\) is a minimum turning point of \(C\)
2. Show that the \(x\) coordinate of \(P\) is \(\ln ( q + \sqrt { q } )\) where \(q = \frac { 1 } { 2 } ( 1 + \sqrt { k } )\) and \(k\) is an integer to be determined.

1. The curve \(C\) has equation

$$y = 9 \cosh x + 3 \sinh x + 7 x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm.

1. The curve \(C\) has equation

$$y = 9 \cosh x + 3 \sinh x + 7 x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm.

4. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation $$y = 40 \operatorname { arcosh } x - 9 x , \quad x \geqslant 1$$ Use calculus to find the exact coordinates of the turning point of the curve, giving your answer in the form \(\left( \frac { p } { q } , r \ln 3 + s \right)\), where \(p , q , r\) and \(s\) are integers.

4

1. Find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } \mathrm {~d} x\), giving your answer in an exact logarithmic form.
2. 1. Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that \(\sinh 2 x = 2 \sinh x \cosh x\).
   2. Show that one of the stationary points on the curve $$y = 20 \cosh x - 3 \cosh 2 x$$ has coordinates \(\left( \ln 3 , \frac { 59 } { 3 } \right)\), and find the coordinates of the other two stationary points.
   3. Show that \(\int _ { - \ln 3 } ^ { \ln 3 } ( 20 \cosh x - 3 \cosh 2 x ) \mathrm { d } x = 40\).

4

1. Prove, using exponential functions, that $$\sinh 2 x = 2 \sinh x \cosh x$$ Differentiate this result to obtain a formula for \(\cosh 2 x\).
2. Sketch the curve with equation \(y = \cosh x - 1\). The region bounded by this curve, the \(x\)-axis, and the line \(x = 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find, correct to 3 decimal places, the volume generated. (You must show your working; numerical integration by calculator will receive no credit.)
3. Show that the curve with equation $$y = \cosh 2 x + \sinh x$$ has exactly one stationary point.
   Determine, in exact logarithmic form, the \(x\)-coordinate of the stationary point.

8 The curve with equation \(y = \frac { \sinh x } { x ^ { 2 } }\), for \(x > 0\), has one turning point.

1. Show that the \(x\)-coordinate of the turning point satisfies the equation \(x - 2 \tanh x = 0\).
2. Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2\), to find the next two approximations, \(x _ { 2 }\) and \(x _ { 3 }\), to the positive root of \(x - 2 \tanh x = 0\).
3. By considering the approximate errors in \(x _ { 1 }\) and \(x _ { 2 }\), estimate the error in \(x _ { 3 }\). (You are not expected to evaluate \(x _ { 4 }\).)

4

1. Sketch, on the same diagram, the curves with equations \(y = \operatorname { sech } x\) and \(y = x ^ { 2 }\).
2. By using the definition of \(\operatorname { sech } x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that the \(x\)-coordinates of the points at which these curves meet are solutions of the equation $$x ^ { 2 } = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 } .$$
3. The iteration $$x _ { n + 1 } = \sqrt { \frac { 2 \mathrm { e } ^ { x _ { n } } } { \mathrm { e } ^ { 2 x _ { n } } + 1 } }$$ can be used to find the positive root of the equation in part (ii). With initial value \(x _ { 1 } = 1\), the approximations \(x _ { 2 } = 0.8050 , x _ { 3 } = 0.8633 , x _ { 4 } = 0.8463\) and \(x _ { 5 } = 0.8513\) are obtained, correct to 4 decimal places. State with a reason whether, in this case, the iteration produces a 'staircase' or a ‘cobweb’ diagram.

4

1. Show that \(\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
2. Find \(\int _ { 2.5 } ^ { 3.9 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 9 } } \mathrm {~d} x\), giving your answer in the form \(a \ln b\), where \(a\) and \(b\) are rational numbers.
3. There are two points on the curve \(y = \frac { \cosh x } { 2 + \sinh x }\) at which the gradient is \(\frac { 1 } { 9 }\). Show that one of these points is \(\left( \ln ( 1 + \sqrt { 2 } ) , \frac { 1 } { 3 } \sqrt { 2 } \right)\), and find the coordinates of the other point, in a similar form.

4

1. Solve the equation $$\sinh t + 7 \cosh t = 8$$ expressing your answer in exact logarithmic form. A curve has equation \(y = \cosh 2 x + 7 \sinh 2 x\).
2. Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16 . Show that there is no point on the curve at which the gradient is zero.
   Sketch the curve.
3. Find, in an exact form, the positive value of \(a\) for which the area of the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = a\) is \(\frac { 1 } { 2 }\).

9

1. Given that \(y = \tanh ^ { - 1 } x\), for \(- 1 < x < 1\), prove that \(y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
2. It is given that \(\mathrm { f } ( x ) = a \cosh x - b \sinh x\), where \(a\) and \(b\) are positive constants.
   (a) Given that \(b \geqslant a\), show that the curve with equation \(y = \mathrm { f } ( x )\) has no stationary points.
   (b) In the case where \(a > 1\) and \(b = 1\), show that \(\mathrm { f } ( x )\) has a minimum value of \(\sqrt { a ^ { 2 } - 1 }\).

5 A function is defined by \(\mathrm { f } ( x ) = \sinh ^ { - 1 } x + \sinh ^ { - 1 } \left( \frac { 1 } { x } \right)\), for \(x \neq 0\).

1. When \(x > 0\), show that the value of \(\mathrm { f } ( x )\) for which \(\mathrm { f } ^ { \prime } ( x ) = 0\) is \(2 \ln ( 1 + \sqrt { 2 } )\).
2. \includegraphics[max width=\textwidth, alt={}, center]{72a1330a-c6dc-4f3a-9b0e-333b099f4509-3_497_659_520_708} The diagram shows the graph of \(y = \mathrm { f } ( x )\) for \(x > 0\). Sketch the graph of \(y = \mathrm { f } ( x )\) for \(x < 0\) and state the range of values that \(\mathrm { f } ( x )\) can take for \(x \neq 0\).

8 It is given that \(\mathrm { f } ( x ) = 2 \sinh x + 3 \cosh x\).

1. Show that the curve \(y = \mathrm { f } ( x )\) has a stationary point at \(x = - \frac { 1 } { 2 } \ln 5\) and find the value of \(y\) at this point.
2. Solve the equation \(\mathrm { f } ( x ) = 5\), giving your answers exactly. \section*{Question 9 begins on page 4.}

10 The curve, \(C\), has equation \(y = \frac { x } { \cosh x }\) 10

1. Show that the \(x\)-coordinates of any stationary points of \(C\) satisfy the equation \(\tanh x = \frac { 1 } { x }\) [0pt] [3 marks] 10
2. 1. Sketch the graphs of \(y = \tanh x\) and \(y = \frac { 1 } { x }\) on the axes below.
      [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{a155b39a-6835-4d62-a481-41ef822bbd5f-14_1151_1226_1461_358} 10
3. (ii) Hence determine the number of stationary points of the curve \(C\). 10
4. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = 0\) at each of the stationary points of the curve \(C\).
   [0pt] [4 marks]

7

1. By using the definitions of \(\cosh u\) and \(\sinh u\) in terms of \(\mathrm { e } ^ { u }\) and \(\mathrm { e } ^ { - u }\), show that \(\sinh 2 u \equiv 2 \sinh u \cosh u\). The equation of a curve, \(C\), is \(\mathrm { y } = 16 \cosh \mathrm { x } - \sinh 2 \mathrm { x }\).
2. Show that there is only one solution to the equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = 0\) You are now given that \(C\) has exactly one point of inflection.
3. Use your answer to part (b) to determine the exact coordinates of this point of inflection. Give your answer in a logarithmic form where appropriate.

8 You are given that \(\mathrm { f } ( x ) = 4 \sinh x + 3 \cosh x\).

1. Show that the curve \(y = f ( x )\) has no turning points.
2. Determine the exact solution of the equation \(\mathrm { f } ( x ) = 5\).

8 The equation of a curve is \(y = \cosh ^ { 2 } x - 3 \sinh x\). Show that \(\left( \ln \left( \frac { 3 + \sqrt { 13 } } { 2 } \right) , - \frac { 5 } { 4 } \right)\) is the only stationary point on the curve.

4

1. Prove that the curve $$y = 12 \cosh x - 8 \sinh x - x$$ has exactly one stationary point.
2. Given that the coordinates of this stationary point are \(( a , b )\), show that \(a + b = 9\).

2

1. 1. Sketch on the axes below the graphs of \(y = \sinh x\) and \(y = \cosh x\).
   2. Use your graphs to explain why the equation $$( k + \sinh x ) \cosh x = 0$$ where \(k\) is a constant, has exactly one solution.
2. A curve \(C\) has equation \(y = 6 \sinh x + \cosh ^ { 2 } x\). Show that \(C\) has only one stationary point and show that its \(y\)-coordinate is an integer. \includegraphics[max width=\textwidth, alt={}, center]{53d742f4-923b-478c-8ae6-ada6c0bb4a7e-2_560_704_1416_171} \includegraphics[max width=\textwidth, alt={}, center]{53d742f4-923b-478c-8ae6-ada6c0bb4a7e-2_560_711_1416_964}

1. (a) By writing \(y = \sinh ^ { - 1 } ( 4 x + 3 )\) as \(\sinh y = 4 x + 3\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { \sqrt { 16 x ^ { 2 } + 24 x + 10 } }\).
   (b) Show that the graph of \(\mathrm { e } ^ { - 3 x } y = \sinh 2 x\) has only one stationary point.

\section*{}

1. The curve \(C\) has equation

$$y = 31 \sinh x - 2 \sinh 2 x \quad x \in \mathbb { R }$$ Determine, in terms of natural logarithms, the exact \(x\) coordinates of the stationary points of \(C\).

6 In this question you must show detailed reasoning.

1. Find the coordinates of all stationary points on the graph of \(y = 6 \sinh ^ { 2 } x - 13 \cosh x\), giving your answers in an exact, simplified form.
2. By finding the second derivative, classify the stationary points found in part (i).