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.

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.

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.}

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.

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. 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 \(C\).

8

1. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that \(\sinh 2 x = 2 \sinh x \cosh x\). You are given the function \(\mathrm { f } ( x ) = a \cosh x - \cosh 2 x\), where \(a\) is a positive constant.
2. Verify that, for any value of \(a\), the curve \(y = \mathrm { f } ( x )\) has a stationary point on the \(y\)-axis.
3. Find the coordinates of the stationary point found in part (ii).
4. Determine the maximum value of \(a\) for which the stationary point found in part (ii) is the only stationary point on the curve \(y = \mathrm { f } ( x )\). You are given that for any value of \(a\) greater than the value found in part (iv) there are three stationary points, the one found in part (ii) and two others, one of which satisfies \(x > 0\).
5. Find the coordinates of this point when \(a = 6\). Give your answer in the form \(\left( \cosh ^ { - 1 } p , q \right)\).

9

1. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), prove that $$\sinh 2 x = 2 \sinh x \cosh x$$
2. Show that the curve with equation $$y = \cosh 2 x - 6 \sinh x$$ has just one stationary point, and find its \(x\)-coordinate in logarithmic form. Determine the nature of the stationary point.

The curve \(C\) has equation $$y = 9 \cosh x + 3 \sinh x + 7x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm. [6]

1. Show that, for \(x = \ln k\), where \(k\) is a positive constant, $$\cosh 2x = \frac{k^4 + 1}{2k^2}.$$ [3]

Given that \(f(x) = px - \tanh 2x\), where \(p\) is a constant,

1. find the value of \(p\) for which \(f(x)\) has a stationary value at \(x = \ln 2\), giving your answer as an exact fraction. [4]

(Total 7 marks)

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

1. Show that the \(x\)-coordinates of any stationary points of \(C\) satisfy the equation \(\tanh x = \frac{1}{x}\) [3 marks]
2. 1. Sketch the graphs of \(y = \tanh x\) and \(y = \frac{1}{x}\) on the axes below. [2 marks]
   2. Hence determine the number of stationary points of the curve \(C\). [1 mark]
3. Show that \(\frac{d^2y}{dx^2} + y = 0\) at each of the stationary points of the curve \(C\). [4 marks]

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. [8]

1. A curve is in the first quadrant. It has parametric equations \(x = \cosh t + \sinh t\), \(y = \cosh t - \sinh t\) where \(t \in \mathbb{R}\). Show that the cartesian equation of the curve is \(xy = 1\). [2]

Fig. 6 shows the curve from part (i). P is a point on the curve. O is the origin. Point A lies on the \(x\)-axis, point B lies on the \(y\)-axis and OAPB is a rectangle. \includegraphics{figure_6}

1. Find the smallest possible value of the perimeter of rectangle OAPB. Justify your answer. [4]

A function \(f\) has domain \((-\infty,\infty)\) and is defined by \(f(x) = \cosh^3 x - 3\cosh x\).

1. Show that the graph of \(y = f(x)\) has only one stationary point. [5]
2. Find the nature of this stationary point. [2]
3. State the largest possible range of \(f(x)\). [1]