4.07b Hyperbolic graphs: sketch and properties

8

1. Sketch the graph of \(\mathrm { y } = \operatorname { coth } \mathrm { x }\) for \(x > 0\) and state the equations of the asymptotes.
2. Starting from the definitions of coth and cosech in terms of exponentials, prove that $$\operatorname { coth } ^ { 2 } x - \operatorname { cosech } ^ { 2 } x = 1$$ The curve \(C\) has equation \(\mathrm { y } = \ln \operatorname { coth } \left( \frac { 1 } { 2 } \mathrm { x } \right)\) for \(x > 0\).
3. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \operatorname { cosechx }\).
4. It is given that the arc length of \(C\) from \(\mathrm { x } = \mathrm { a }\) to \(\mathrm { x } = 2 \mathrm { a }\) is \(\ln 4\), where \(a\) is a positive constant. Show that \(\cosh a = 2\) and find, in logarithmic form, the exact value of \(a\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1. Solve the equation

$$18 \cosh x + 14 \sinh x = 11 + \mathrm { e } ^ { x }$$ Give your answers in the form \(\ln a\), where \(a\) is rational.

7. The curve \(C\) has parametric equations $$x = \cosh t + t , \quad y = \cosh t - t \quad 0 \leqslant t \leqslant \ln 3$$

1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = 2 \cosh ^ { 2 } t$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).
2. Show that $$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { \ln 3 } \left( \cosh ^ { 2 } t - t \cosh t \right) d t$$
3. Hence find the value of \(S\), giving your answer in the form $$\frac { \pi \sqrt { 2 } } { 9 } ( a + b \ln 3 )$$ where \(a\) and \(b\) are constants to be determined.

7. \begin{figure}[h] \end{figure} The curves shown in Figure 1 have equations $$y = 6 \cosh x \text { and } y = 9 - 2 \sinh x$$

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\), find exact values for the \(x\)-coordinates of the two points where the curves intersect. The finite region between the two curves is shown shaded in Figure 1.
2. Using calculus, find the area of the shaded region, giving your answer in the form \(a \ln b + c\), where \(a , b\) and \(c\) are integers.

4

1. On separate diagrams, sketch the graphs of \(y = \sinh x\) and \(y = \operatorname { cosech } x\).
2. Show that \(\operatorname { cosech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }\), and hence, using the substitution \(u = \mathrm { e } ^ { x }\), find \(\int \operatorname { cosech } x \mathrm {~d} x\).

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.

7

1. Sketch the graph of \(y = \tanh x\) and state the value of the gradient when \(x = 0\). On the same axes, sketch the graph of \(y = \tanh ^ { - 1 } x\). Label each curve and give the equations of the asymptotes.
2. Find \(\int _ { 0 } ^ { k } \tanh x \mathrm {~d} x\), where \(k > 0\).
3. Deduce, or show otherwise, that \(\int _ { 0 } ^ { \tanh k } \tanh ^ { - 1 } x \mathrm {~d} x = k \tanh k - \ln ( \cosh k )\).

7

1. Sketch the graph of \(y = \operatorname { coth } x\), and give the equations of any asymptotes.
2. It is given that \(\mathrm { f } ( x ) = x \tanh x - 2\). Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2\), to find the next three approximations \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) to a root of \(\mathrm { f } ( x ) = 0\). Give the answers correct to 4 decimal places.
3. If \(\mathrm { f } ( x ) = 0\), show that \(\operatorname { coth } x = \frac { 1 } { 2 } x\). Hence write down the roots of \(\mathrm { f } ( x ) = 0\), correct to 4 decimal places.

9 Two thin poles, \(O A\) and \(B C\), are fixed vertically on horizontal ground. A chain is fixed at \(A\) and \(C\) such that it touches the ground at point \(D\) as shown in the diagram. On a coordinate system the coordinates of \(A\), \(B\) and \(D\) are \(( 0,3 ) , ( 5,0 )\) and \(( 2,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{c07ba83a-75fa-42dc-9bfd-6fc2f9226a23-5_805_1554_452_258} It is required to find the height of pole \(B C\) by modelling the shape of the curve that the chain forms.
Jofra models the curve using the equation \(\mathrm { y } = \mathrm { k } \cosh ( \mathrm { ax } - \mathrm { b } ) - 1\) where \(k , a\) and \(b\) are positive constants.

1. Determine the value of \(k\).
2. Find the exact value of \(a\) and the exact value of \(b\), giving your answers in logarithmic form. Holly models the curve using the equation \(y = \frac { 3 } { 4 } x ^ { 2 } - 3 x + 3\).
3. Write down the coordinates of the point, \(( u , v )\) where \(u\) and \(v\) are both non-zero, at which the two models will agree.
4. Show that Jofra's model and Holly's model disagree in their predictions of the height of pole \(B C\) by 3.32 m to 3 significant figures.

1

1. Use the definitions \(\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) and \(\sinh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right)\) to show that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
2. 1. Express $$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x$$ in terms of \(\cosh x\).
   2. Sketch the curve \(y = \cosh x\).
   3. Hence solve the equation $$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x = 9.5$$ giving your answers in logarithmic form.

1

1. Show, by means of a sketch, that the curves with equations $$y = \sinh x$$ and $$y = \operatorname { sech } x$$ have exactly one point of intersection.
2. Find the \(x\)-coordinate of this point of intersection, giving your answer in the form \(a \ln b\).

1

1. Sketch the curve \(y = \cosh x\).
2. Solve the equation $$6 \cosh ^ { 2 } x - 7 \cosh x - 5 = 0$$ giving your answers in logarithmic form.

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}

4

1. Sketch the graph of \(y = \tanh x\).
2. Given that \(u = \tanh x\), use the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that $$x = \frac { 1 } { 2 } \ln \left( \frac { 1 + u } { 1 - u } \right)$$
3. 1. Show that the equation $$3 \operatorname { sech } ^ { 2 } x + 7 \tanh x = 5$$ can be written as $$3 \tanh ^ { 2 } x - 7 \tanh x + 2 = 0$$
   2. Show that the equation $$3 \tanh ^ { 2 } x - 7 \tanh x + 2 = 0$$ has only one solution for \(x\).
      Find this solution in the form \(\frac { 1 } { 2 } \ln a\), where \(a\) is an integer.

2

1. Sketch the graph of \(y = \tanh x\) and state the equations of its asymptotes.
2. Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that $$\operatorname { sech } ^ { 2 } x + \tanh ^ { 2 } x = 1$$
3. Solve the equation \(6 \operatorname { sech } ^ { 2 } x = 4 + \tanh x\), giving your answers in terms of natural logarithms.
   [0pt] [5 marks] \section*{Answer space for question 2}
   1. \includegraphics[max width=\textwidth, alt={}, center]{bc3aaed2-4aef-4aec-b657-098b1e581e55-04_855_1447_920_324}

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.

6

1. On the axes below, sketch the graph of $$y = \cosh x$$ Indicate the value of any intercept of the curve with the axes. \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-05_1114_1121_552_447} 6
2. Solve the equation $$\cosh x = 2$$ Give your answers to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-06_2491_1755_173_123}

4 Which of the following graphs intersects the graph of \(y = \sinh x\) at exactly one point? Circle your answer. \(y = \operatorname { cosech } x\) \(y = \cosh x\) \(y = \operatorname { coth } x\) \(y = \operatorname { sech } x\)

12

1. Let \(I _ { n } = \int \frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } } \mathrm {~d} x\), for integers \(n \geqslant 0\).
   By writing \(\frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } }\) as \(x ^ { n - 1 } \times \frac { x } { \sqrt { x ^ { 2 } + 1 } }\), or otherwise, show that, for \(n \geqslant 2\), $$n I _ { n } = x ^ { n - 1 } \sqrt { x ^ { 2 } + 1 } - ( n - 1 ) I _ { n - 2 } .$$
2. The diagram shows a sketch of the hyperbola \(H\) with equation \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 16 } = 1\). \includegraphics[max width=\textwidth, alt={}, center]{32ed7cc8-3456-4cf0-952a-ee04eada1298-6_593_666_776_776}
   1. Find the coordinates of the points where \(H\) crosses the \(x\)-axis.
   2. The curve \(J\) has parametric equations \(x = 2 \cosh \theta , y = 4 \sinh \theta\), for \(\theta \geqslant 0\). Show that these parametric equations satisfy the cartesian equation of \(H\), and indicate on a copy of the above diagram which part of \(H\) is \(J\).
   3. The arc of the curve \(J\) between the points where \(x = 2\) and \(x = 34\) is rotated once completely about the \(x\)-axis to form a surface of revolution with area \(S\). Show that $$S = 16 \pi \int _ { \alpha } ^ { \beta } \sinh \theta \sqrt { 5 \cosh ^ { 2 } \theta - 1 } \mathrm {~d} \theta$$ for suitable constants \(\alpha\) and \(\beta\).
   4. Use the substitution \(u ^ { 2 } = 5 \cosh ^ { 2 } \theta - 1\) to show that $$S = \frac { 8 \pi } { \sqrt { 5 } } ( 644 \sqrt { 5 } - \ln ( 9 + 4 \sqrt { 5 } ) )$$

3 Solve the equation $$5 \cosh x - \sinh x = 7$$ giving your answers in an exact logarithmic form.

The curve \(C_1\) has equation \(y = 3\sinh 2x\), and the curve \(C_2\) has equation \(y = 13 - 3e^{2x}\).

1. Sketch the graph of the curves \(C_1\) and \(C_2\) on one set of axes, giving the equation of any asymptote and the coordinates of points where the curves cross the axes. [4]
2. Solve the equation \(3\sinh 2x = 13 - 3e^{2x}\), giving your answer in the form \(\frac{1}{2}\ln k\), where \(k\) is an integer. [5]

Find the values of \(x\) for which $$9 \cosh x - 6 \sinh x = 7$$ giving your answers as natural logarithms. (Total 6 marks)

The function g is defined by $$g(x) = \text{sech } x \quad\quad (x \in \mathbb{R})$$ Which one of the following is the range of g? Tick (\(\checkmark\)) one box. [1 mark] \(-\infty < g(x) \leq -1\) \quad \(\square\) \(-1 \leq g(x) < 0\) \quad \(\square\) \(0 < g(x) \leq 1\) \quad \(\square\) \(1 \leq g(x) \leq \infty\) \quad \(\square\)