Hyperbolic functions

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. 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.

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.

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

1. Show that $$12 \cosh x - 4 \sinh x = 4\text{e}^x + 8\text{e}^{-x}$$ [2 marks]
2. Solve the equation $$12 \cosh x - 4 \sinh x = 33$$ giving your answers in the form \(k \ln 2\). [5 marks]

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.

By writing \(x = \sinh y\), show that \(\sinh^{-1} x = \ln\left(x + \sqrt{x^2 + 1}\right)\). [6]

1. (a) Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, show that, for \(x \in \mathbb { R }\)

$$\tanh x = \frac { \mathrm { e } ^ { 2 x } - 1 } { \mathrm { e } ^ { 2 x } + 1 }$$ (b) Hence, given that \(- 1 < \theta < 1\), prove that $$\operatorname { artanh } \theta = \frac { 1 } { 2 } \ln \left( \frac { 1 + \theta } { 1 - \theta } \right)$$ uestion 1 continued \(\_\_\_\_\) 7

1. Given that \(|x| < 1\), prove that $$\tanh^{-1}x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$$ [4 marks]
2. Solve the equation $$20\operatorname{sech}^2x - 11\tanh x = 16$$ Give your answer in logarithmic form. [4 marks]

1. Solve the equation

$$2 \cosh ^ { 2 } x - 3 \sinh x = 1$$ giving your answers in terms of natural logarithms.

1. Find the exact values of \(x\) for which

$$\cosh 2 x - 7 \sinh x = 5$$ giving your answers as natural logarithms.

1

1. Use the definitions of hyperbolic functions in terms of exponentials to prove that $$8 \cosh ^ { 4 } x = \cosh 4 x + p \cosh 2 x + q$$ where \(p\) and \(q\) are constants to be determined.
2. Hence, or otherwise, solve the equation $$\cosh 4 x - 17 \cosh 2 x + 9 = 0$$ giving your answers in exact simplified form in terms of natural logarithms.

2. A curve has equation $$y = \cosh x , \quad 1 \leqslant x \leqslant \ln 5$$ Find the exact length of this curve. Give your answer in terms of e .

2. A curve has equation $$y = \cosh x , \quad 1 \leqslant x \leqslant \ln 5$$ Find the exact length of this curve. Give your answer in terms of e .

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

\begin{figure}[h] \end{figure} Figure 1 shows the curve with equation $$y = \ln \left( \tanh \frac { x } { 2 } \right) \quad 1 \leqslant x \leqslant 2$$

1. Show that the length, \(s\), of the curve is given by $$s = \int _ { 1 } ^ { 2 } \operatorname { coth } x \mathrm {~d} x$$
2. Hence show that $$s = \ln \left( \mathrm { e } + \frac { 1 } { \mathrm { e } } \right)$$

16

1. Show using exponentials that \(\cosh 2 u = 1 + 2 \sinh ^ { 2 } u\).
2. Show that \(\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { \sqrt { 4 + x ^ { 2 } } } \mathrm {~d} x = 2 \sqrt { 2 } - 2 \ln ( 1 + \sqrt { 2 } )\).

5

1. Given that \(u = \cosh ^ { 2 } x\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \sinh 2 x\).
2. Hence show that $$\int _ { 0 } ^ { 1 } \frac { \sinh 2 x } { 1 + \cosh ^ { 4 } x } \mathrm {~d} x = \tan ^ { - 1 } \left( \cosh ^ { 2 } 1 \right) - \frac { \pi } { 4 }$$

7

1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$2 \sinh ^ { 2 } A = \cosh 2 A - 1$$ \includegraphics[max width=\textwidth, alt={}, center]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-12_79_1556_358_347} \includegraphics[max width=\textwidth, alt={}]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-12_69_1575_466_328} ....................................................................................................................................... ........................................................................................................................................
2. A curve has equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\), for \(0 \leqslant x \leqslant \frac { 2 } { 3 }\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\).
   Use the substitution \(\mathrm { X } = \frac { 1 } { 2 } \operatorname { sinhu }\) to show that \(S = \frac { 1 } { 32 } \pi \left( \frac { 820 } { 81 } - \ln 3 \right)\).

Find the exact solution to the equation $$\sinh \theta(\sinh \theta + \cosh \theta) = 1$$ [4 marks]

4

1. Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
2. Solve the equation \(4 \cosh ^ { 2 } x + 9 \sinh x = 13\), giving the answers in exact logarithmic form.
3. Show that there is only one stationary point on the curve $$y = 4 \cosh ^ { 2 } x + 9 \sinh x$$ and find the \(y\)-coordinate of the stationary point.
4. Show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 \cosh ^ { 2 } x + 9 \sinh x \right) \mathrm { d } x = 2 \ln 2 + \frac { 33 } { 8 }\).

1. Using the definition of \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$4\sinh^3 x = \sinh 3x - 3\sinh x.$$ [3]
2. In this question you must show detailed reasoning. By making a suitable substitution, find the real root of the equation $$16u^3 + 12u = 3.$$ Give your answer in the form \(\frac{(a^{\frac{1}{b}} - a^{-\frac{1}{b}})}{c}\) where \(a\), \(b\) and \(c\) are integers. [5]

Prove that \(\sinh(i\pi - \theta) = \sinh \theta\). [4]

1 Express \(\cosh ^ { 2 } x\) in terms of \(\sinh x\) Circle your answer. \(1 + \sinh ^ { 2 } x\) \(1 - \sinh ^ { 2 } x\) \(\sinh ^ { 2 } x - 1\) \(- 1 - \sinh ^ { 2 } x\)

14

1. Given that $$\sinh ( A + B ) = \sinh A \cosh B + \cosh A \sinh B$$ express \(\sinh ( m + 1 ) x\) and \(\sinh ( m - 1 ) x\) in terms of \(\sinh m x , \cosh m x , \sinh x\) and \(\cosh x\) 14
2. Hence find the sum of the series $$C _ { n } = \cosh x + \cosh 2 x + \cdots + \cosh n x$$ in terms of \(\sinh x , \sinh n x\) and \(\sinh ( n + 1 ) x\) Do not write \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-30_2491_1736_219_139}

2 A curve has equation \(\mathrm { y } = \cosh \mathrm { x }\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 }\).
Find, in terms of \(\pi\) and e, the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.

2 A curve has equation \(\mathrm { y } = \cosh \mathrm { x }\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 }\).
Find, in terms of \(\pi\) and e, the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.

8 The curve \(C\) has parametric equations $$\mathbf { x } = 2 \cosh t , \quad \mathbf { y } = \frac { 3 } { 2 } \mathbf { t } - \frac { 1 } { 4 } \sinh 2 \mathbf { t } , \text { for } 0 \leqslant t \leqslant 1$$

1. Find \(\frac { \mathrm { dx } } { \mathrm { dt } }\) and show that \(\frac { \mathrm { dy } } { \mathrm { dt } } = 1 - \sinh ^ { 2 } \mathrm { t }\).
   The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).
2. 1. Show that \(\mathrm { A } = \pi \int _ { 0 } ^ { 1 } \left( \frac { 3 } { 2 } \mathrm { t } - \frac { 1 } { 4 } \sinh 2 \mathrm { t } \right) ( 1 + \cosh 2 \mathrm { t } ) \mathrm { dt }\).
   2. Hence find \(A\) in terms of \(\pi , \sinh 2\) and \(\cosh 2\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4. $$y = \operatorname { artanh } \left( \frac { \cos x + a } { \cos x - a } \right)$$ where \(a\) is a non-zero constant.
Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \tan x$$ where \(k\) is a constant to be determined.

Given that \(y = \operatorname{sech}x\), find \(\frac{dy}{dx}\) Tick (\(\checkmark\)) one box. [1 mark] \(\operatorname{sech}x\tanh x\) \(\square\) \(-\operatorname{sech}x\tanh x\) \(\square\) \(\operatorname{cosech}x\coth x\) \(\square\) \(-\operatorname{cosech}x\coth x\) \(\square\)

Given that \(y = \text{artanh}(\cos x)\)

1. show that $$\frac{dy}{dx} = -\text{cosec } x$$ [2]
2. Hence find the exact value of $$\int_{0}^{\frac{\pi}{4}} \cos x \, \text{artanh}(\cos x) \, dx$$ giving your answer in the form \(a \ln\left(b + c\sqrt{3}\right) + d\pi\), where \(a\), \(b\), \(c\) and \(d\) are rational numbers to be found. [5]

Given that \(y = \sin x + \sinh x\), find \(\frac{d^2y}{dx^2} + y\) Circle your answer. [1 mark] \(2\sin x\) \quad \(-2\sin x\) \quad \(2\sinh x\) \quad \(-2\sinh x\)

Given that \(y = \sin x + \sinh x\), find \(\frac{d^2y}{dx^2} + y\) Circle your answer. [1 mark] \(2\sin x\) \quad \(-2\sin x\) \quad \(2\sinh x\) \quad \(-2\sinh x\)

1. Given that

$$y = \mathrm { e } ^ { 2 x } \sinh x$$ prove by induction that for \(n \in \mathbb { N }\) $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = \mathrm { e } ^ { 2 x } \left( \frac { 3 ^ { n } + 1 } { 2 } \sinh x + \frac { 3 ^ { n } - 1 } { 2 } \cosh x \right)$$

1. Solve the equation

$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form \(\ln a\) where \(a\) is a rational number.

Which one of these functions has the set \(\{x : |x| < 1\}\) as its greatest possible domain? Circle your answer. [1 mark] \(\cosh x\) \quad \(\cosh^{-1} x\) \quad \(\tanh x\) \quad \(\tanh^{-1} x\)

1. Show that, for \(0 < x \leq 1\), $$\ln \left(\frac{1 - \sqrt{1-x^2}}{x}\right) = -\ln \left(\frac{1 + \sqrt{1-x^2}}{x}\right).$$ [3]
2. Using the definition of \(\cosh x\) or \(\operatorname{sech} x\) in terms of exponentials, show that, for \(0 < x \leq 1\), $$\operatorname{arsech} x = \ln \left(\frac{1 + \sqrt{1-x^2}}{x}\right).$$ [5]
3. Solve the equation $$3 \tanh^2 x - 4 \operatorname{sech} x + 1 = 0,$$ giving exact answers in terms of natural logarithms. [5]

(Total 13 marks)

It is given that \(f(x) = \cosh^{-1}(x - 3)\) Which of the sets listed below is the greatest possible domain of the function \(f\)? Circle your answer. [1 mark] \(\{x : x \geq 4\}\) \quad \(\{x : x \geq 3\}\) \quad \(\{x : x \geq 1\}\) \quad \(\{x : x \geq 0\}\)

Find the number of solutions of the equation \(\tanh x = \cosh x\) Circle your answer. [1 mark] \(0 \quad 1 \quad 2 \quad 3\)

1 Express sech \(2 x\) in terms of exponentials and hence, by using the substitution \(u = e ^ { 2 x }\), find \(\int \operatorname { sech } 2 x \mathrm {~d} x\).

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find $$\int_1^{\infty} \frac{1}{\cosh u} du,$$ giving your answer in an exact form. [7 marks]

3 It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } \left( \frac { 1 - x } { 3 + x } \right)\) for \(x > - 1\).

1. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 1 } { 2 ( x + 1 ) ^ { 2 } }\).
2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\).

In this question you must show detailed reasoning. The region in the first quadrant bounded by curve \(y = \cosh\frac{1}{2}x^2\), the \(y\)-axis, and the line \(y = 2\) is rotated through \(360°\) about the \(y\)-axis. Find the exact volume of revolution generated, expressing your answer in a form involving a logarithm. [7]

1. (a) Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials to show that

$$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$ (b) Hence find the value of \(x\) for which $$\cosh ( x + \ln 2 ) = 5 \sinh x$$ giving your answer in the form \(\frac { 1 } { 2 } \ln k\), where \(k\) is a rational number to be determined.
(5)

4

1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1 .$$
2. Show that \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \tan ^ { - 1 } ( \sinh x ) \right) = \operatorname { sech } x\).
3. Sketch the graph of \(y = \operatorname { sechx }\), stating the equation of the asymptote.
4. By considering a suitable set of \(n\) rectangles of unit width, use your sketch to show that $$\sum _ { r = 1 } ^ { n } \operatorname { sechr } < \tan ^ { - 1 } ( \operatorname { sinhn } )$$
5. Hence state an upper bound, in terms of \(\pi\), for \(\sum _ { r = 1 } ^ { \infty }\) sech \(r\).

6 In this question you must show detailed reasoning. The power output, \(p\) watts, of a machine at time \(t\) hours after it is switched on can be modelled by the equation \(\mathrm { p } = 20 - 20 \tanh ( 1.44 \mathrm { t } )\) for \(t \geqslant 0\). Determine, according to the model, the mean power output of the machine over the first half hour after it is switched on. Give your answer correct to \(\mathbf { 2 }\) decimal places.

1. The hyperbola \(H\) is given by the equation \(x ^ { 2 } - y ^ { 2 } = 1\)
   1. Write down the equations of the two asymptotes of \(H\).
   2. Show that an equation of the tangent to \(H\) at the point \(P ( \cosh t , \sinh t )\) is
   $$y \sinh t = x \cosh t - 1$$ The tangent at \(P\) meets the asymptotes of \(H\) at the points \(Q\) and \(R\).
2. Show that \(P\) is the midpoint of \(Q R\).
3. Show that the area of the triangle \(O Q R\), where \(O\) is the origin, is independent of \(t\).

5

1. Using the identities $$\cosh ^ { 2 } t - \sinh ^ { 2 } t = 1 , \quad \tanh t = \frac { \sinh t } { \cosh t } \quad \text { and } \quad \operatorname { sech } t = \frac { 1 } { \cosh t }$$ show that:
   1. \(\tanh ^ { 2 } t + \operatorname { sech } ^ { 2 } t = 1\);
   2. \(\frac { \mathrm { d } } { \mathrm { d } t } ( \tanh t ) = \operatorname { sech } ^ { 2 } t\);
   3. \(\frac { \mathrm { d } } { \mathrm { d } t } ( \operatorname { sech } t ) = - \operatorname { sech } t \tanh t\).
2. A curve \(C\) is given parametrically by $$x = \operatorname { sech } t , y = 4 - \tanh t$$
   1. Show that the arc length, \(s\), of \(C\) between the points where \(t = 0\) and \(t = \frac { 1 } { 2 } \ln 3\) is given by $$s = \int _ { 0 } ^ { \frac { 1 } { 2 } \ln 3 } \operatorname { sech } t \mathrm {~d} t$$
   2. Using the substitution \(u = \mathrm { e } ^ { t }\), find the exact value of \(s\).

      REFERENCE
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