4.07d Differentiate/integrate: hyperbolic functions

103 questions

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CAIE Further Paper 2 2021 June Q8
13 marks Challenging +1.8
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 \(A\).
    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.
CAIE Further Paper 2 2023 November Q5
10 marks Hard +2.3
5
[diagram]
The diagram shows part of the curve \(\mathrm { y } = \mathrm { xsech } ^ { 2 } \mathrm { x }\) and its maximum point \(M\).
  1. Show that, at \(M\), $$2 x \tanh x - 1 = 0$$ and verify that this equation has a root between 0.7 and 0.8 .
  2. By considering a suitable set of rectangles, use the diagram to show that \(\sum _ { r = 2 } ^ { n } r \operatorname { sech } ^ { 2 } r < n \tanh n + \operatorname { lnsechn } - \tanh 1 - \operatorname { lnsech } 1\).
Edexcel F3 2021 January Q2
6 marks Standard +0.8
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 \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\)
Edexcel F3 2024 January Q7
9 marks Challenging +1.8
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.
Edexcel F3 2016 June Q1
6 marks Standard +0.3
  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.
Edexcel F3 2016 June Q8
10 marks Challenging +1.8
8. $$I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 1\)
  2. Hence show that $$\int _ { 0 } ^ { \ln 2 } \tanh ^ { 4 } x \mathrm {~d} x = p + \ln 2$$ where \(p\) is a rational number to be found.
    8. \(\quad I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0\)
    1. Show that, for \(n \geqslant 1\) $$I _ { n } = I _ { n - 1 } - \frac { 1 } { 2 n - 1 } \left( \frac { 3 } { 5 } \right) ^ { 2 n - 1 }$$
Edexcel F3 2018 June Q6
7 marks Challenging +1.8
6. The curve \(C\) has parametric equations $$x = \theta - \tanh \theta , \quad y = \operatorname { sech } \theta , \quad 0 \leqslant \theta \leqslant \ln 3$$
  1. Find
    1. \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\)
    2. \(\frac { \mathrm { d } y } { \mathrm {~d} \theta }\) The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Find the exact area of the curved surface formed, giving your answer as a multiple of \(\pi\).
Edexcel F3 2020 June Q7
12 marks Challenging +1.3
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.
Edexcel F3 2023 June Q7
9 marks Challenging +1.8
7. $$I _ { n } = \int \cosh ^ { n } 2 x \mathrm {~d} x \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 2\) $$I _ { n } = \frac { \cosh ^ { n - 1 } 2 x \sinh 2 x } { 2 n } + \frac { n - 1 } { n } I _ { n - 2 }$$
  2. Hence determine $$\int ( 1 + \cosh 2 x ) ^ { 3 } d x$$ collecting any like terms in your answer.
Edexcel FP3 2013 June Q8
11 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{094b3c91-1460-44a2-b9d6-4de90d3adfa0-15_590_855_210_548} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The curve \(C\), shown in Figure 2, has equation $$y = 2 x ^ { \frac { 1 } { 2 } } , \quad 1 \leqslant x \leqslant 8$$
  1. Show that the length \(s\) of curve \(C\) is given by the equation $$s = \int _ { 1 } ^ { 8 } \sqrt { } \left( 1 + \frac { 1 } { x } \right) \mathrm { d } x$$
  2. Using the substitution \(x = \sinh ^ { 2 } u\), or otherwise, find an exact value for \(s\). Give your answer in the form \(a \sqrt { } 2 + \ln ( b + c \sqrt { } 2 )\) where \(a , b\) and \(c\) are integers.
Edexcel FP3 2014 June Q3
9 marks Challenging +1.2
  1. The curve \(C\) has equation
$$y = \frac { 1 } { 2 } \ln ( \operatorname { coth } x ) , \quad x > 0$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosech } 2 x$$ The points \(A\) and \(B\) lie on \(C\). The \(x\) coordinates of \(A\) and \(B\) are \(\ln 2\) and \(\ln 3\) respectively.
  2. Find the length of the arc \(A B\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.
    (6)
Edexcel FP3 2014 June Q7
12 marks Challenging +1.2
7. The curve \(C\) has equation $$y = \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$ The part of the curve \(C\) between \(x = 0\) and \(x = \ln 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  1. Show that the area \(S\) of the curved surface generated is given by $$S = 2 \pi \int _ { 0 } ^ { \ln 3 } \mathrm { e } ^ { - x } \sqrt { 1 + \mathrm { e } ^ { - 2 x } } \mathrm {~d} x$$
  2. Use the substitution \(\mathrm { e } ^ { - x } = \sinh u\) to show that $$S = 2 \pi \int _ { \operatorname { arsinh } \alpha } ^ { \operatorname { arsinh } \beta } \cosh ^ { 2 } u \mathrm {~d} u$$ where \(\alpha\) and \(\beta\) are constants to be determined.
  3. Show that $$2 \int \cosh ^ { 2 } u \mathrm {~d} u = \frac { 1 } { 2 } \sinh 2 u + u + k$$ where \(k\) is an arbitrary constant.
  4. Hence find the value of \(S\), giving your answer to 3 decimal places.
Edexcel FP3 2017 June Q7
10 marks Challenging +1.8
7. $$I _ { n } = \int _ { 0 } ^ { \ln 2 } \cosh ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 2\), $$I _ { n } = \frac { 3 a ^ { n - 1 } } { n b ^ { n } } + \frac { n - 1 } { n } I _ { n - 2 }$$ where \(a\) and \(b\) are integers to be found.
  2. Hence, or otherwise, find the exact value of $$\int _ { 0 } ^ { \ln 2 } \cosh ^ { 4 } x \mathrm {~d} x$$
OCR MEI FP2 2006 June Q4
18 marks Challenging +1.2
4
  1. Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$1 + 2 \sinh ^ { 2 } x = \cosh 2 x$$
  2. Solve the equation $$2 \cosh 2 x + \sinh x = 5 ,$$ giving the answers in an exact logarithmic form.
  3. Show that \(\int _ { 0 } ^ { \ln 3 } \sinh ^ { 2 } x \mathrm {~d} x = \frac { 10 } { 9 } - \frac { 1 } { 2 } \ln 3\).
  4. Find the exact value of \(\int _ { 3 } ^ { 5 } \sqrt { x ^ { 2 } - 9 } \mathrm {~d} x\).
OCR MEI FP2 2007 June Q4
18 marks Standard +0.8
4
  1. Find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } \mathrm {~d} x\), giving your answer in an exact logarithmic form.
    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 \(\int _ { - \ln 3 } ^ { \ln 3 } ( 20 \cosh x - 3 \cosh 2 x ) \mathrm { d } x = 40\).
OCR MEI FP2 2008 June Q4
18 marks Standard +0.3
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 \(\int _ { 0 } ^ { \ln 2 } \left( 4 \cosh ^ { 2 } x + 9 \sinh x \right) \mathrm { d } x = 2 \ln 2 + \frac { 33 } { 8 }\).
OCR MEI FP2 2010 June Q4
18 marks Challenging +1.2
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.
OCR FP2 2011 June Q7
10 marks Standard +0.8
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 \(\int _ { 0 } ^ { \tanh k } \tanh ^ { - 1 } x \mathrm {~d} x = k \tanh k - \ln ( \cosh k )\).
OCR FP2 Specimen Q7
13 marks Standard +0.8
7 The curve with equation $$y = \frac { x } { \cosh x }$$ has one stationary point for \(x > 0\).
  1. Show that the \(x\)-coordinate of this stationary point satisfies the equation \(x \tanh x - 1 = 0\). The positive root of the equation \(x \tanh x - 1 = 0\) is denoted by \(\alpha\).
  2. Draw a sketch showing (for positive values of \(x\) ) the graph of \(y = \tanh x\) and its asymptote, and the graph of \(y = \frac { 1 } { x }\). Explain how you can deduce from your sketch that \(\alpha > 1\).
  3. Use the Newton-Raphson method, taking first approximation \(x _ { 1 } = 1\), to find further approximations \(x _ { 2 }\) and \(x _ { 3 }\) for \(\alpha\).
  4. By considering the approximate errors in \(x _ { 1 }\) and \(x _ { 2 }\), estimate the error in \(x _ { 3 }\).
OCR MEI FP2 2006 January Q4
18 marks Standard +0.8
4
  1. Solve the equation $$\sinh x + 4 \cosh x = 8$$ giving the answers in an exact logarithmic form.
  2. Find the exact value of \(\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x } \sinh x \mathrm {~d} x\).
    1. Differentiate \(\operatorname { arsinh } \left( \frac { 2 } { 3 } x \right)\) with respect to \(x\).
    2. Use integration by parts to show that \(\int _ { 0 } ^ { 2 } \operatorname { arsinh } \left( \frac { 2 } { 3 } x \right) \mathrm { d } x = 2 \ln 3 - 1\).
OCR MEI FP2 2007 January Q4
18 marks Challenging +1.2
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.
OCR MEI FP2 2008 January Q4
18 marks Standard +0.8
4
  1. Given that \(k \geqslant 1\) and \(\cosh x = k\), show that \(x = \pm \ln \left( k + \sqrt { k ^ { 2 } - 1 } \right)\).
  2. Find \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 1 } } \mathrm {~d} x\), giving the answer in an exact logarithmic form.
  3. Solve the equation \(6 \sinh x - \sinh 2 x = 0\), giving the answers in an exact form, using logarithms where appropriate.
  4. Show that there is no point on the curve \(y = 6 \sinh x - \sinh 2 x\) at which the gradient is 5 .
OCR MEI FP2 2009 January Q4
18 marks Standard +0.8
4
    1. Prove, from definitions involving exponentials, that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
    2. Given that \(\sinh x = \tan y\), where \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\), show that
      (A) \(\tanh x = \sin y\),
      (B) \(x = \ln ( \tan y + \sec y )\).
    1. Given that \(y = \operatorname { artanh } x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). Hence show that \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x = 2 \operatorname { artanh } \frac { 1 } { 2 }\).
    2. Express \(\frac { 1 } { 1 - x ^ { 2 } }\) in partial fractions and hence find an expression for \(\int \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x\) in terms of logarithms.
    3. Use the results in parts (i) and (ii) to show that \(\operatorname { artanh } \frac { 1 } { 2 } = \frac { 1 } { 2 } \ln 3\).
OCR MEI FP2 2010 January Q4
18 marks Standard +0.8
4
  1. Prove, using exponential functions, that $$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$ Differentiate this result to obtain a formula for \(\sinh 2 x\).
  2. Solve the equation $$2 \cosh 2 x + 3 \sinh x = 3$$ expressing your answers in exact logarithmic form.
  3. Given that \(\cosh t = \frac { 5 } { 4 }\), show by using exponential functions that \(t = \pm \ln 2\). Find the exact value of the integral $$\int _ { 4 } ^ { 5 } \frac { 1 } { \sqrt { x ^ { 2 } - 16 } } \mathrm {~d} x$$
OCR MEI FP2 2013 January Q4
18 marks Challenging +1.8
4
  1. Show that the curve with equation $$y = 3 \sinh x - 2 \cosh x$$ has no turning points.
    Show that the curve crosses the \(x\)-axis at \(x = \frac { 1 } { 2 } \ln 5\). Show that this is also the point at which the gradient of the curve has a stationary value.
  2. Sketch the curve.
  3. Express \(( 3 \sinh x - 2 \cosh x ) ^ { 2 }\) in terms of \(\sinh 2 x\) and \(\cosh 2 x\). Hence or otherwise, show that the volume of the solid of revolution formed by rotating the region bounded by the curve and the axes through \(360 ^ { \circ }\) about the \(x\)-axis is $$\pi \left( 3 - \frac { 5 } { 4 } \ln 5 \right) .$$ Option 2: Investigation of curves \section*{This question requires the use of a graphical calculator.}