Hyperbolic power functions

Questions involving hyperbolic functions raised to powers (e.g., cosh^n x, sinh x sin x) where the derivative relationship involves recurrence relations or products of hyperbolic and trigonometric functions.

4 questions

CAIE Further Paper 2 2020 June Q6
6
  1. Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh ^ { 2 } \theta = \operatorname { sech } ^ { 2 } \theta$$ \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1552_374_347}
    \includegraphics[max width=\textwidth, alt={}]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_67_1569_466_328} ......................................................................................................................................... ........................................................................................................................................
    \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1573_735_324}
    \includegraphics[max width=\textwidth, alt={}]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1573_826_324} .......................................................................................................................................... . ........................................................................................................................................ . The variables \(x\) and \(y\) are such that \(\tanh y = \cos \left( x + \frac { 1 } { 4 } \pi \right)\), for \(- \frac { 1 } { 4 } \pi < x < \frac { 3 } { 4 } \pi\).
  2. By differentiating the equation \(\tanh y = \cos \left( x + \frac { 1 } { 4 } \pi \right)\) with respect to \(x\), show that $$\frac { \mathrm { dy } } { \mathrm { dx } } = - \operatorname { cosec } \left( \mathrm { x } + \frac { 1 } { 4 } \pi \right)$$
  3. Hence find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \cos \left( x + \frac { 1 } { 4 } \pi \right) \right)\) in the form \(\frac { 1 } { 2 } \ln a + b x + c x ^ { 2 }\), giving the exact values of the constants \(a , b\) and \(c\).
Edexcel CP2 2022 June Q9
9. $$y = \cosh ^ { n } x \quad n \geqslant 5$$
    1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = n ^ { 2 } \cosh ^ { n } x - n ( n - 1 ) \cosh ^ { n - 2 } x$$
    2. Determine an expression for \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
  2. Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form.
Edexcel CP2 Specimen Q5
5. $$y = \sin x \sinh x$$
  1. Show that \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = - 4 y\)
  2. Hence find the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in its simplest form.
  3. Find an expression for the \(n\)th non-zero term of the Maclaurin series for \(y\).
SPS SPS FM Pure 2024 February Q15
15. $$y = \cosh ^ { n } x \quad n \geqslant 5$$
    1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = n ^ { 2 } \cosh ^ { n } x - n ( n - 1 ) \cosh ^ { n - 2 } x$$
    2. Determine an expression for \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
  2. Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for \(y\), simplifying each coefficient and justifying your answer.
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