Parametric curves with hyperbolic functions

A question is this type if and only if it involves parametric equations where x and/or y are defined using hyperbolic functions, requiring differentiation or integration in parametric form.

1 questions · Standard +0.8

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AQA FP2 2010 June Q5
18 marks Standard +0.8
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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