Maths with David

    9231. FP2. Hyperbolic Functions. Past Exam Questions

    9231/21/M/J/25q6 (includes differentiation)

    (a) Starting from the definitions of tanh and sech in terms of exponentials, prove that 1 – tanh2u = sech2u. [3 marks]

    (b) Show that \frac{d}{dt} (sech^{-1}t) = - \frac{1}{t \sqrt{1-t^2} } [4 marks]

    It is given that x = tanh-1t and y = tsech-1t, for 0 < t < 1.

    (c) Show that \frac{dy}{dx} = - \sqrt{1-t^2} + (1-t^2)sech^{-1}t . [4 marks]

    (d) Find \frac{d^2y}{dx^2} in terms of t. [4 marks]