4.07d Differentiate/integrate: hyperbolic functions

The curve \(C\) has equation $$y = 9 \cosh x + 3 \sinh x + 7x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm. [6]

Using calculus, find the exact value of

1. \(\int_1^2 \frac{1}{\sqrt{x^2 - 2x + 3}} \, dx\) [4]
2. \(\int_0^1 e^{-x} \sinh x \, dx\) [4]

\includegraphics{figure_13} A rope is hung from points \(P\) and \(Q\) on the same horizontal level, as shown in Fig. 2. The curve formed by the rope is modelled by the equation $$y = a \cosh\left(\frac{x}{a}\right), \quad -ka \leq x \leq ka,$$ where \(a\) and \(k\) are positive constants.

1. Prove that the length of the rope is \(2a \sinh k\). [5]

Given that the length of the rope is \(8a\),

1. find the coordinates of \(Q\), leaving your answer in terms of natural logarithms and surds, where appropriate. [4]

Given that \(y = \sinh^{n-1} x \cosh x\),

1. show that \(\frac{dy}{dx} = (n-1) \sinh^{n-2} x + n \sinh^n x\). [3]

The integral \(I_n\) is defined by \(I_n = \int_0^{\operatorname{arsinh} 1} \sinh^n x \, dx\), \(n \geq 0\).

1. Using the result in part (a), or otherwise, show that $$nI_n = \sqrt{2} - (n-1)I_{n-2}, \quad n \geq 2$$ [2]
2. Hence find the value of \(I_4\). [4]

1. Show that, for \(x = \ln k\), where \(k\) is a positive constant, $$\cosh 2x = \frac{k^4 + 1}{2k^2}.$$ [3]

Given that \(f(x) = px - \tanh 2x\), where \(p\) is a constant,

1. find the value of \(p\) for which \(f(x)\) has a stationary value at \(x = \ln 2\), giving your answer as an exact fraction. [4]

(Total 7 marks)

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

The curve, \(C\), has equation \(y = \frac{x}{\cosh x}\)

1. Show that the \(x\)-coordinates of any stationary points of \(C\) satisfy the equation \(\tanh x = \frac{1}{x}\) [3 marks]
2. 1. Sketch the graphs of \(y = \tanh x\) and \(y = \frac{1}{x}\) on the axes below. [2 marks]
   2. Hence determine the number of stationary points of the curve \(C\). [1 mark]
3. Show that \(\frac{d^2y}{dx^2} + y = 0\) at each of the stationary points of the curve \(C\). [4 marks]

A curve has equation \(y = \cosh x\) Show that the arc length of the curve from \(x = a\) to \(x = b\), where \(0 < a < b\), is equal to $$\sinh b - \sinh a$$ [4 marks]

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 \(-2\sinh x\)

You are given that \(\mathrm{f}(x) = 4 \sinh x + 3 \cosh x\).

1. Show that the curve \(y = \mathrm{f}(x)\) has no turning points. [3]
2. Determine the exact solution of the equation \(\mathrm{f}(x) = 5\). [5]

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]

A particle \(P\) of mass \(3\) kg is moving on a smooth horizontal surface under the influence of a variable horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\), \(\mathbf{v}\) m s\(^{-1}\), is given by $$\mathbf{v} = (32\sinh(2t))\mathbf{i} + (32\cosh(2t) - 257)\mathbf{j}.$$

1. 1. By considering kinetic energy, determine the work done by \(\mathbf{F}\) over the interval \(0 \leqslant t \leqslant \ln 2\). [5]
   2. Explain the significance of the sign of the answer to part (a)(i). [1]
2. Determine the rate at which \(\mathbf{F}\) is working at the instant when \(P\) is moving parallel to the \(\mathbf{i}\)-direction. [6]

A function \(f\) has domain \((-\infty,\infty)\) and is defined by \(f(x) = \cosh^3 x - 3\cosh x\).

1. Show that the graph of \(y = f(x)\) has only one stationary point. [5]
2. Find the nature of this stationary point. [2]
3. State the largest possible range of \(f(x)\). [1]

Evaluate the integrals

1. \(\int_{-2}^{0} e^{2x} \sinh x \, \mathrm{d}x\), [5]
2. \(\int_{\frac{1}{2}}^{3} \frac{5}{(x-1)(x^2+9)} \, \mathrm{d}x\). [9]

Find each of the following integrals.

1. \(\int \frac{3-x}{x(x^2+1)} \mathrm{d}x\) [8]
2. \(\int \frac{\sinh 2x}{\sqrt{\cosh^4 x - 9\cosh^2 x}} \mathrm{d}x\) [6]

You are given that \(f(x) = 4\sinh x + 3\cosh x\).

1. Show that the curve \(y = f(x)\) has no turning points. [3]
2. Determine the exact solution of the equation \(f(x) = 5\). [5]

The curve with equation $$y = -x + \tanh(36x), \quad x \geq 0,$$ has a maximum turning point \(A\).

1. Find, in exact logarithmic form, the \(x\)-coordinate of \(A\). [4 marks]
2. Show that the \(y\)-coordinate of \(A\) is $$\frac{\sqrt{35}}{6} - \frac{1}{36}\ln(6 + \sqrt{35}).$$ [3 marks]

\(y = \cosh^n x\) \quad \(n \geq 5\)

1. 1. Show that $$\frac{d^2y}{dx^2} = n^2\cosh^n x - n(n-1)\cosh^{n-2}x$$ [4]
   2. Determine an expression for \(\frac{d^4y}{dx^4}\) [2]
2. Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for \(y\), simplifying each coefficient and justifying your answer. [2]