4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials

Sketch the polar graph of $$r = \sinh \theta + \cosh \theta$$ for \(0 \leq \theta \leq 2\pi\) \includegraphics{figure_11} [3 marks]

Which one of these functions has the set \(\{x : |x| < 1\}\) as its greatest possible domain? Circle your answer. [1 mark] \(\cosh x\) \quad \(\cosh^{-1} x\) \quad \(\tanh x\) \quad \(\tanh^{-1} x\)

Show that the solutions to the equation $$3\tanh^2 x - 2\operatorname{sech} x = 2$$ can be expressed in the form $$x = \pm \ln(a + \sqrt{b})$$ where \(a\) and \(b\) are integers to be found. You may use without proof the result \(\cosh^{-1} y = \ln(y + \sqrt{y^2 - 1})\) [5 marks]

Find the number of solutions of the equation \(\tanh x = \cosh x\) Circle your answer. [1 mark] \(0 \quad 1 \quad 2 \quad 3\)

Solve the differential equation $$\frac{dy}{dx} + y\tanh x = \sinh^3 x$$ given that \(y = 3\) when \(x = \ln 2\) Give your answer in an exact form. [7 marks]

Use the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\) to show that \(\cosh^2 x - \sinh^2 x = 1\) [2 marks]

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]

The function \(f(x) = \cosh(ix)\) is defined over the domain \(\{x \in \mathbb{R} : -a\pi \leq x \leq a\pi\}\), where \(a\) is a positive integer. By considering the graph of \(y = [f(x)]^n\), find the mean value of \([f(x)]^n\), when \(n\) is an odd positive integer. Fully justify your answer. [3 marks]

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]

In polar coordinates, the equation of a curve, \(C\), is \(r = 6\sin(2\theta)\sinh\left(\frac{1}{3}\theta\right)\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). The pole of the polar coordinate system corresponds to the origin of the cartesian system and the initial line corresponds to the positive \(x\)-axis.

1. Explain how you can tell that \(C\) comprises a single loop in the first quadrant, passing through the pole. [3]

The incomplete table below shows values of \(r\) for various values of \(\theta\).

\(\theta\)	0	\(\frac{1}{12}\pi\)	\(\frac{1}{6}\pi\)	\(\frac{1}{4}\pi\)	\(\frac{1}{3}\pi\)	\(\frac{5}{12}\pi\)	\(\frac{1}{2}\pi\)
\(r\)	0	0.262			1.851

1. Use the copy of the table and the polar coordinate system diagram given in the Printed Answer Booklet to complete the table and sketch \(C\). [3]

The point on \(C\) which is furthest away from the pole is denoted by \(A\) and the value of \(\theta\) at \(A\) is denoted by \(\phi\).

1. Show that \(\phi\) satisfies the equation \(\phi = \frac{3}{4}\ln\left(\frac{6-\tan 2\phi}{6+\tan 2\phi}\right)\) [4]
2. You are given that the relevant solution of the equation given in part (c) is \(\phi = 1.0207\) correct to 5 significant figures. Find the distance from \(A\) to the pole. Give your answer correct to 3 significant figures. [1]

1. Express \(17\cosh x - 15\sinh x\) in the form \(e^{-x}(ae^{bx} + c)\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]

A function is defined by \(f(x) = \frac{1}{\sqrt{17\cosh x - 15\sinh x}}\). The region bounded by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \ln 3\) is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).

1. In this question you must show detailed reasoning. Use a suitable substitution, together with known results from the formula book, to show that the volume of \(S\) is given by \(k\pi\tan^{-1} q\) where \(k\) and \(q\) are rational numbers to be determined. [7]

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]

1. A curve is in the first quadrant. It has parametric equations \(x = \cosh t + \sinh t\), \(y = \cosh t - \sinh t\) where \(t \in \mathbb{R}\). Show that the cartesian equation of the curve is \(xy = 1\). [2]

Fig. 6 shows the curve from part (i). P is a point on the curve. O is the origin. Point A lies on the \(x\)-axis, point B lies on the \(y\)-axis and OAPB is a rectangle. \includegraphics{figure_6}

1. Find the smallest possible value of the perimeter of rectangle OAPB. Justify your answer. [4]

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]

The functions \(f\) and \(g\) have domains \((-1, \infty)\) and \((0, \infty)\) respectively and are defined by $$f(x) = \cosh x, \qquad g(x) = x^2 - 1.$$

1. State the domain and range of \(fg\). [2]
2. Solve the equation \(fg(x) = 3\). Give your answer correct to three decimal places. [3]

The function \(f\) is defined by \(f(x) = \cosh\left(\frac{x}{2}\right)\).

1. State the Maclaurin series expansion for \(\cosh\left(\frac{x}{2}\right)\) up to and including the term in \(x^4\). [2]

Another function \(g\) is defined by \(g(x) = x^2 - 2\). The diagram below shows parts of the graphs of \(y = f(x)\) and \(y = g(x)\). \includegraphics{figure_2}

1. The two graphs intersect at the point A, as shown in the diagram. Use your answer from part (a) to find an approximation for the \(x\)-coordinate of A, giving your answer correct to two decimal places. [5]
2. Using your answer to part (b), find an approximation for the area of the shaded region enclosed by the two graphs, the \(x\)-axis and the \(y\)-axis. [6]