Sketch graphs of hyperbolic functions

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1. On separate diagrams, sketch the graphs of \(y = \sinh x\) and \(y = \operatorname { cosech } x\).
2. Show that \(\operatorname { cosech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }\), and hence, using the substitution \(u = \mathrm { e } ^ { x }\), find \(\int \operatorname { cosech } x \mathrm {~d} x\).

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1. Sketch the graph of \(y = \tanh x\) and state the value of the gradient when \(x = 0\). On the same axes, sketch the graph of \(y = \tanh ^ { - 1 } x\). Label each curve and give the equations of the asymptotes.
2. Find \(\int _ { 0 } ^ { k } \tanh x \mathrm {~d} x\), where \(k > 0\).
3. Deduce, or show otherwise, that \(\int _ { 0 } ^ { \tanh k } \tanh ^ { - 1 } x \mathrm {~d} x = k \tanh k - \ln ( \cosh k )\).

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1. Sketch the graph of \(y = \operatorname { coth } x\), and give the equations of any asymptotes.
2. It is given that \(\mathrm { f } ( x ) = x \tanh x - 2\). Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2\), to find the next three approximations \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) to a root of \(\mathrm { f } ( x ) = 0\). Give the answers correct to 4 decimal places.
3. If \(\mathrm { f } ( x ) = 0\), show that \(\operatorname { coth } x = \frac { 1 } { 2 } x\). Hence write down the roots of \(\mathrm { f } ( x ) = 0\), correct to 4 decimal places.

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1. On the axes below, sketch the graph of $$y = \cosh x$$ Indicate the value of any intercept of the curve with the axes. \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-05_1114_1121_552_447} 6
2. Solve the equation $$\cosh x = 2$$ Give your answers to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-06_2491_1755_173_123}

It is given that \(f(x) = \cosh^{-1}(x - 3)\) Which of the sets listed below is the greatest possible domain of the function \(f\)? Circle your answer. [1 mark] \(\{x : x \geq 4\}\) \quad \(\{x : x \geq 3\}\) \quad \(\{x : x \geq 1\}\) \quad \(\{x : x \geq 0\}\)

The function g is defined by $$g(x) = \text{sech } x \quad\quad (x \in \mathbb{R})$$ Which one of the following is the range of g? Tick (\(\checkmark\)) one box. [1 mark] \(-\infty < g(x) \leq -1\) \quad \(\square\) \(-1 \leq g(x) < 0\) \quad \(\square\) \(0 < g(x) \leq 1\) \quad \(\square\) \(1 \leq g(x) \leq \infty\) \quad \(\square\)