Sketch graphs of hyperbolic functions

4

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 \(\int \operatorname { cosech } x \mathrm {~d} x\).

7

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 \(\int _ { 0 } ^ { \tanh k } \tanh ^ { - 1 } x \mathrm {~d} x = k \tanh k - \ln ( \cosh k )\).

7

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.

6

1. On the axes provided, sketch the graph of $$x = \cosh ( y + b )$$ where \(b\) is a positive constant.
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-06_1148_1317_1347_358} 6
2. Determine the minimum distance between the graph of \(x = \cosh ( y + b )\) and the \(y\)-axis.

6

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}

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

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

3 The function \(g\) is defined by $$g ( x ) = \operatorname { sech } x \quad ( x \in \mathbb { R } )$$ Which one of the following is the range of \(g\) ?
Tick \(( \checkmark )\) one box.
\(- \infty < \mathrm { g } ( x ) \leq - 1\)
\includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-03_117_117_635_854}
\(- 1 \leq \mathrm { g } ( x ) < 0\)
\includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-03_113_113_785_854}
\(0 < \mathrm { g } ( x ) \leq 1\)
\includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-03_117_117_927_854}
\(1 \leq g ( x ) \leq \infty\) □