Intersection points of hyperbolic curves

5 The curves \(C _ { 1 } : y = \cosh x\) and \(C _ { 2 } : y = \sinh 2 x\) intersect at the point where \(x = a\).

1. Find the exact value of \(a\), giving your answer in logarithmic form.
2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
3. Find the exact value of the length of the arc of \(C _ { 1 }\) from \(x = 0\) to \(\mathrm { x } = \mathrm { a }\).

1. (a) Use the definition of \(\sinh x\) in terms of exponentials to show that

$$\sinh 3 x \equiv 4 \sinh ^ { 3 } x + 3 \sinh x$$ (b) Hence determine the exact coordinates of the points of intersection of the curve with equation \(y = \sinh 3 x\) and the curve with equation \(y = 19 \sinh x\), giving your answers as simplified logarithms where necessary.

7. \begin{figure}[h] \end{figure} The curves shown in Figure 1 have equations $$y = 6 \cosh x \text { and } y = 9 - 2 \sinh x$$

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\), find exact values for the \(x\)-coordinates of the two points where the curves intersect. The finite region between the two curves is shown shaded in Figure 1.
2. Using calculus, find the area of the shaded region, giving your answer in the form \(a \ln b + c\), where \(a , b\) and \(c\) are integers.

1

1. Show, by means of a sketch, that the curves with equations $$y = \sinh x$$ and $$y = \operatorname { sech } x$$ have exactly one point of intersection.
2. Find the \(x\)-coordinate of this point of intersection, giving your answer in the form \(a \ln b\).

1 Find the number of solutions of the equation \(\tanh x = \cosh x\)
Circle your answer.
0
1

4 Which of the following graphs intersects the graph of \(y = \sinh x\) at exactly one point? Circle your answer.
\(y = \operatorname { cosech } x\)
\(y = \cosh x\)
\(y = \operatorname { coth } x\)
\(y = \operatorname { sech } x\)