11. Fig. 15 shows the graph of \(\mathrm { f } ( x ) = 2 x + \frac { 1 } { x } + \ln x - 4\).
\begin{figure}[h]
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\caption{Fig. 15}
\end{figure}
- Show that the equation
$$2 x + \frac { 1 } { x } + \ln x - 4 = 0$$
has a root, \(\alpha\), such that \(0.1 < \alpha < 0.9\).
- Obtain the following Newton-Raphson iteration for the equation in part (i).
$$x _ { r + 1 } = x _ { r } - \frac { 2 x _ { r } ^ { 3 } + x _ { r } + x _ { r } ^ { 2 } \left( \ln x _ { r } - 4 \right) } { 2 x _ { r } ^ { 2 } - 1 + x _ { r } }$$
- Explain why this iteration fails to find \(\alpha\) using each of the following starting values.
(A) \(x _ { 0 } = 0.4\)
(B) \(x _ { 0 } = 0.5\)
(C) \(x _ { 0 } = 0.6\)
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\section*{12. In this question you must show detailed reasoning.}
\includegraphics[max width=\textwidth, alt={}]{14f14bf3-88ee-413c-a62d-0914f41a485d-26_819_589_173_826}
The curve \(C\) has parametric equations
$$x = \frac { 1 } { \sqrt { 2 + t } } , \quad y = \ln ( 1 + t ) , \quad 2 \leq t < \infty$$
The point \(P\) on curve \(C\) has \(x\)-coordinate \(\frac { 1 } { 2 }\).
(a) Find the exact \(y\)-coordinate of \(P\).
The tangent to \(C\) at \(P\) meets the \(y\)-axis at point \(Y\).
(b) Determine the exact coordinates of \(Y\).
The curve \(C\) and the line segment \(P Y\) are rotated \(2 \pi\) radians about the \(y\)-axis.
(c) Determine the exact volume of the solid generated.
Give your answer in the form \(\pi ( \ln p + q )\), where \(p\) and \(q\) are rational numbers.
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[You are given that the volume of a cone with radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) ]
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