Newton-Raphson with trigonometric or exponential functions

2 It is given that \(\mathrm { f } ( x ) = x ^ { 2 } - \tan ^ { - 1 } x\).

1. Show by calculation that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval \(0.8 < x < 0.9\).
2. Use the Newton-Raphson method, with a first approximation 0.8, to find the next approximation to this root. Give your answer correct to 3 decimal places.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$f ( x ) = 4 \cos 2 x - 2 x + 1 \quad x > 0$$ and where \(x\) is measured in radians.
The curve crosses the \(x\)-axis at the point \(A\), as shown in figure 1 .
Given that \(x\)-coordinate of \(A\) is \(\alpha\) a. show that \(\alpha\) lies between 0.7 and 0.8 Given that \(x\)-coordinates of \(B\) and \(C\) are \(\beta\) and \(\gamma\) respectively and they are two smallest values of \(x\) at which local maxima occur
b. find, using calculus, the value of \(\beta\) and the value of \(\gamma\), giving your answers to 3 significant figures.
c. taking \(x _ { 0 } = 0.7\) or 0.8 as a first approximation to \(\alpha\), apply the Newton-Raphson method once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Show, your method and give your answer to 2 significant figures.

2 Use the Newton-Raphson method to find the root of the equation \(\mathrm { e } ^ { - x } = x\) which is close to \(x = 0.5\). Give the root correct to 3 decimal places.

7

1. Describe the transformation which maps the graph of \(y = \ln x\) onto the graph of \(y = \ln ( 1 + x )\).
2. By sketching the curves \(y = \ln ( 1 + x )\) and \(y = 4 - x\) on a single diagram, show that the equation $$\ln ( 1 + x ) = 4 - x$$ has exactly one root.
3. Use the Newton-Raphson method with \(x _ { 0 } = 2\) to find the root of the equation \(\ln ( 1 + x ) = 4 - x\) correct to 3 decimal places. Show the result of each iteration.

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1. Find the coordinates of the stationary point of the curve with equation $$y = \ln x - k x , \text { where } k > 0 \text { and } x > 0$$ and determine its nature.
2. Hence show that the equation \(\ln x - k x = 0\) has real roots if \(0 < k \leqslant \frac { 1 } { \mathrm { e } }\).
3. In the particular case that \(k = \frac { 1 } { 3 }\), the equation \(\ln x - k x = 0\) has two roots, one of which is near \(x = 5\). Use the Newton-Raphson process to find, correct to 3 significant figures, the root of the equation \(\ln x - \frac { 1 } { 3 } x = 0\) which is near \(x = 5\).
4. Show that the equation \(\ln x - k x = 0\) has one real root if \(k \leqslant 0\).
5. Explain why the equation \(\ln x - k x = 0\) has two distinct real roots if \(0 < k < \frac { 1 } { \mathrm { e } }\).

By drawing suitable graphs, show that \(x - 1 = \cos x\) has only one root. Starting with \(x_0 = 1\), use the Newton-Raphson method to find the value of this root correct to two decimal places. [6]