Newton-Raphson method

3. $$f ( x ) = x ^ { 2 } + \frac { 3 } { 4 \sqrt { } x } - 3 x - 7 , \quad x > 0$$ A root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 3,5 ]\).
Taking 4 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 2 decimal places.

5
\includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-2_556_844_1731_648} The diagram shows the curve \(y = \frac { 4 \ln x } { x ^ { 2 } + 1 }\) and its stationary point \(M\). The \(x\)-coordinate of \(M\) is \(m\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m = \mathrm { e } ^ { 0.5 \left( 1 + m ^ { - 2 } \right) }\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(m\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

3. $$\mathrm { f } ( x ) = x + \tan \left( \frac { 1 } { 2 } x \right) \quad \pi < x < \frac { 3 \pi } { 2 }$$ Given that the equation \(\mathrm { f } ( x ) = 0\) has a single root \(\alpha\)

1. show that \(\alpha\) lies in the interval [3.6, 3.7]
2. Find \(\mathrm { f } ^ { \prime } ( x )\)
3. Using 3.7 as a first approximation for \(\alpha\), apply the Newton-Raphson method once to obtain a second approximation for \(\alpha\). Give your answer to 3 decimal places.

1. The equation \(2 x ^ { 3 } + x ^ { 2 } - 1 = 0\) has exactly one real root.
   1. Show that, for this equation, the Newton-Raphson formula can be written
   $$x _ { n + 1 } = \frac { 4 x _ { n } ^ { 3 } + x _ { n } ^ { 2 } + 1 } { 6 x _ { n } ^ { 2 } + 2 x _ { n } }$$ Using the formula given in part (a) with \(x _ { 1 } = 1\)
2. find the values of \(x _ { 2 }\) and \(x _ { 3 }\)
3. Explain why, for this question, the Newton-Raphson method cannot be used with \(x _ { 1 } = 0\)

10. The function f is defined by \(\mathrm { f } ( x ) = \arccos x\) for \(0 \leq x \leq a\)
The curve with equation \(y = \mathrm { f } ( x )\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-22_769_771_317_648}

1. State the value of \(a\)
2. 1. On the diagram above, sketch the curve with equation $$y = \cos x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ and
      sketch the line with equation $$y = x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$
   2. Explain why the solution to the equation $$x - \cos x = 0$$ must also be a solution to the equation $$\cos x = \arccos x$$
3. Use the Newton-Raphson method with \(x _ { 0 } = 0\) to find an approximate solution, \(x _ { 3 }\), to the equation $$x - \cos x = 0$$ Give your answer to four decimal places. CONTINUE YOUR ANSWER HERE CONTINUE YOUR ANSWER HERE

4 The positive integers \(x , y\) and \(z\) are the first, second and third terms, respectively, of an arithmetic progression with common difference - 4 . Also, \(x , \frac { 15 } { y }\) and \(z\) are the first, second and third terms, respectively, of a geometric progression.

1. Show that \(y\) satisfies the equation \(y ^ { 4 } - 16 y ^ { 2 } - 225 = 0\).
2. Hence determine the sum to infinity of the geometric progression.

8. $$f ( x ) = 2 x + \sin x - 3 \cos x$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [0.7, 0.8].
2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where it crosses the \(y\)-axis.
3. Find the values of the constants \(a , b\) and \(c\), where \(b > 0\) and \(0 < c < \frac { \pi } { 2 }\), such that $$f ^ { \prime } ( x ) = a + b \cos ( x - c )$$
4. Hence find the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\), giving your answers to 2 decimal places.

1. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x - 4$$

1. Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \(1.4 < x < 1.5\)
2. Taking 1.4 as a first approximation to \(\alpha\), use the Newton-Raphson procedure once to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.

8. (i) Solve the equation $$\pi - 3 \cos ^ { - 1 } \theta = 0$$ (ii) Sketch on the same diagram the curves \(y = \cos ^ { - 1 } ( x - 1 ) , 0 \leq x \leq 2\) and \(y = \sqrt { x + 2 } , x \geq - 2\). Given that \(\alpha\) is the root of the equation $$\cos ^ { - 1 } ( x - 1 ) = \sqrt { x + 2 }$$ (iii) show that \(0 < \alpha < 1\),
(iv) use the iterative formula $$x _ { n + 1 } = 1 + \cos \sqrt { x _ { n } + 2 }$$ with \(x _ { 0 } = 1\) to find \(\alpha\) correct to 3 decimal places.
You should show the result of each iteration.

1. $$f ( x ) = 3 x ^ { 2 } - \frac { 5 } { 3 \sqrt { x } } - 6 , \quad x > 0$$ The single root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [1.5, 1.6].

1. Taking 1.5 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
   [0pt]
2. Use linear interpolation once on the interval [1.5, 1.6] to find another approximation to \(\alpha\). Give your answer to 3 decimal places.

5 The equation $$x ^ { 3 } - 5 x + 3 = 0$$ may be solved by the Newton-Raphson method. Successive approximations to a root are denoted by \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots\).

1. Show that the Newton-Raphson formula can be written in the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), where $$\mathrm { F } ( x ) = \frac { 2 x ^ { 3 } - 3 } { 3 x ^ { 2 } - 5 }$$
2. Find \(\mathrm { F } ^ { \prime } ( x )\) and hence verify that \(\mathrm { F } ^ { \prime } ( \alpha ) = 0\), where \(\alpha\) is any one of the roots of equation (A).
3. Use the Newton-Raphson method to find the root of equation (A) which is close to 2 . Write down sufficient approximations to find the root correct to 4 decimal places.

6. $$f ( x ) = x ^ { 3 } - \frac { 1 } { 2 x } + x ^ { \frac { 3 } { 2 } } , \quad x > 0$$ The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [0.6, 0.7].

1. Taking 0.6 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
2. Show that your answer to part (a) is correct to 3 decimal places.

4. $$f ( x ) = 1 - \frac { 1 } { 8 x ^ { 4 } } + \frac { 2 } { 7 \sqrt { x ^ { 7 } } } \quad x > 0$$ The equation \(\mathrm { f } ( x ) = 0\) has a single root, \(\alpha\), that lies in the interval \([ 0.15,0.25 ]\)

1. 1. Determine \(\mathrm { f } ^ { \prime } ( x )\)
   2. Explain why 0.25 cannot be used as an initial approximation for \(\alpha\) in the Newton-Raphson process.
   3. Taking 0.15 as a first approximation to \(\alpha\) apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\) Give your answer to 3 decimal places.
2. Use linear interpolation once on the interval \([ 0.15,0.25 ]\) to find another approximation to \(\alpha\) Give your answer to 3 decimal places.

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { x _ { n } \left( x _ { n } ^ { 3 } + 100 \right) } { 2 \left( x _ { n } ^ { 3 } + 25 \right) }$$ with initial value \(x _ { 1 } = 3.5\), converges to \(\alpha\).

1. Use this formula to calculate \(\alpha\) correct to 4 decimal places, showing the result of each iteration to 6 decimal places.
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

5
\includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-3_661_734_267_703} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 4.36$$ The curve has turning points at \(x = 1\) and \(x = 2\) and crosses the \(x\)-axis at \(x = \alpha , x = \beta\) and \(x = \gamma\), where \(0 < \alpha < \beta < \gamma\).

1. The Newton-Raphson method is to be used to find the roots of the equation \(\mathrm { f } ( x ) = 0\), with \(x _ { 1 } = k\).
   (a) To which root, if any, would successive approximations converge in each of the cases \(k < 0\) and \(k = 1\) ?
   (b) What happens if \(1 < k < 2\) ?
2. Sketch the curve with equation \(y ^ { 2 } = \mathrm { f } ( x )\). State the coordinates of the points where the curve crosses the \(x\)-axis and the coordinates of any turning points.
3. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$1 + 2 \sinh ^ { 2 } x \equiv \cosh 2 x .$$
4. Solve the equation $$\cosh 2 x - 5 \sinh x = 4$$ giving your answers in logarithmic form.

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.

14 The function f is defined by $$f ( x ) = 3 ^ { x } \sqrt { x } - 1 \quad \text { where } x \geq 0$$ 14

1. \(\quad \mathrm { f } ( x ) = 0\) has a single solution at the point \(x = \alpha\)
   By considering a suitable change of sign, show that \(\alpha\) lies between 0 and 1
   14
2. 1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 3 ^ { x } ( 1 + x \ln 9 ) } { 2 \sqrt { x } }$$

      14 (ii) Use the Newton-Raphson method with \(x _ { 1 } = 1\) to find \(x _ { 3 }\), an approximation for \(\alpha\). Give your answer to five decimal places. [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 14 (iii) Explain why the Newton-Raphson method fails to find \(\alpha\) with \(x _ { 1 } = 0\) [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

7
\includegraphics[max width=\textwidth, alt={}, center]{074597e7-5bb1-4249-9cfa-784974a6fd2b-3_531_1065_1208_539} The line \(y = x\) and the curve \(y = 2 \ln ( 3 x - 2 )\) meet where \(x = \alpha\) and \(x = \beta\), as shown in the diagram.

1. Use the iteration \(x _ { n + 1 } = 2 \ln \left( 3 x _ { n } - 2 \right)\), with initial value \(x _ { 1 } = 5.25\), to find the value of \(\beta\) correct to 2 decimal places. Show all your working.
2. With the help of a 'staircase' diagram, explain why this iteration will not converge to \(\alpha\), whatever value of \(x _ { 1 }\) (other than \(\alpha\) ) is used.
3. Show that the equation \(x = 2 \ln ( 3 x - 2 )\) can be rewritten as \(x = \frac { 1 } { 3 } \left( \mathrm { e } ^ { \frac { 1 } { 2 } x } + 2 \right)\). Use the NewtonRaphson method, with \(\mathrm { f } ( x ) = \frac { 1 } { 3 } \left( \mathrm { e } ^ { \frac { 1 } { 2 } x } + 2 \right) - x\) and \(x _ { 1 } = 1.2\), to find \(\alpha\) correct to 2 decimal places. Show all your working.
4. Given that \(x _ { 1 } = \ln 36\), explain why the Newton-Raphson method would not converge to a root of \(\mathrm { f } ( x ) = 0\).

6 The following flow chart has been written to find a root of the cubic equation \(x ^ { 3 } + A x ^ { 2 } + B x + C = 0\), given a starting value \(X\) that is thought to be near the root.
\includegraphics[max width=\textwidth, alt={}, center]{ccb12789-cd5f-40dc-9f10-f8bb45399580-8_1410_1648_324_212}

1. Work through the algorithm, recording the values of \(X , Y , Z\) and \(W\) each time they change, for the equation \(x ^ { 3 } - 4 x ^ { 2 } + 5 x + 1 = 0\), with a starting value of \(X = 0\).
2. Show what happens when the algorithm is used for the equation \(x ^ { 3 } - 4 x ^ { 2 } + 5 x + 1 = 0\), with a starting value of \(X = 1\).
3. Show what happens when the algorithm is used for the equation \(x ^ { 3 } - 4 x ^ { 2 } + 5 x + 1 = 0\), with a starting value of \(X = - 1\).
4. Identify a possible problem with using this algorithm.

3. $$f ( x ) = x ^ { 2 } + 5 x - 2 \sec x , \quad x \in \mathbb { R } , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } .$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [1,1.5]. A more accurate estimate of this root is to be found using iterations of the form $$x _ { n + 1 } = \arccos \mathrm { g } \left( x _ { n } \right) .$$
2. Find a suitable form for \(\mathrm { g } ( x )\) and use this formula with \(x _ { 0 } = 1.25\) to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 3 decimal places. The curve \(y = \mathrm { f } ( x )\) has a stationary point at \(P\).
3. Show that the \(x\)-coordinate of \(P\) is 1.0535 correct to 5 significant figures.