Show convergence to specific root

6

1. By sketching a suitable pair of graphs on the same diagram, show that the equation $$\ln x = 2 \mathrm { e } ^ { - x }$$ has exactly one root.
2. Verify by calculation that the root lies between 1.5 and 1.6.
3. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { 2 \mathrm { e } ^ { - x _ { n } } }$$ converges, then it converges to the root of the equation in part (a).
4. Use the iterative formula in part (c) to determine the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

6

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) that is a root of the equation \(x = 9 \mathrm { e } ^ { - 2 x }\).
2. Verify, by calculation, that this root lies between 1 and 2 .
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( \ln 9 - \ln x _ { n } \right)$$ converges, then it converges to the root of the equation given in part (i).
4. Use the iterative formula, with \(x _ { 1 } = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7

1. By sketching a suitable pair of graphs, show that the equation $$\mathrm { e } ^ { 2 x } = 2 - x$$ has only one root.
2. Verify by calculation that this root lies between \(x = 0\) and \(x = 0.5\).
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 2 - x _ { n } \right)$$ converges, then it converges to the root of the equation in part (i).
4. Use this iterative formula, with initial value \(x _ { 1 } = 0.25\), to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
5. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
6. By expressing \(\cot ^ { 2 } x\) in terms of \(\operatorname { cosec } ^ { 2 } x\) and using the result of part (i), show that $$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { 2 } x \mathrm {~d} x = 1 - \frac { 1 } { 4 } \pi$$
7. Express \(\cos 2 x\) in terms of \(\sin ^ { 2 } x\) and hence show that \(\frac { 1 } { 1 - \cos 2 x }\) can be expressed as \(\frac { 1 } { 2 } \operatorname { cosec } ^ { 2 } x\). Hence, using the result of part (i), find $$\int \frac { 1 } { 1 - \cos 2 x } \mathrm {~d} x$$

7

1. The equation \(x ^ { 3 } + x + 1 = 0\) has one real root. Show by calculation that this root lies between - 1 and 0 .
2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } - 1 } { 3 x _ { n } ^ { 2 } + 1 }$$ converges, then it converges to the root of the equation given in part (i).
3. Use this iterative formula, with initial value \(x _ { 1 } = - 0.5\), to determine the root correct to 2 decimal places, showing the result of each iteration.

4 The equation \(x ^ { 3 } - 2 x - 2 = 0\) has one real root.

1. Show by calculation that this root lies between \(x = 1\) and \(x = 2\).
2. Prove that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 2 } { 3 x _ { n } ^ { 2 } - 2 }$$ converges, then it converges to this root.
3. Use this iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4 The equation \(x = \frac { 10 } { \mathrm { e } ^ { 2 x } - 1 }\) has one positive real root, denoted by \(\alpha\).

1. Show that \(\alpha\) lies between \(x = 1\) and \(x = 2\).
2. Show that if a sequence of positive values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 1 + \frac { 10 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
3. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. By sketching a suitable pair of graphs, show that the equation $$5 \mathrm { e } ^ { - x } = \sqrt { } x$$ has one root.
2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 25 } { x _ { n } } \right)$$ converges, then it converges to the root of the equation in part (i).
3. Use this iterative formula, with initial value \(x _ { 1 } = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 The equation \(\cot x = 1 - x\) has one root in the interval \(0 < x < \pi\), denoted by \(\alpha\).

1. Show by calculation that \(\alpha\) is greater than 2.5.
2. Show that, if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula \(x _ { n + 1 } = \pi + \tan ^ { - 1 } \left( \frac { 1 } { 1 - x _ { n } } \right)\) converges, then it converges to \(\alpha\).
3. Use this iterative formula to determine \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

5

1. By sketching suitable graphs, show that the equation $$\sec x = 3 - x ^ { 2 }$$ has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } ^ { 2 } } \right)$$ converges, then it converges to a root of the equation given in part (i).
3. Use this iterative formula, with initial value \(x _ { 1 } = 1\), to determine the root in the interval \(0 < x < \frac { 1 } { 2 } \pi\) correct to 2 decimal places, showing the result of each iteration.

4 The equation \(x ^ { 3 } - x ^ { 2 } - 6 = 0\) has one real root, denoted by \(\alpha\).

1. Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( x _ { n } + \frac { 6 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
3. Use this iterative formula to determine \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6

1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } \frac { 1 } { 2 } x = \frac { 1 } { 3 } x + 1$$ has one root in the interval \(0 < x \leqslant \pi\).
2. Show by calculation that this root lies between 1.4 and 1.6.
3. Show that, if a sequence of values in the interval \(0 < x \leqslant \pi\) given by the iterative formula $$x _ { n + 1 } = 2 \sin ^ { - 1 } \left( \frac { 3 } { x _ { n } + 3 } \right)$$ converges, then it converges to the root of the equation in part (i).
4. Use this iterative formula to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

3

1. By sketching a suitable pair of graphs, show that the equation \(x ^ { 3 } = 3 - x\) has exactly one real root.
2. Show that if a sequence of real values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 3 } { 3 x _ { n } ^ { 2 } + 1 }$$ converges, then it converges to the root of the equation in part (i).
3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

4 The equation \(x ^ { 3 } - x ^ { 2 } - 6 = 0\) has one real root, denoted by \(\alpha\).

1. Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( x _ { n } + \frac { 6 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
3. Use this iterative formula to determine \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

7

1. By sketching a suitable pair of graphs, show that the equation $$\cos x = 2 - 2 x$$ where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 0.5 and 1 .
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = 1 - \frac { 1 } { 2 } \cos x _ { n }$$ converges, then it converges to the root of the equation in part (i).
4. Use this iterative formula, with initial value \(x _ { 1 } = 0.6\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. By sketching a suitable pair of graphs, show that the equation \(x ^ { 5 } = 2 + x\) has exactly one real root.
2. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 4 x _ { n } ^ { 5 } + 2 } { 5 x _ { n } ^ { 4 } - 1 }$$ converges, then it converges to the root of the equation in part (a).
3. Use the iterative formula with initial value \(x _ { 1 } = 1.5\) to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. \(7 \quad\) Let \(\mathrm { f } ( x ) = \frac { 2 } { ( 2 x - 1 ) ( 2 x + 1 ) }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{c1fbc9ef-2dc6-43c3-bc58-179f683c9acf-18_471_686_276_717} The curve \(y = x \sqrt { \sin x }\) has one stationary point in the interval \(0 < x < \pi\), where \(x = a\) (see diagram).

1. Show that \(\tan a = - \frac { 1 } { 2 } a\).
2. Verify by calculation that \(a\) lies between 2 and 2.5.
3. Show that if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula \(x _ { n + 1 } = \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } \right)\) converges, then it converges to \(a\), the root of the equation in part (a). [2]
4. Use the iterative formula given in part (c) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 The equation \(\cot \frac { 1 } { 2 } x = 3 x\) has one root in the interval \(0 < x < \pi\), denoted by \(\alpha\).

1. Show by calculation that \(\alpha\) lies between 0.5 and 1 .
2. Show that, if a sequence of positive values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( x _ { n } + 4 \tan ^ { - 1 } \left( \frac { 1 } { 3 x _ { n } } \right) \right)$$ converges, then it converges to \(\alpha\).
3. Use this iterative formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. By sketching a suitable pair of graphs, show that the equation \(\operatorname { cosec } \frac { 1 } { 2 } x = \mathrm { e } ^ { x } - 3\) has exactly one root, denoted by \(\alpha\), in the interval \(0 < x < \pi\).
2. Verify by calculation that \(\alpha\) lies between 1 and 2 .
3. Show that if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula $$x _ { n + 1 } = \ln \left( \operatorname { cosec } \frac { 1 } { 2 } x _ { n } + 3 \right)$$ converges, then it converges to \(\alpha\).
4. Use this iterative formula with an initial value of 1.4 to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
5. State the minimum number of calculated iterations needed with this initial value to determine \(\alpha\) correct to 2 decimal places.

5

1. It is given that the equation \(\mathrm { e } ^ { 2 x } = 5 + \cos 3 x\) has only one root.
   Show by calculation that this root lies in the interval \(0.7 < x < 0.8\).
2. Show that if a sequence of values in the interval \(0.7 < x < 0.8\) given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 5 + \cos 3 x _ { n } \right)$$ converges then it converges to the root of the equation in part (a).
3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

8

1. By sketching a suitable pair of graphs, show that the equation $$\sqrt { x } = \mathrm { e } ^ { x } - 3$$ has only one root.
2. Show by calculation that this root lies between 1 and 2 .
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \ln \left( 3 + \sqrt { x _ { n } } \right)$$ converges, then it converges to the root of the equation in (a).
4. Use the iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2

1. By sketching a suitable pair of graphs, show that the equation \(\cot 2 x = \sec x\) has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Show that if a sequence of real values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \cos x _ { n } \right)$$ converges, then it converges to the root in part (a).