1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams

9 \includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-3_481_483_1434_831} The diagram shows the curves \(y = x \cos x\) and \(y = \frac { k } { x }\), where \(k\) is a constant, for \(0 < x \leqslant \frac { 1 } { 2 } \pi\). The curves touch at the point where \(x = a\).

1. Show that \(a\) satisfies the equation \(\tan a = \frac { 2 } { a }\).
2. Use the iterative formula \(a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { a _ { n } } \right)\) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
3. Hence find the value of \(k\) correct to 2 decimal places.

9 It is given that \(\int _ { 1 } ^ { a } x ^ { \frac { 1 } { 2 } } \ln x \mathrm {~d} x = 2\), where \(a > 1\).

1. Show that \(a ^ { \frac { 3 } { 2 } } = \frac { 7 + 2 a ^ { \frac { 3 } { 2 } } } { 3 \ln a }\).
2. Show by calculation that \(a\) lies between 2 and 4 .
3. Use the iterative formula $$a _ { n + 1 } = \left( \frac { 7 + 2 a _ { n } ^ { \frac { 3 } { 2 } } } { 3 \ln a _ { n } } \right) ^ { \frac { 2 } { 3 } }$$ to determine \(a\) 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.

5 The curve with equation \(y = \mathrm { e } ^ { - 2 x } \ln ( x - 1 )\) has a stationary point when \(x = p\).

1. Show that \(p\) satisfies the equation \(x = 1 + \exp \left( \frac { 1 } { 2 ( x - 1 ) } \right)\), where \(\exp ( x )\) denotes \(\mathrm { e } ^ { x }\).
2. Verify by calculation that \(p\) lies between 2.2 and 2.6.
3. Use an iterative formula based on the equation in part (i) to determine \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

9 It is given that \(\int _ { 0 } ^ { a } x \cos \frac { 1 } { 3 } x \mathrm {~d} x = 3\), where the constant \(a\) is such that \(0 < a < \frac { 3 } { 2 } \pi\).

1. Show that \(a\) satisfies the equation $$a = \frac { 4 - 3 \cos \frac { 1 } { 3 } a } { \sin \frac { 1 } { 3 } a }$$
2. Verify by calculation that \(a\) lies between 2.5 and 3 .
3. Use an iterative formula based on the equation in part (i) to calculate \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

5

1. By sketching a suitable pair of graphs, show that the equation \(\ln ( x + 2 ) = 4 \mathrm { e } ^ { - x }\) has exactly one real root.
2. Show by calculation that this root lies between \(x = 1\) and \(x = 1.5\).
3. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 4 } { \ln \left( x _ { n } + 2 \right) } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 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.

6 \includegraphics[max width=\textwidth, alt={}, center]{6694ccc1-c8b1-42a7-8b21-829a89af74c9-08_732_807_258_667} The diagram shows the curve with equation \(y = \frac { 8 + x ^ { 3 } } { 2 - 5 x }\). The maximum point is denoted by \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine the gradient of the curve at the point where the curve crosses the \(x\)-axis.
2. Show that the \(x\)-coordinate of the point \(M\) satisfies the equation \(x = \sqrt { } \left( 0.6 x + 4 x ^ { - 1 } \right)\).
3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

6 \includegraphics[max width=\textwidth, alt={}, center]{f5e0b088-73db-405b-a832-aa01d9fcba64-08_396_716_260_712} The diagram shows the curve with parametric equations $$x = 3 t - 6 \mathrm { e } ^ { - 2 t } , \quad y = 4 t ^ { 2 } \mathrm { e } ^ { - t }$$ for \(0 \leqslant t \leqslant 2\). At the point \(P\) on the curve, the \(y\)-coordinate is 1 .

1. Show that the value of \(t\) at the point \(P\) satisfies the equation \(t = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t }\).
2. Use the iterative formula \(t _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t _ { n } }\) with \(t _ { 1 } = 0.7\) to find the value of \(t\) at \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
3. Find the gradient of the curve at \(P\), giving the answer correct to 2 significant figures.

6 \includegraphics[max width=\textwidth, alt={}, center]{0d15e5a1-d05f-48bc-8613-198804ff605c-08_396_716_260_712} The diagram shows the curve with parametric equations $$x = 3 t - 6 \mathrm { e } ^ { - 2 t } , \quad y = 4 t ^ { 2 } \mathrm { e } ^ { - t }$$ for \(0 \leqslant t \leqslant 2\). At the point \(P\) on the curve, the \(y\)-coordinate is 1 .

1. Show that the value of \(t\) at the point \(P\) satisfies the equation \(t = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t }\).
2. Use the iterative formula \(t _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t _ { n } }\) with \(t _ { 1 } = 0.7\) to find the value of \(t\) at \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
3. Find the gradient of the curve at \(P\), giving the answer correct to 2 significant figures.

4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } + 4 x _ { n } ^ { - 3 } \right)$$ with initial value \(x _ { 1 } = 1.5\), converges to \(\alpha\).

1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
2. State an equation that is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

5 It is given that \(a\) is a positive constant such that $$\int _ { 0 } ^ { a } \left( 1 + 2 x + 3 \mathrm { e } ^ { 3 x } \right) \mathrm { d } x = 250$$

1. Show that \(a = \frac { 1 } { 3 } \ln \left( 251 - a - a ^ { 2 } \right)\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 The equation of a curve is \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 x + 1 }\) and the point \(P\) on the curve has \(y\)-coordinate 10 .

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 2 } \ln ( 40 x + 10 )\).
2. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 40 x _ { n } + 10 \right)\) with \(x _ { 1 } = 2.3\) to find the \(x\)-coordinate of \(P\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
3. Find the gradient of the curve at \(P\), giving the answer correct to 3 significant figures.

4

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) in the interval \(0 < x < \frac { 1 } { 2 } \pi\) that is a root of the equation $$\sin x = \frac { 1 } { x ^ { 2 } }$$
2. Verify by calculation that this root lies between 1 and 1.5.
3. Show that this value of \(x\) is also a root of the equation $$x = \sqrt { } ( \operatorname { cosec } x )$$
4. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \operatorname { cosec } x _ { n } \right)$$ to determine this root correct to 3 significant figures, showing the value of each approximation that you calculate.

6

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) in the interval \(0 < x < \frac { 1 } { 2 } \pi\) that is a root of the equation $$\cot x = x$$
2. Verify by calculation that this root lies between 0.8 and 0.9 radians.
3. Show that this value of \(x\) is also a root of the equation $$x = \tan ^ { - 1 } \left( \frac { 1 } { x } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { x _ { n } } \right)$$ to determine this root correct to 2 decimal places, showing the result of each iteration.

5

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) that is a root of the equation $$\frac { 1 } { x } = \ln x$$
2. Verify by calculation that this root lies between 1 and 2 .
3. Show that this root also satisfies the equation $$x = \mathrm { e } ^ { \frac { 1 } { x } }$$
4. Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \frac { 1 } { x _ { n } } }$$ with initial value \(x _ { 1 } = 1.8\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5 \includegraphics[max width=\textwidth, alt={}, center]{4029c46c-50a1-4d23-bc29-589417a6b7f5-2_396_392_1603_879} The diagram shows a chord joining two points, \(A\) and \(B\), on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \pi\). The area of the shaded segment is one sixth of the area of the circle.

1. Show that \(\alpha\) satisfies the equation $$x = \frac { 1 } { 3 } \pi + \sin x .$$
2. Verify by calculation that \(\alpha\) lies between \(\frac { 1 } { 2 } \pi\) and \(\frac { 2 } { 3 } \pi\).
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \pi + \sin x _ { n } ,$$ with initial value \(x _ { 1 } = 2\), to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } } { 3 } + \frac { 4 } { x _ { n } ^ { 2 } }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. State an equation that is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

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.

7 \includegraphics[max width=\textwidth, alt={}, center]{67a12825-d7ce-4853-ada4-b8d3009331b5-3_531_759_262_694} The diagram shows the curve \(y = \mathrm { e } ^ { - x }\). The shaded region \(R\) is bounded by the curve and the lines \(y = 1\) and \(x = p\), where \(p\) is a constant.

1. Find the area of \(R\) in terms of \(p\).
2. Show that if the area of \(R\) is equal to 1 then $$p = 2 - \mathrm { e } ^ { - p }$$
3. Use the iterative formula $$p _ { n + 1 } = 2 - \mathrm { e } ^ { - p _ { n } }$$ with initial value \(p _ { 1 } = 2\), to calculate the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{729aa2f6-2b62-445f-a2aa-a63b45cb6b64-3_604_971_262_587} The diagram shows the curve \(y = x ^ { 2 } \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Show by differentiation that the \(x\)-coordinate of \(M\) satisfies the equation $$\tan x = \frac { 2 } { x }$$
2. Verify by calculation that this equation has a root (in radians) between 1 and 1.2.
3. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { x _ { n } } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 The curve with equation \(y = \frac { 6 } { x ^ { 2 } }\) intersects the line \(y = x + 1\) at the point \(P\).

1. Verify by calculation that the \(x\)-coordinate of \(P\) lies between 1.4 and 1.6.
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt { } \left( \frac { 6 } { x + 1 } \right)$$
3. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 6 } { x _ { n } + 1 } \right)$$ with initial value \(x _ { 1 } = 1.5\), to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 7 x _ { n } } { 8 } + \frac { 5 } { 2 x _ { n } ^ { 4 } }$$ with initial value \(x _ { 1 } = 1.7\), converges to \(\alpha\).

1. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. State an equation that is satisfied by \(\alpha\) and hence show that \(\alpha = \sqrt [ 5 ] { 20 }\).

6

1. Verify by calculation that the cubic equation $$x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3 = 0$$ has a root that lies between \(x = 0.7\) and \(x = 0.8\).
2. Show that this root also satisfies an equation of the form $$x = \frac { a x ^ { 2 } + 3 } { x ^ { 2 } + b }$$ where the values of \(a\) and \(b\) are to be found.
3. With these values of \(a\) and \(b\), use the iterative formula $$x _ { n + 1 } = \frac { a x _ { n } ^ { 2 } + 3 } { x _ { n } ^ { 2 } + b }$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5

1. By sketching a suitable pair of graphs, show that the equation $$\frac { 1 } { x } = \sin x$$ where \(x\) is in radians, has only one root for \(0 < x \leqslant \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between \(x = 1.1\) and \(x = 1.2\).
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { \sin x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.