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

7 \includegraphics[max width=\textwidth, alt={}, center]{e82fee05-0c55-4fe2-b781-e5e82186c153-2_608_999_1430_571} The diagram shows the curve \(y = ( x - 4 ) \mathrm { e } ^ { \frac { 1 } { 2 } x }\). The curve has a gradient of 3 at the point \(P\).

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = 2 + 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x }$$
2. Verify that the equation in part (i) has a root between \(x = 3.1\) and \(x = 3.3\).
3. Use the iterative formula \(x _ { n + 1 } = 2 + 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } }\) 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]{9e1bd528-e7c4-4936-a05a-dde1d1ace7c2-2_512_775_1318_683} The diagram shows the curve \(y = \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). A rectangle \(O A B C\) is drawn, where \(B\) is the point on the curve with \(x\)-coordinate \(\theta\), and \(A\) and \(C\) are on the axes, as shown. The shaded region \(R\) is bounded by the curve and by the lines \(x = \theta\) and \(y = 0\).

1. Find the area of \(R\) in terms of \(\theta\).
2. The area of the rectangle \(O A B C\) is equal to the area of \(R\). Show that $$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
3. Use the iterative formula \(\theta _ { n + 1 } = \frac { 1 - \sin \theta _ { n } } { \cos \theta _ { n } }\), with initial value \(\theta _ { 1 } = 0.5\), to determine the value of \(\theta\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4 \includegraphics[max width=\textwidth, alt={}, center]{0355f624-3a35-4b9e-8520-af011a0fb6db-2_499_787_922_678} The diagram shows the part of the curve \(y = \sqrt { } ( 2 - \sin x )\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { } ( 2 - \sin x ) \mathrm { d } x$$ giving your answer correct to 2 decimal places.
2. The line \(y = x\) intersects the curve \(y = \sqrt { } ( 2 - \sin x )\) at the point \(P\). Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \sin x _ { n } \right)$$ to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5 \includegraphics[max width=\textwidth, alt={}, center]{96a4df57-b3c7-4dbf-9bea-bb00ed6a4a16-2_512_775_1318_683} The diagram shows the curve \(y = \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). A rectangle \(O A B C\) is drawn, where \(B\) is the point on the curve with \(x\)-coordinate \(\theta\), and \(A\) and \(C\) are on the axes, as shown. The shaded region \(R\) is bounded by the curve and by the lines \(x = \theta\) and \(y = 0\).

1. Find the area of \(R\) in terms of \(\theta\).
2. The area of the rectangle \(O A B C\) is equal to the area of \(R\). Show that $$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
3. Use the iterative formula \(\theta _ { n + 1 } = \frac { 1 - \sin \theta _ { n } } { \cos \theta _ { n } }\), with initial value \(\theta _ { 1 } = 0.5\), to determine the value of \(\theta\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 \includegraphics[max width=\textwidth, alt={}, center]{faf83d93-40b6-4557-bfd5-f94c67470dfa-2_449_639_388_753} The diagram shows the curve \(y = x ^ { 4 } + 2 x - 9\). The curve cuts the positive \(x\)-axis at the point \(P\).

1. Verify by calculation that the \(x\)-coordinate of \(P\) lies between 1.5 and 1.6.
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt [ 3 ] { \left( \frac { 9 } { x } - 2 \right) }$$
3. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { \left( \frac { 9 } { x _ { n } } - 2 \right) }$$ to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{0900b607-6136-4bf7-a42e-6824d1a21e43-3_451_451_255_845} The diagram shows part of the curve \(y = 8 x + \frac { 1 } { 2 } \mathrm { e } ^ { x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = a\), where \(a\) is positive. The area of \(R\) is equal to \(\frac { 1 } { 2 }\).

1. Find an equation satisfied by \(a\), and show that the equation can be written in the form $$a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)$$
2. Verify by calculation that the equation \(a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)\) has a root between 0.2 and 0.3.
3. Use the iterative formula \(a _ { n + 1 } = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a _ { n } } } { 8 } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 \includegraphics[max width=\textwidth, alt={}, center]{a3e778cb-9f95-4750-ba49-a57ee22af018-2_449_639_388_753} The diagram shows the curve \(y = x ^ { 4 } + 2 x - 9\). The curve cuts the positive \(x\)-axis at the point \(P\).

1. Verify by calculation that the \(x\)-coordinate of \(P\) lies between 1.5 and 1.6.
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt [ 3 ] { \left( \frac { 9 } { x } - 2 \right) }$$
3. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { \left( \frac { 9 } { x _ { n } } - 2 \right) }$$ to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{72d50061-ead5-466a-96fc-2203438d1407-3_296_675_945_735} The diagram shows part of the curve \(y = \frac { x ^ { 2 } } { 1 + \mathrm { e } ^ { 3 x } }\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(m\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m\) satisfies the equation \(x = \frac { 2 } { 3 } \left( 1 + \mathrm { e } ^ { - 3 x } \right)\).
2. Show by calculation that \(m\) lies between 0.7 and 0.8 .
3. Use an iterative formula based on the equation in part (i) to find \(m\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{c703565b-8aa8-424b-9684-6592d4effdf8-3_597_931_260_607} The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 4 } - 3 x ^ { 3 } + 3 x ^ { 2 } - 25 x + 48 .$$ The diagram shows the curve \(y = \mathrm { p } ( x )\) which crosses the \(x\)-axis at ( \(\alpha , 0\) ) and ( 3,0 ).

1. Divide \(\mathrm { p } ( x )\) by a suitable linear factor and hence show that \(\alpha\) is a root of the equation \(x = \sqrt [ 3 ] { } ( 16 - 3 x )\).
2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 16 - 3 x _ { n } \right)\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{293e1e27-77e9-4b19-a152-96d71b75346e-3_296_675_945_735} The diagram shows part of the curve \(y = \frac { x ^ { 2 } } { 1 + \mathrm { e } ^ { 3 x } }\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(m\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m\) satisfies the equation \(x = \frac { 2 } { 3 } \left( 1 + \mathrm { e } ^ { - 3 x } \right)\).
2. Show by calculation that \(m\) lies between 0.7 and 0.8 .
3. Use an iterative formula based on the equation in part (i) to find \(m\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

4

1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
2. Verify by calculation that \(4.5 < \alpha < 5.0\).
3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) 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 $$x _ { n + 1 } = 2 + \frac { 4 } { x _ { n } ^ { 2 } + 2 x _ { n } + 4 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Determine the value of \(\alpha\) correct to 3 decimal places, giving the result of each iteration to 5 decimal places.
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

5 It is given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 3 x } + 5 \mathrm { e } ^ { x } \right) \mathrm { d } x = 100\), where \(a\) is a positive constant.

1. Show that \(a = \frac { 1 } { 3 } \ln \left( 106 - 5 \mathrm { e } ^ { a } \right)\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 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.

7 \includegraphics[max width=\textwidth, alt={}, center]{ebfaafca-3279-44f8-a910-0756ffa25c70-10_542_789_260_676} The diagram shows the curve $$y = x ^ { 2 } + 3 x + 1 + 5 \cos \frac { 1 } { 2 } x .$$ The curve crosses the \(y\)-axis at the point \(P\) and the gradient of the curve at \(P\) is \(m\). The point \(Q\) on the curve has \(x\)-coordinate \(q\) and the gradient of the curve at \(Q\) is \(- m\).

1. Find the value of \(m\) and hence show that \(q\) satisfies the equation $$x = a \sin \frac { 1 } { 2 } x + b ,$$ where the values of the constants \(a\) and \(b\) are to be determined.
2. Show by calculation that \(- 4.5 < q < - 4.0\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{da5162f3-b5d5-417f-9b6c-5ae0024f22d9-10_542_789_260_676} The diagram shows the curve $$y = x ^ { 2 } + 3 x + 1 + 5 \cos \frac { 1 } { 2 } x .$$ The curve crosses the \(y\)-axis at the point \(P\) and the gradient of the curve at \(P\) is \(m\). The point \(Q\) on the curve has \(x\)-coordinate \(q\) and the gradient of the curve at \(Q\) is \(- m\).

1. Find the value of \(m\) and hence show that \(q\) satisfies the equation $$x = a \sin \frac { 1 } { 2 } x + b ,$$ where the values of the constants \(a\) and \(b\) are to be determined.
2. Show by calculation that \(- 4.5 < q < - 4.0\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

4 \includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-06_652_789_260_676} The diagram shows the curve with equation $$y = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 12 x - 32$$ The curve crosses the \(x\)-axis at points with coordinates \(( \alpha , 0 )\) and \(( \beta , 0 )\).

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of $$x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 12 x - 32$$
2. Show that \(\beta\) satisfies an equation of the form \(x = \sqrt [ 3 ] { } ( p + q x )\), and state the values of \(p\) and \(q\). [3]
3. Use an iterative formula based on the equation in part (ii) to find the value of \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 The curve with equation $$y = 5 \mathrm { e } ^ { 2 x } - 8 x ^ { 2 } - 20$$ crosses the \(x\)-axis at only one point. This point has coordinates \(( p , 0 )\).

1. Show that \(p\) satisfies the equation \(x = \frac { 1 } { 2 } \ln \left( 1.6 x ^ { 2 } + 4 \right)\).
2. Show by calculation that \(0.75 < p < 0.85\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 5 significant figures. Give the result of each iteration to 7 significant figures.
4. Find the gradient of the curve at the point \(( p , 0 )\).

5 It is given that \(\int _ { 0 } ^ { a } \left( 3 x ^ { 2 } + 4 \cos 2 x - \sin x \right) \mathrm { d } x = 2\), where \(a\) is a constant.

1. Show that \(a = \sqrt [ 3 ] { } ( 3 - 2 \sin 2 a - \cos a )\).
2. Using the equation in part (i), show by calculation that \(0.5 < a < 0.75\).
3. Use an iterative formula, based on the equation in part (i), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

4 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { x _ { n } } { \ln \left( 2 x _ { n } \right) }$$ converges to the value \(\alpha\).

1. Use the iterative formula to find the value of \(\alpha\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
2. State an equation satisfied by \(\alpha\) and hence determine the exact value of \(\alpha\).

4

1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
2. Verify by calculation that \(4.5 < \alpha < 5.0\).
3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{da8dedae-4714-408e-a983-90ece63d9e91-08_501_1086_262_525} The diagram shows a circle with centre \(O\) and radius \(r\). The tangents to the circle at the points \(A\) and \(B\) meet at \(T\), and angle \(A O B\) is \(2 x\) radians. The shaded region is bounded by the tangents \(A T\) and \(B T\), and by the minor \(\operatorname { arc } A B\). The area of the shaded region is equal to the area of the circle.

1. Show that \(x\) satisfies the equation \(\tan x = \pi + x\).
2. This equation has one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Verify by calculation that this root lies between 1 and 1.4.
3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \pi + x _ { n } \right)$$ to determine the 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 \(\mathrm { f } ( x ) = \frac { 2 } { ( 2 x - 1 ) ( 2 x + 1 ) }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{a257f49d-5c8f-4c23-be78-46619b746fde-10_353_689_262_726} The diagram shows the curve \(y = \frac { \tan ^ { - 1 } x } { \sqrt { x } }\) and its maximum point \(M\) where \(x = a\).

1. Show that \(a\) satisfies the equation $$a = \tan \left( \frac { 2 a } { 1 + a ^ { 2 } } \right)$$
2. Verify by calculation that \(a\) lies between 1.3 and 1.5.
3. Use an iterative formula based on the equation in part (a) to determine \(a\) 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 \(\cot \frac { 1 } { 2 } x = 1 + \mathrm { e } ^ { - x }\) has exactly one root in the interval \(0 < x \leqslant \pi\).
2. Verify by calculation that this root lies between 1 and 1.5.
3. Use the iterative formula \(x _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 1 } { 1 + \mathrm { e } ^ { - x _ { n } } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.