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

5 \includegraphics[max width=\textwidth, alt={}, center]{8bdd1285-9e39-465a-8c09-bbe410504f9d-06_442_698_260_721} The diagram shows part of the curve with equation \(y = x ^ { 3 } \cos 2 x\). The curve has a maximum at the point \(M\).

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \sqrt [ 3 ] { 1.5 x ^ { 2 } \cot 2 x }\).
2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(M\) lies between 0.59 and 0.60.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

7 It is given that \(\int _ { 0 } ^ { a } \left( \frac { 4 } { 2 x + 1 } + 8 x \right) \mathrm { d } x = 10\), where \(a\) is a positive constant.

1. Show that \(a = \sqrt { 2.5 - 0.5 \ln ( 2 a + 1 ) }\).
2. Using the equation in part (a), show by calculation that \(1 < a < 2\).
3. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{2d6fc4c5-70ec-4cd8-9b48-59d5ce0e39b7-08_575_618_262_762} The diagram shows the curve with equation \(y = \frac { 3 x + 2 } { \ln x }\). The curve has a minimum point \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { 3 x + 2 } { 3 \ln x }\). [3]
2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(M\) lies between 3 and 4.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 5 significant figures. Give the result of each iteration to 7 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{388d7076-636c-417d-84cb-e6e2a3e9a6a0-10_465_785_260_680} The diagram shows the curve with parametric equations $$x = 4 t + \mathrm { e } ^ { 2 t } , \quad y = 6 t \sin 2 t$$ for \(0 \leqslant t \leqslant 1\). The point \(P\) on the curve has parameter \(p\) and \(y\)-coordinate 3 .

1. Show that \(p = \frac { 1 } { 2 \sin 2 p }\).
2. Show by calculation that the value of \(p\) lies between 0.5 and 0.6 .
3. Use an iterative formula, based on the equation in part (a), to find the value of \(p\) correct to 3 significant figures. Use an initial value of 0.55 and give the result of each iteration to 5 significant figures.
4. Find the gradient of the curve at \(P\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 \includegraphics[max width=\textwidth, alt={}, center]{61df367d-741f-4906-8ab9-2f32e8711aa6-10_465_785_260_680} The diagram shows the curve with parametric equations $$x = 4 t + \mathrm { e } ^ { 2 t } , \quad y = 6 t \sin 2 t$$ for \(0 \leqslant t \leqslant 1\). The point \(P\) on the curve has parameter \(p\) and \(y\)-coordinate 3 .

1. Show that \(p = \frac { 1 } { 2 \sin 2 p }\).
2. Show by calculation that the value of \(p\) lies between 0.5 and 0.6 .
3. Use an iterative formula, based on the equation in part (a), to find the value of \(p\) correct to 3 significant figures. Use an initial value of 0.55 and give the result of each iteration to 5 significant figures.
4. Find the gradient of the curve at \(P\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5

1. By sketching the graphs of $$y = | 5 - 2 x | \quad \text { and } \quad y = 3 \ln x$$ on the same diagram, show that the equation \(| 5 - 2 x | = 3 \ln x\) has exactly two roots.
2. Show that the value of the larger root satisfies the equation \(x = 2.5 + 1.5 \ln x\).
3. Show by calculation that the value of the larger root lies between 4.5 and 5.0.
4. Use an iterative formula, based on the equation in part (b), to find the value of the larger root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

6 \includegraphics[max width=\textwidth, alt={}, center]{1b9c6b41-69dd-4132-92c7-9507cbd741dd-08_542_661_269_731} The diagram shows the curve \(y = 3 \mathrm { e } ^ { 2 x - 1 }\). The shaded region is bounded by the curve and the lines \(x = a , x = a + 1\) and \(y = 0\), where \(a\) is a constant. It is given that the area of the shaded region is 120 square units.

1. Show that \(a = \frac { 1 } { 2 } \ln \left( 80 + \mathrm { e } ^ { 2 a - 1 } \right) - \frac { 1 } { 2 }\).
2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

6 \includegraphics[max width=\textwidth, alt={}, center]{712be8e6-e1e9-4662-b1f1-51c39c2c9df1-08_542_661_269_731} The diagram shows the curve \(y = 3 \mathrm { e } ^ { 2 x - 1 }\). The shaded region is bounded by the curve and the lines \(x = a , x = a + 1\) and \(y = 0\), where \(a\) is a constant. It is given that the area of the shaded region is 120 square units.

1. Show that \(a = \frac { 1 } { 2 } \ln \left( 80 + \mathrm { e } ^ { 2 a - 1 } \right) - \frac { 1 } { 2 }\).
2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

3 It is given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 2 x } - 1 \right) \mathrm { d } x = 12\), where \(a\) is a positive constant.

1. Show that \(a = \frac { 1 } { 2 } \ln \left( 9 + \frac { 2 } { 3 } a \right)\).
2. Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 4 significant figures. Use an initial value of 1 and give the result of each iteration to 6 significant figures. [3]

4

1. \includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-05_753_944_278_630} The diagram shows the graph of \(y = 3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x }\).
   On the diagram, sketch the graph of \(y = | 5 x - 4 |\), and show that the equation \(3 - e ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) has exactly two real roots. It is given that the two roots of \(3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
2. Show by calculation that \(\alpha\) lies between 0.36 and 0.37 .
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 7 - \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to find \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

4

1. \includegraphics[max width=\textwidth, alt={}, center]{a1ea242a-c7f4-46b0-b4b8-bd13b3880557-05_753_944_278_630} The diagram shows the graph of \(y = 3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x }\).
   On the diagram, sketch the graph of \(y = | 5 x - 4 |\), and show that the equation \(3 - e ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) has exactly two real roots. It is given that the two roots of \(3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
2. Show by calculation that \(\alpha\) lies between 0.36 and 0.37 .
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 7 - \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to find \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 A curve has equation \(\mathrm { y } = \frac { 1 + \mathrm { e } ^ { 2 \mathrm { x } } } { 1 + 3 \mathrm { x } }\). The curve has exactly one stationary point \(P\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) and hence show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 6 } + \frac { 1 } { 2 } \mathrm { e } ^ { - 2 x }\).
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.35 and 0.45 .
3. Use an iterative formula based on the equation in part (a) to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-10_451_647_258_699} The diagram shows the curve with equation \(\mathrm { y } = \sqrt { \sin 2 \mathrm { x } + \sin ^ { 2 } 2 \mathrm { x } }\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\). The shaded region is bounded by the curve and the straight lines \(x = \frac { 1 } { 6 } \pi\) and \(y = 0\).

6 \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-10_417_700_310_685} The diagram shows the curve with equation \(y = \frac { \ln ( 2 x + 1 ) } { x + 3 }\). The curve has a maximum point \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { x + 3 } { \ln ( 2 x + 1 ) } - 0.5\).
3. Show by calculation that the \(x\)-coordinate of \(M\) lies between 2.5 and 3.0 .
4. Use an iterative formula based on the equation in part (b) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

6 A curve has equation \(y = x ^ { 3 } \mathrm { e } ^ { 0.2 x }\) where \(x \geqslant 0\). At the point \(P\) on the curve, the gradient of the curve is 15 .

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \sqrt { \frac { 75 \mathrm { e } ^ { - 0.2 x } } { 15 + x } }\).
2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(P\) lies between 1.7 and 1.8.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5

1. Given that \(2 \ln ( x + 1 ) + \ln x = \ln ( x + 9 )\), show that \(x = \sqrt { \frac { 9 } { x + 2 } }\).
2. It is given that the equation \(x = \sqrt { \frac { 9 } { x + 2 } }\) has a single root. Show by calculation that this root lies between 1.5 and 2.0.
3. Use an iterative formula, based on the equation in part (b), to find the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

7 A curve has equation \(\mathrm { e } ^ { 2 x } y - \mathrm { e } ^ { y } = 100\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \mathrm { e } ^ { 2 x } y } { \mathrm { e } ^ { y } - \mathrm { e } ^ { 2 x } }\).
2. Show that the curve has no stationary points.
   It is required to find the \(x\)-coordinate of \(P\), the point on the curve at which the tangent is parallel to the \(y\)-axis.
3. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \ln 10 - \frac { 1 } { 2 } \ln ( 2 x - 1 )$$
4. Use an iterative formula, based on the equation in part (c), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The sequence of values given by the iterative formula \(x _ { n + 1 } = \frac { 6 + 8 x _ { n } } { 8 + x _ { n } ^ { 2 } }\) with initial value \(x _ { 1 } = 2\) converges to \(\alpha\).

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 \(\alpha\).

7 \includegraphics[max width=\textwidth, alt={}, center]{c473f577-1e96-4d11-a0d5-cdfa4873c295-12_650_720_260_708} A curve has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } + 5 x - 15\). As shown in the diagram, the curve crosses the \(x\)-axis at the points \(A\) and \(B\) with coordinates \(( a , 0 )\) and \(( b , 0 )\) respectively.

1. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. By first finding the quotient when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\), show that $$a = - \sqrt { \frac { 5 } { 2 - a } } .$$
3. Use an iterative formula, based on the equation in part (b), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The curve with equation \(y = x \mathrm { e } ^ { 2 x } + 5 \mathrm { e } ^ { - x }\) has a minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { 1 } { 3 } \ln 5 - \frac { 1 } { 3 } \ln ( 1 + 2 x )\).
2. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Use an initial value of 0.35 and give the result of each iteration to 5 significant figures.

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.

4

1. By sketching a suitable pair of graphs on the same diagram, show that the equation $$\mathrm { e } ^ { - \frac { 1 } { 2 } x } = x ^ { 5 }$$ has exactly one real root.
2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } }\) to determine the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 The curve with equation \(y = x \ln ( 4 x + 1 ) - 3 x\) has one stationary point \(P\).

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \frac { 2 x + 0.75 } { \ln ( 4 x + 1 ) } - 0.25$$
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 1.8 and 1.9.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

7 The curve with equation \(\mathrm { e } ^ { 2 x } - 18 x + y ^ { 3 } + y = 11\) has a stationary point at \(( p , q )\).

1. Find the exact value of \(p\).
2. Show that \(q = \sqrt [ 3 ] { 2 + 18 \ln 3 - q }\).
3. Show by calculation that the value of \(q\) lies between 2.5 and 3.0.
4. Use an iterative formula, based on the equation in (b), to find the value of \(q\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
   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 \includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-10_803_394_269_863} The diagram shows the curve with parametric equations $$x = 3 \ln ( 2 t - 3 ) , \quad y = 4 t \ln t$$ The curve crosses the \(y\)-axis at the point \(A\). At the point \(B\), the gradient of the curve is 12 .

1. Find the exact gradient of the curve at \(A\).
2. Show that the value of the parameter \(t\) at \(B\) satisfies the equation $$t = \frac { 9 } { 1 + \ln t } + \frac { 3 } { 2 }$$
3. Use an iterative formula, based on the equation in (b), to find the value of \(t\) at \(B\), giving your answer correct to 3 significant figures. Use an initial value of 5 and give the result of each iteration to 5 significant figures.

5 It is given that \(\int _ { a } ^ { a ^ { 3 } } \frac { 10 } { 2 x + 1 } \mathrm {~d} x = 7\), where \(a\) is a constant greater than 1 .

1. Show that \(a = \sqrt [ 3 ] { 0.5 \mathrm { e } ^ { 1.4 } ( 2 a + 1 ) - 0.5 }\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-08_2718_35_107_2011} \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-09_2725_35_99_20}
2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.