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

8 The equation of a curve is \(y = \ln x + \frac { 2 } { x }\), where \(x > 0\).

1. Find the coordinates of the stationary point of the curve and determine whether it is a maximum or a minimum point.
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 - \ln x _ { n } }$$ with initial value \(x _ { 1 } = 1\), converges to \(\alpha\). State an equation satisfied by \(\alpha\), and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 3\).
3. Use this iterative formula to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.

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.

7

1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } x = \frac { 1 } { 2 } x + 1$$ where \(x\) is in radians, has a root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify, by calculation, that this root lies between 0.5 and 1 .
3. Show that this root also satisfies the equation $$x = \sin ^ { - 1 } \left( \frac { 2 } { x + 2 } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 } { x _ { n } + 2 } \right)$$ with initial value \(x _ { 1 } = 0.75\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-2_355_601_1562_772} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \pi\). The area of triangle \(A O B\) is half the area of the sector.

1. Show that \(\alpha\) satisfies the equation $$x = 2 \sin x$$
2. Verify by calculation that \(\alpha\) lies between \(\frac { 1 } { 2 } \pi\) and \(\frac { 2 } { 3 } \pi\).
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( x _ { n } + 4 \sin x _ { n } \right)$$ converges, then it converges to a root of the equation in part (i).
4. Use this iterative formula, with initial value \(x _ { 1 } = 1.8\), to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 Solve, correct to 3 significant figures, the equation $$\mathrm { e } ^ { x } + \mathrm { e } ^ { 2 x } = \mathrm { e } ^ { 3 x }$$

3 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-2_337_828_657_657} In the diagram, \(A B C D\) is a rectangle with \(A B = 3 a\) and \(A D = a\). A circular arc, with centre \(A\) and radius \(r\), joins points \(M\) and \(N\) on \(A B\) and \(C D\) respectively. The angle \(M A N\) is \(x\) radians. The perimeter of the sector \(A M N\) is equal to half the perimeter of the rectangle.

1. Show that \(x\) satisfies the equation $$\sin x = \frac { 1 } { 4 } ( 2 + x ) \text {. }$$
2. This equation has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Use the iterative formula $$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 + x _ { n } } { 4 } \right) ,$$ with initial value \(x _ { 1 } = 0.8\), to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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.

6 \includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-3_380_641_258_751} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The angle \(B O C\) is \(x\) radians. The area of the shaded segment is a quarter of the area of the semicircle.

1. Show that \(x\) satisfies the equation $$x = \frac { 3 } { 4 } \pi - \sin x$$
2. This equation has one root. Verify by calculation that the root lies between 1.3 and 1.5.
3. Use the iterative formula $$x _ { n + 1 } = \frac { 3 } { 4 } \pi - \sin x _ { n }$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4 \includegraphics[max width=\textwidth, alt={}, center]{20de5ba6-9426-4431-99af-6e8e62607f3e-2_513_895_1055_625} The diagram shows the curve \(y = \frac { \sin x } { x }\) for \(0 < x \leqslant 2 \pi\), and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation $$x = \tan x$$
2. The iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } \right) + \pi$$ can be used to determine the \(x\)-coordinate of \(M\). Use this formula to determine the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{cc85b13a-7f15-4025-a545-373cda454de8-3_456_495_255_824} The diagram shows a circle with centre \(O\) and radius 10 cm . The chord \(A B\) divides the circle into two regions whose areas are in the ratio \(1 : 4\) and it is required to find the length of \(A B\). The angle \(A O B\) is \(\theta\) radians.

1. Show that \(\theta = \frac { 2 } { 5 } \pi + \sin \theta\).
2. Showing all your working, use an iterative formula, based on the equation in part (i), with an initial value of 2.1 , to find \(\theta\) correct to 2 decimal places. Hence find the length of \(A B\) in centimetres correct to 1 decimal place.

6

1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 1 + x ^ { 2 }$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 0.5 and 0.8.
3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + x _ { n } ^ { 2 } } \right)$$ 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]{d3f0b201-3004-497a-9b29-30c94d0bec5b-2_300_767_518_689} In the diagram, \(A B C\) is a triangle in which angle \(A B C\) is a right angle and \(B C = a\). A circular arc, with centre \(C\) and radius \(a\), joins \(B\) and the point \(M\) on \(A C\). The angle \(A C B\) is \(\theta\) radians. The area of the sector \(C M B\) is equal to one third of the area of the triangle \(A B C\).

1. Show that \(\theta\) satisfies the equation $$\tan \theta = 3 \theta .$$
2. This equation has one root in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\). Use the iterative formula $$\theta _ { n + 1 } = \tan ^ { - 1 } \left( 3 \theta _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{e2cc23d2-f3ac-488b-97e1-79e2a98a87ba-3_421_885_251_628} The diagram shows part of the curve \(y = \cos ( \sqrt { } x )\) for \(x \geqslant 0\), where \(x\) is in radians. The shaded region between the curve, the axes and the line \(x = p ^ { 2 }\), where \(p > 0\), is denoted by \(R\). The area of \(R\) is equal to 1 .

1. Use the substitution \(x = u ^ { 2 }\) to find \(\int _ { 0 } ^ { p ^ { 2 } } \cos ( \sqrt { } x ) \mathrm { d } x\). Hence show that \(\sin p = \frac { 3 - 2 \cos p } { 2 p }\).
2. Use the iterative formula \(p _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 3 - 2 \cos p _ { n } } { 2 p _ { n } } \right)\), with initial value \(p _ { 1 } = 1\), to find the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

10 Liquid is flowing into a small tank which has a leak. Initially the tank is empty and, \(t\) minutes later, the volume of liquid in the tank is \(V \mathrm {~cm} ^ { 3 }\). The liquid is flowing into the tank at a constant rate of \(80 \mathrm {~cm} ^ { 3 }\) per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to \(k V \mathrm {~cm} ^ { 3 }\) per minute where \(k\) is a positive constant.

1. Write down a differential equation describing this situation and solve it to show that $$V = \frac { 1 } { k } \left( 80 - 80 \mathrm { e } ^ { - k t } \right)$$
2. It is observed that \(V = 500\) when \(t = 15\), so that \(k\) satisfies the equation $$k = \frac { 4 - 4 e ^ { - 15 k } } { 25 }$$ Use an iterative formula, based on this equation, to find the value of \(k\) correct to 2 significant figures. Use an initial value of \(k = 0.1\) and show the result of each iteration to 4 significant figures.
3. Determine how much liquid there is in the tank 20 minutes after the liquid started flowing, and state what happens to the volume of liquid in the tank after a long time.

8

1. By sketching each of the graphs \(y = \operatorname { cosec } x\) and \(y = x ( \pi - x )\) for \(0 < x < \pi\), show that the equation $$\operatorname { cosec } x = x ( \pi - x )$$ has exactly two real roots in the interval \(0 < x < \pi\).
2. Show that the equation \(\operatorname { cosec } x = x ( \pi - x )\) can be written in the form \(x = \frac { 1 + x ^ { 2 } \sin x } { \pi \sin x }\).
3. The two real roots of the equation \(\operatorname { cosec } x = x ( \pi - x )\) in the interval \(0 < x < \pi\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
   1. Use the iterative formula $$x _ { n + 1 } = \frac { 1 + x _ { n } ^ { 2 } \sin x _ { n } } { \pi \sin x _ { n } }$$ to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
   2. Deduce the value of \(\beta\) correct to 2 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.

10 \includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_515_508_1105_815} The diagram shows part of the curve with parametric equations $$x = 2 \ln ( t + 2 ) , \quad y = t ^ { 3 } + 2 t + 3$$

1. Find the gradient of the curve at the origin.
2. At the point \(P\) on the curve, the value of the parameter is \(p\). It is given that the gradient of the curve at \(P\) is \(\frac { 1 } { 2 }\).
   1. Show that \(p = \frac { 1 } { 3 p ^ { 2 } + 2 } - 2\).
   2. By first using an iterative formula based on the equation in part (a), determine the coordinates of the point \(P\). Give the result of each iteration to 5 decimal places and each coordinate of \(P\) correct to 2 decimal places.

5 \includegraphics[max width=\textwidth, alt={}, center]{d1377d66-73c8-4d97-9cae-d784b41fb0a8-2_519_800_1359_669} 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 the 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 perimeter of the shaded region is equal to the circumference 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 1.3.
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 It is given that \(\int _ { 0 } ^ { a } x \cos x \mathrm {~d} x = 0.5\), where \(0 < a < \frac { 1 } { 2 } \pi\).

1. Show that \(a\) satisfies the equation \(\sin a = \frac { 1.5 - \cos a } { a }\).
2. Verify by calculation that \(a\) is greater than 1 .
3. Use the iterative formula $$a _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 1.5 - \cos a _ { n } } { a _ { n } } \right)$$ to determine the value of \(a\) correct to 4 decimal places, giving the result of each iteration to 6 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.

8 \includegraphics[max width=\textwidth, alt={}, center]{9d3af34a-670f-425e-8156-0ad4d08fbdc0-3_499_552_258_792} The diagram shows the curve \(y = \operatorname { cosec } x\) for \(0 < x < \pi\) and part of the curve \(y = \mathrm { e } ^ { - x }\). When \(x = a\), the tangents to the curves are parallel.

1. By differentiating \(\frac { 1 } { \sin x }\), show that if \(y = \operatorname { cosec } x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } x \cot x\).
2. By equating the gradients of the curves at \(x = a\), show that $$a = \tan ^ { - 1 } \left( \frac { \mathrm { e } ^ { a } } { \sin a } \right)$$
3. Verify by calculation that \(a\) lies between 1 and 1.5.
4. Use an iterative formula based on the equation in part (ii) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6 The curve with equation \(y = x ^ { 2 } \cos \frac { 1 } { 2 } x\) has a stationary point at \(x = p\) in the interval \(0 < x < \pi\).

1. Show that \(p\) satisfies the equation \(\tan \frac { 1 } { 2 } p = \frac { 4 } { p }\).
2. Verify by calculation that \(p\) lies between 2 and 2.5.
3. Use the iterative formula \(p _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 4 } { p _ { n } } \right)\) to determine the value of \(p\) 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.

7 \includegraphics[max width=\textwidth, alt={}, center]{98ee8d3e-9aba-46a2-aa9c-b1e2093f393e-10_702_597_258_772} The diagram shows the curves \(y = 4 \cos \frac { 1 } { 2 } x\) and \(y = \frac { 1 } { 4 - x }\), for \(0 \leqslant x < 4\). When \(x = a\), the tangents to the curves are perpendicular.

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

6 \includegraphics[max width=\textwidth, alt={}, center]{772393d7-6e81-4b99-913a-63c9f87d1af2-08_492_812_260_664} In the diagram, \(A\) is the mid-point of the semicircle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the semicircle at \(B\) and \(C\). The angle \(O A B\) is equal to \(x\) radians. The area of the shaded region bounded by \(A B , A C\) and the arc with centre \(A\) is equal to half the area of the semicircle.

1. Use triangle \(O A B\) to show that \(A B = 2 r \cos x\).
2. Hence show that \(x = \cos ^ { - 1 } \sqrt { } \left( \frac { \pi } { 16 x } \right)\).
3. Verify by calculation that \(x\) lies between 1 and 1.5.
4. Use an iterative formula based on the equation in part (ii) to determine \(x\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.