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

\includegraphics{figure_8} The diagram shows part of each of the curves \(y = e^{\frac{1}{3}x}\) and \(y = \sqrt[3]{(3x + 8)}\). The curves meet, as shown in the diagram, at the point \(P\). The region \(R\), shaded in the diagram, is bounded by the two curves and by the \(y\)-axis.

1. Show by calculation that the \(x\)-coordinate of \(P\) lies between 5.2 and 5.3. [3]
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac{2}{3} \ln(3x + 8)\). [2]
3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(P\) correct to 2 decimal places. [3]
4. Use integration, and your answer to part (iii), to find an approximate value of the area of the region \(R\). [5]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = \cos^{-1} x\).

1. Sketch the curve with equation \(y = 3 \cos^{-1}(x - 1)\), showing the coordinates of the points where the curve meets the axes. [3]
2. By drawing an appropriate straight line on your sketch in part (i), show that the equation \(3 \cos^{-1}(x - 1) = x\) has exactly one root. [1]
3. Show by calculation that the root of the equation \(3 \cos^{-1}(x - 1) = x\) lies between 1.8 and 1.9. [2]
4. The sequence defined by $$x_1 = 2, \quad x_{n+1} = 1 + \cos(\frac{1}{3}x_n)$$ converges to a number \(\alpha\). Find the value of \(\alpha\) correct to 2 decimal places and explain why \(\alpha\) is the root of the equation \(3 \cos^{-1}(x - 1) = x\). [5]

The equation \(2x^3 + 4x - 35 = 0\) has one real root.

1. Show by calculation that this real root lies between 2 and 3. [3]
2. Use the iterative formula $$x_{n+1} = \sqrt[3]{17.5 - 2x_n},$$ with a suitable starting value, to find the real root of the equation \(2x^3 + 4x - 35 = 0\) correct to 2 decimal places. You should show the result of each iteration. [3]

The sequence defined by $$x_1 = 3, \quad x_{n+1} = \sqrt{31 - \frac{5}{2}x_n}$$ converges to the number \(\alpha\).

1. Find the value of \(\alpha\) correct to 3 decimal places, showing the result of each iteration. [3]
2. Find an equation of the form \(ax^3 + bx + c = 0\), where \(a\), \(b\) and \(c\) are integers, which has \(\alpha\) as a root. [3]

It is given that \(\int_a^{3a} (e^{5x} + e^x) dx = 100\), where \(a\) is a positive constant.

1. Show that \(a = \frac{1}{5}\ln(300 + 3e^a - 2e^{3a})\). [5]
2. Use an iterative process, based on the equation in part (i), to find the value of \(a\) correct to 4 decimal places. Use a starting value of 0.6 and show the result of each step of the process. [4]

$$f(x) = 2x^2 + 3 \ln (2 - x), \quad x \in \mathbb{R}, \quad x < 2.$$

1. Show that the equation \(f(x) = 0\) can be written in the form $$x = 2 - e^{kx^2},$$ where \(k\) is a constant to be found. [3]

The root, \(\alpha\), of the equation \(f(x) = 0\) is \(1.9\) correct to \(1\) decimal place.

1. Use the iteration formula $$x_{n+1} = 2 - e^{kx_n^2},$$ with \(x_0 = 1.9\) and your value of \(k\), to find \(\alpha\) to \(3\) decimal places and justify the accuracy of your answer. [5]
2. Solve the equation \(f'(x) = 0\). [5]

A curve has the equation \(y = (2x + 3)e^{-x}\).

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies in the interval \([-2, -1]\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3e^{x_n}}{e^{x_n} - 2}$$ with \(x_0 = -1\), to find \(x_1\), \(x_2\), \(x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [3]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]

$$\text{f}(x) = x^2 + 5x - 2 \sec x, \quad x \in \mathbb{R}, \quad -\frac{\pi}{2} < x < \frac{\pi}{2}.$$

1. Show that the equation \(\text{f}(x) = 0\) has a root, \(\alpha\), such that \(1 < \alpha < 1.5\) [2]
2. Show that a suitable rearrangement of the equation \(\text{f}(x) = 0\) leads to the iterative formula $$x_{n+1} = \cos^{-1} \left( \frac{2}{x_n^2 + 5x_n} \right).$$ [3]
3. Use the iterative formula in part (ii) with a starting value of 1.25 to find \(\alpha\) correct to 3 decimal places. You should show the result of each iteration. [3]

A curve has the equation \(y = (2x + 3)\mathrm{e}^{-x}\).

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies between \(-2\) and \(-1\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3\mathrm{e}^{x_n}}{\mathrm{e}^{x_n} - 2},$$ with \(x_0 = -1\), to find \(x_1, x_2, x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [2]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]

It is given that \(\alpha\) is the only real root of the equation \(x^3 + 2x - 28 = 0\) and that \(1.8 < \alpha < 2\).

1. The iteration \(x_{n+1} = \sqrt[3]{28 - 2x_n}\), with \(x_1 = 1.9\), is to be used to find \(\alpha\). Find the values of \(x_2\), \(x_3\) and \(x_4\), giving the answers correct to 7 decimal places. [3]
2. The error \(e_n\) is defined by \(e_n = \alpha - x_n\). Given that \(\alpha = 1.891 574 9\), correct to 7 decimal places, evaluate \(\frac{e_3}{e_2}\) and \(\frac{e_4}{e_3}\). Comment on these values in relation to the gradient of the curve with equation \(y = \sqrt[3]{28 - 2x}\) at \(x = \alpha\). [3]

The equation \(6\arcsin(2x - 1) - x^2 = 0\) has exactly one real root.

1. Show by calculation that the root lies between 0.5 and 0.6. [2]

In order to find the root, the iterative formula \(x_{n+1} = p + q\sin(rx_n^2)\), with initial value \(x_0 = 0.5\), is to be used.

1. Determine the values of the constants \(p\), \(q\) and \(r\). [2]
2. Hence find the root correct to 4 significant figures. Show the result of each step of the iteration process. [2]

In this question you must show detailed reasoning. \includegraphics{figure_5} The diagram shows the curve with equation \(y = \frac{2x - 3}{4x^2 + 1}\). The tangent to the curve at the point \(P\) has gradient 2.

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$4x^3 + 3x - 3 = 0.$$ [5]
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.5 and 1. [2]
3. Show that the iteration $$x_{n+1} = \frac{3 - 4x_n^3}{3}$$ cannot converge to the \(x\)-coordinate of \(P\) whatever starting value is used. [2]
4. Use the Newton-Raphson method, with initial value 0.5, to determine the coordinates of \(P\) correct to 5 decimal places. [5]

A car \(C\) is moving horizontally in a straight line with velocity \(v \text{ms}^{-1}\) at time \(t\) seconds, where \(v > 0\) and \(t \geq 0\). The acceleration, \(a \text{ms}^{-2}\), of \(C\) is modelled by the equation $$a = v\left(\frac{8t}{7 + 4t^2} - \frac{1}{2}\right).$$

1. In this question you must show detailed reasoning. Find the times when the acceleration of \(C\) is zero. [3] At \(t = 0\) the velocity of \(C\) is \(17.5 \text{ms}^{-1}\) and at \(t = T\) the velocity of \(C\) is \(5 \text{ms}^{-1}\).
2. By setting up and solving a differential equation, show that \(T\) satisfies the equation $$T = 2 \ln\left(\frac{7 + 4T^2}{2}\right).$$ [6]
3. Use an iterative formula, based on the equation in part (b), to find the value of \(T\), giving your answer correct to 4 significant figures. Use an initial value of 11.25 and show the result of each step of the iteration process. [2]
4. The diagram below shows the velocity-time graph for the motion of \(C\). \includegraphics{figure_7d} Find the time taken for \(C\) to decelerate from travelling at its maximum speed until it is travelling at \(5 \text{ms}^{-1}\). [1]

A curve passes through the point \((-2, 4.73)\) and satisfies the differential equation $$\frac{dy}{dx} = \frac{y^2 - x^2}{2x + 3y}$$ Use Euler's step by step method once, and then the midpoint formula $$y_{r+1} = y_{r-1} + 2hf(x_r, y_r), \quad x_{r+1} = x_r + h$$ once, each with a step length of \(0.02\), to estimate the value of \(y\) when \(x = -1.96\) Give your answer to five significant figures. [4 marks]

The equation $$1 + 5x - x^4 = 0$$ has a positive root \(\alpha\).

1. Show that \(\alpha\) lies between 1 and 2. [2]
2. Use the iterative sequence based on the arrangement $$x = \sqrt[4]{1+5x}$$ with starting value 1.5 to find \(\alpha\) correct to two decimal places. [3]
3. Use the Newton-Raphson method to find \(\alpha\) correct to six decimal places. [6]

It is given that there is exactly one value of \(x\), where \(0 < x < \pi\), that satisfies the equation $$3\tan 2x - 8\tan x = 4.$$

1. Show that \(t = \sqrt[3]{\frac{1}{2} + \frac{1}{3}t - \frac{1}{3}t^2}\), where \(t = \tan x\). [3]
2. Show by calculation that the value of \(t\) satisfying the equation in part (i) lies between 0.7 and 0.8. [2]
3. Use an iterative process based on the equation in part (i) to find the value of \(t\) correct to 4 significant figures. Use a starting value of 0.75 and show the result of each iteration. [3]
4. Solve the equation \(3\tan 4y - 8\tan 2y = 4\) for \(0 < y < \frac{1}{4}\pi\). [2]

The curve with equation \(y = f(x)\) where $$f(x) = x^2 + \ln(2x^2 - 4x + 5)$$ has a single turning point at \(x = \alpha\)

1. Show that \(\alpha\) is a solution of the equation $$2x^3 - 4x^2 + 7x - 2 = 0$$ [4]

The iterative formula $$x_{n+1} = \frac{1}{7}(2 + 4x_n^2 - 2x_n^3)$$ is used to find an approximate value for \(\alpha\). Starting with \(x_1 = 0.3\)

1. calculate, giving each answer to 4 decimal places,
   1. the value of \(x_2\)
   2. the value of \(x_4\)
   [2]

Using a suitable interval and a suitable function that should be stated,

1. show that \(\alpha\) is 0.341 to 3 decimal places. [2]

A curve \(C\) has equation \(y = f(x)\) where $$f(x) = x + 2\ln(e - x)$$

1. 1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left(\frac{e}{2-e}\right)x + 2$$ [6 marks]
   2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer. [3 marks]

1. The equation \(f(x) = 0\) has one positive root, \(\alpha\).
   1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer. [3 marks]
   2. Show that the roots of \(f(x) = 0\) satisfy the equation $$x = e - e^{-\frac{x}{2}}$$ [2 marks]
   3. Use the recurrence relation $$x_{n+1} = e - e^{-\frac{x_n}{2}}$$ with \(x_1 = 2\) to find the values of \(x_2\) and \(x_3\) giving your answers to three decimal places. [2 marks]
   4. Figure 1 below shows a sketch of the graphs of \(y = e - e^{-\frac{x}{2}}\) and \(y = x\), and the position of \(x_1\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\) and \(x_3\) on the \(x\)-axis. [2 marks] \includegraphics{figure_1}