Apply iteration to find root

4 The diagram shows part of the graph of \(y = \cos x\), where \(x\) is measured in radians. \includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-05_609_846_294_607}

1. Use the copy of this diagram in the Printed Answer Booklet to find an approximate solution to the equation \(x = \cos x\).
2. Use an iterative method to find the solution to the equation \(x = \cos x\) correct to 3 significant figures. You should show your first, second and last two iterations, writing down all the figures on your calculator.

10 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-7_579_764_255_651} The diagram shows the graph of \(\mathrm { f } ( x ) = \ln ( 3 x + 1 ) - x\), which has a stationary point at \(x = \alpha\). A student wishes to find the non-zero root \(\beta\) of the equation \(\ln ( 3 x + 1 ) - x = 0\) using the Newton-Raphson method.

1. (a) Determine the value of \(\alpha\).
   (b) Explain why the Newton-Raphson method will fail if \(\alpha\) is used as the initial value.
2. Show that the Newton-Raphson iterative formula for finding \(\beta\) can be written as $$x _ { n + 1 } = \frac { 3 x _ { n } - \left( 3 x _ { n } + 1 \right) \ln \left( 3 x _ { n } + 1 \right) } { 2 - 3 x _ { n } } .$$
3. Apply the iterative formula in part (ii) with initial value \(x _ { 1 } = 1\) to find the value of \(\beta\) correct to 5 significant figures. You should show the result of each iteration.
4. Use a change of sign method to verify that the value of \(\beta\) found in part (iii) is correct to 5 significant figures.

4. $$\mathrm { f } ( x ) = 3 \mathrm { e } ^ { x } - \frac { 1 } { 2 } \ln x - 2 , \quad x > 0 .$$

1. Differentiate to find \(\mathrm { f } ^ { \prime } ( x )\). The curve with equation \(y = \mathrm { f } ( x )\) has a turning point at \(P\). The \(x\)-coordinate of \(P\) is \(\alpha\).
2. Show that \(\alpha = \frac { 1 } { 6 } \mathrm { e } ^ { - \alpha }\). The iterative formula $$x _ { n + 1 } = \frac { 1 } { 6 } \mathrm { e } ^ { - x _ { n } } , \quad x _ { 0 } = 1$$ is used to find an approximate value for \(\alpha\).
3. Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 decimal places.
4. By considering the change of sign of \(\mathrm { f } ^ { \prime } ( x )\) in a suitable interval, prove that \(\alpha = 0.1443\) correct to 4 decimal places.

3. The root of the equation \(\mathrm { f } ( x ) = 0\), where $$f ( x ) = x + \ln 2 x - 4$$ is to be estimated using the iterative formula \(x _ { n + 1 } = 4 - \ln 2 x _ { n }\), with \(x _ { 0 } = 2.4\).

1. Showing your values of \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\), obtain the value, to 3 decimal places, of the root.
2. By considering the change of sign of \(\mathrm { f } ( x )\) in a suitable interval, justify the accuracy of your answer to part (a).

6 [Figure 1, printed on the insert, is provided for use in this question.]
The curve \(y = x ^ { 3 } + 4 x - 3\) intersects the \(x\)-axis at the point \(A\) where \(x = \alpha\).

1. Show that \(\alpha\) lies between 0.5 and 1.0.
2. Show that the equation \(x ^ { 3 } + 4 x - 3 = 0\) can be rearranged into the form \(x = \frac { 3 - x ^ { 3 } } { 4 }\).
   (1 mark)
3. 1. Use the iteration \(x _ { n + 1 } = \frac { 3 - x _ { n } { } ^ { 3 } } { 4 }\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\), giving your answer to two decimal places.
   2. The sketch on Figure 1 shows parts of the graphs of \(y = \frac { 3 - x ^ { 3 } } { 4 }\) 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.
      (3 marks)

6 [Figure 1, printed on the insert, is provided for use in this question.]
The curve \(y = x ^ { 3 } + 4 x - 3\) intersects the \(x\)-axis at the point \(A\) where \(x = \alpha\).

1. Show that \(\alpha\) lies between 0.5 and 1.0.
2. Show that the equation \(x ^ { 3 } + 4 x - 3 = 0\) can be rearranged into the form \(x = \frac { 3 - x ^ { 3 } } { 4 }\).
   (1 mark)
3. 1. Use the iteration \(x _ { n + 1 } = \frac { 3 - x _ { n } { } ^ { 3 } } { 4 }\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\), giving your answer to two decimal places.
      (3 marks)
   2. The sketch on Figure 1 shows parts of the graphs of \(y = \frac { 3 - x ^ { 3 } } { 4 }\) 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.
      (3 marks)

3 [Figure 1, printed on the insert, is provided for use in this question.]
The curve with equation \(y = x ^ { 3 } + 5 x - 4\) intersects the \(x\)-axis at the point \(A\), where \(x = \alpha\).

1. Show that \(\alpha\) lies between 0.5 and 1 .
2. Show that the equation \(x ^ { 3 } + 5 x - 4 = 0\) can be rearranged into the form $$x = \frac { 1 } { 5 } \left( 4 - x ^ { 3 } \right)$$
3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 4 - x _ { n } { } ^ { 3 } \right)\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\), giving your answer to three decimal places.
4. The sketch on Figure 1 shows parts of the graphs of \(y = \frac { 1 } { 5 } \left( 4 - x ^ { 3 } \right)\) 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.

4 [Figure 1, printed on the insert, is provided for use in this question.]

1. Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to \(\int _ { 1 } ^ { 2 } 3 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
2. The curve \(y = 3 ^ { x }\) intersects the line \(y = x + 3\) at the point where \(x = \alpha\).
   1. Show that \(\alpha\) lies between 0.5 and 1.5.
   2. Show that the equation \(3 ^ { x } = x + 3\) can be rearranged into the form $$x = \frac { \ln ( x + 3 ) } { \ln 3 }$$
   3. Use the iteration \(x _ { n + 1 } = \frac { \ln \left( x _ { n } + 3 \right) } { \ln 3 }\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\) to two significant figures.
   4. The sketch on Figure 1 shows part of the graphs of \(y = \frac { \ln ( x + 3 ) } { \ln 3 }\) 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 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x$$ where the angle \(2 x\) is measured in radians.
Starting at the point \(( 0.5,1 )\) on the curve, use a step-by-step method with a step length of 0.1 to estimate the value of \(y\) at \(x = 0.7\). Give your answer to three significant figures.
(6 marks)

2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \log _ { 10 } x$$ Starting at the point \(( 2,3 )\) on the curve, use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 2.4\). Give your answer to three decimal places.

5 A student is using the algorithm below to find an approximate value of \(\sqrt { 2 }\).
Line 10 Let \(A = 1 , B = 3 , C = 0\) Line \(20 \quad\) Let \(D = 1 , E = 2 , F = 0\) Line 30 Let \(G = B / E\) Line \(40 \quad\) Let \(H = G ^ { 2 }\) Line 50 If \(( H - 2 ) ^ { 2 } < 0.0001\) then go to Line 130
Line 60 Let \(C = 2 B + A\) Line 70 Let \(A = B\) Line 80 Let \(B = C\) Line 90 Let \(F = 2 E + D\) Line 100 Let \(D = E\) Line 110 Let \(E = F\) Line 120 Go to Line 30
Line 130 Print ' \(\sqrt { 2 }\) is approximately', \(B / E\) Line 140 Stop
Trace the algorithm.

6 A student is finding a numerical approximation for the area under a curve.
The algorithm that the student is using is as follows:
Line 10 Input \(A , B , N\) Line 20 Let \(T = 0\) Line 30 Let \(D = A\) Line \(40 \quad\) Let \(H = ( B - A ) / N\) Line \(50 \quad\) Let \(E = H / 2\) Line 60 Let \(T = T + A ^ { 3 } + B ^ { 3 }\) Line \(70 \quad\) Let \(D = D + H\) Line 80 If \(D = B\) then go to line 110
Line 90 Let \(T = T + 2 D ^ { 3 }\) Line 100 Go to line 70
Line \(110 \quad\) Print 'Area \(=\), \(T \times E\) Line 120 End
Trace the algorithm in the case where the input values are:

1. \(A = 1 , B = 5 , N = 2\);
2. \(A = 1 , B = 5 , N = 4\).

5 A student is using the following algorithm with different values of \(X\).

1. Trace the algorithm, giving your answers to three decimal places where appropriate:
   1. in the case where the input value of \(X\) is 2 ;
   2. in the case where the input value of \(X\) is - 6 .
2. Another student used the same algorithm but omitted LINE 70. Describe the outcome for this student.

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

1. Show by calculation that the root lies between 0.5 and 0.6 . In order to find the root, the iterative formula $$x _ { n + 1 } = p + q \sin \left( r x _ { n } ^ { 2 } \right)$$ with initial value \(x _ { 0 } = 0.5\), is to be used.
2. Determine the values of the constants \(p , q\) and \(r\).
3. Hence find the root correct to \(\mathbf { 4 }\) significant figures. Show the result of each step of the iteration process.

11 The daily world production of oil can be modelled using $$V = 10 + 100 \left( \frac { t } { 30 } \right) ^ { 3 } - 50 \left( \frac { t } { 30 } \right) ^ { 4 }$$ where \(V\) is volume of oil in millions of barrels, and \(t\) is time in years since 1 January 1980. 11

1. 1. The model is used to predict the time, \(T\), when oil production will fall to zero.
      Show that \(T\) satisfies the equation $$T = \sqrt [ 3 ] { 60 T ^ { 2 } + \frac { 162000 } { T } }$$ 11
2. (ii) Use the iterative formula \(T _ { n + 1 } = \sqrt [ 3 ] { 60 T _ { n } { } ^ { 2 } + \frac { 162000 } { T _ { n } } }\), with \(T _ { 0 } = 38\), to find the values of \(T _ { 1 } , T _ { 2 }\), and \(T _ { 3 }\), giving your answers to three decimal places.
   11
3. (iii) Explain the relevance of using \(T _ { 0 } = 38\) 11
4. From 1 January 1980 the daily use of oil by one technologically developing country can be modelled as $$V = 4.5 \times 1.063 ^ { t }$$ Use the models to show that the country's use of oil and the world production of oil will be equal during the year 2029.
   [0pt] [4 marks] \(12 \quad \mathrm { p } ( x ) = 30 x ^ { 3 } - 7 x ^ { 2 } - 7 x + 2\)

7

1. By sketching the graphs of \(y = \frac { 1 } { x }\) and \(y = \sec 2 x\) on the axes below, show that the equation $$\frac { 1 } { x } = \sec 2 x$$ has exactly one solution for \(x > 0\) \includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-08_675_771_689_639} 7
2. By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6
   7
3. Show that the equation can be rearranged to give $$x = \frac { 1 } { 2 } \cos ^ { - 1 } x$$ 7
4. 1. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \cos ^ { - 1 } x _ { n }$$ with \(x _ { 1 } = 0.4\), to find \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to four decimal places.
      7
5. (ii) On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
   [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-09_954_1600_1717_223}

13 The function f is defined by $$\mathrm { f } ( x ) = \arccos x \text { for } 0 \leq x \leq a$$ The curve with equation \(y = \mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-18_842_837_550_603} 13

1. State the value of \(a\) 13
2. 1. On the diagram above, sketch the curve with equation $$y = \cos x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ and
      sketch the line with equation $$y = x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ 13
3. (ii) Explain why the solution to the equation $$x - \cos x = 0$$ must also be a solution to the equation $$\cos x = \arccos x$$ Question 13 continues on the next page 13
4. Use the Newton-Raphson method with \(x _ { 0 } = 0\) to find an approximate solution, \(x _ { 3 }\), to the equation $$x - \cos x = 0$$ Give your answer to four decimal places. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-21_2491_1716_219_153}

14

1. The equation $$x ^ { 3 } = \mathrm { e } ^ { 6 - 2 x }$$ has a single solution, \(x = \alpha\) By considering a suitable change of sign, show that \(\alpha\) lies between 0 and 4
   14
2. Show that the equation \(x ^ { 3 } = \mathrm { e } ^ { 6 - 2 x }\) can be rearranged to give $$x = 3 - \frac { 3 } { 2 } \ln x$$ 14
3. 1. Use the iterative formula $$x _ { n + 1 } = 3 - \frac { 3 } { 2 } \ln x _ { n }$$ with \(x _ { 1 } = 4\), to find \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) Give your answers to three decimal places.
      14
4. (ii) Figure 1 below shows a sketch of parts of the graphs of $$y = 3 - \frac { 3 } { 2 } \ln x \text { and } y = x$$ On Figure 1, draw a staircase or cobweb diagram to show how convergence takes place.
   Label, on the \(x\)-axis, the positions of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) [0pt] [2 marks] \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0320e0a6-adc0-440a-b1da-d1a49fe06179-22_1328_1390_744_395}
   \end{figure} 14
5. (iii) Explain why the iterative formula $$x _ { n + 1 } = 3 - \frac { 3 } { 2 } \ln x _ { n }$$ fails to converge to \(\alpha\) when the starting value is \(x _ { 1 } = 0\)