Sign Change & Interval Methods

2 By considering a change of sign, show that the equation \(\mathrm { e } ^ { x } - 5 x ^ { 3 } = 0\) has a root between 0 and 1 .

2. (a) Show that \(\mathrm { f } ( x ) = x ^ { 4 } + x - 1\) has a real root \(\alpha\) in the interval [0.5, 1.0].
[0pt] (b) Starting with the interval [0.5, 1.0], use interval bisection twice to find an interval of width 0.125 which contains \(\alpha\).
(c) Taking 0.75 as a first approximation, apply the Newton Raphson process twice to \(\mathrm { f } ( x )\) to obtain an approximate value of \(\alpha\). Give your answer to 3 decimal places.

1. (a) On the same diagram, sketch and clearly label the graphs with equations

$$y = \mathrm { e } ^ { x } \quad \text { and } \quad y = 10 - x$$ Show on your sketch the coordinates of each point at which the graphs cut the axes.
(b) Explain why the equation \(\mathrm { e } ^ { x } - 10 + x = 0\) has only one solution.
(c) Show that the solution of the equation $$\mathrm { e } ^ { x } - 10 + x = 0$$ lies between \(x = 2\) and \(x = 3\)
(d) Use the iterative formula $$x _ { n + 1 } = \ln \left( 10 - x _ { n } \right) , \quad x _ { 1 } = 2$$ to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
Give your answers to 4 decimal places.

\(\mathbf { 4 }\) & 11 & 11 & 7 & 14 & 14
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1 The table shows some values of \(x\), together with the associated values of a function, \(\mathrm { f } ( x )\).

1. Use the information in the table to calculate the most accurate estimate of \(f ^ { \prime } ( 2 )\) possible.
2. Calculate an estimate of the error when \(f ( 2 )\) is used as an estimate of \(f ( 2.05 )\).

2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ^ { x }$$ Starting at the point \(( 1,4 )\) on the curve, use a step-by-step method with a step length of 0.01 to estimate the value of \(y\) at \(x = 1.02\). Give your answer to six significant figures.

1. Use Simpson's rule with 4 intervals to estimate

$$\int _ { 0.4 } ^ { 2 } e ^ { x ^ { 2 } } d x$$

1

1. Solve the following simultaneous equations. $$\begin{aligned} & x + \quad y = 1
   & x + 0.99 y = 2 \end{aligned}$$
2. The coefficient 0.99 is correct to two decimal places. All other coefficients in the equations are exact. With the aid of suitable calculations, explain why your answer to part (i) is unreliable.

1

1. Show that the equation $$x ^ { 3 } + 2 x - 2 = 0$$ has a root between 0.5 and 1 .
2. Use linear interpolation once to find an estimate of this root. Give your answer to two decimal places.

4
\includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-2_358_618_1082_762} The diagram shows a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(O\). The chord \(A B\) divides the sector into a triangle \(A O B\) and a segment \(A X B\). Angle \(A O B\) is \(\theta\) radians.

1. In the case where the areas of the triangle \(A O B\) and the segment \(A X B\) are equal, find the value of the constant \(p\) for which \(\theta = p \sin \theta\).
2. In the case where \(r = 8\) and \(\theta = 2.4\), find the perimeter of the segment \(A X B\).

2 Solve the equation $$\ln ( 1 + x ) = 1 + \ln x$$ giving your answer correct to 2 significant figures.

1.
\(\begin{array} { l l l l l l l l l l } 17 & 9 & 15 & 8 & 20 & 13 & 28 & 4 & 12 & 5 \end{array}\)

2 Fig. 2 shows 3 values of \(x\) and the associated values of a function, \(\mathrm { f } ( x )\). \begin{table}[h] \end{table} Find a polynomial \(p ( x )\) of degree 2 to approximate \(\mathrm { f } ( x )\), giving your answer in the form \(p ( x ) = a x ^ { 2 } + b x + c\), where \(a\), \(b\) and \(c\) are constants to be determined.

2

1. Use the trapezium rule, with four strips each of width 0.5 , to estimate the value of $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x ^ { 2 } } \mathrm {~d} x$$ giving your answer correct to 3 significant figures.
2. Explain how the trapezium rule could be used to obtain a more accurate estimate.

3. This question should be answered on the sheet provided. Arthur is planning a bus journey from town \(A\) to town \(L\). There are various routes he can take but he will have to change buses three times - at \(B , C\) or \(D\), at \(E , F , G\) or \(H\) and at \(I , J\) or \(K\). \begin{figure}[h] \end{figure} Figure 2 shows the bus routes that Arthur can use. The number on each arc shows the average waiting time, in minutes, for a bus to come on that route. As the forecast is for rain, Arthur wishes to plan his journey so that the maximum waiting time at any one stop is as small as possible. Use dynamic programming to find the route that Arthur should use.
(9 marks)

1 Hussain wants to travel by train from Edinburgh to Southampton, leaving Edinburgh after 9 am and arriving in Southampton by 4 pm . He wants to leave Edinburgh as late as possible.
Hussain rings the train company to find out about the train times. Write down a question he might ask that leads to
(A) an existence problem,
(B) an optimisation problem.

2

1. Show that the equation $$x ^ { 3 } + x - 7 = 0$$ has a root between 1.6 and 1.8.
2. Use interval bisection twice, starting with the interval in part (a), to give this root to one decimal place.

2 The diagram shows part of the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic polynomial in \(x\).
\includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-04_437_620_909_274} Explain why one of the roots of the equation \(\mathrm { f } ( x ) = 0\) cannot be found by the sign change method.

1 The flowchart below has positive inputs \(X , Y\) and \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{74b6f747-7045-4902-8b21-0b59c007f7f6-2_1274_643_392_242}

1. Trace through the flowchart above using the inputs \(X = 1 , Y = 2\) and \(M = 2\). You only need to record values when they change.
2. Explain why the process in the flowchart is finite.

8 The graph of \(\mathrm { y } = 0.2 \cosh \mathrm { x } - 0.4 \mathrm { x }\) for values of \(x\) from 0 to 3.32 is shown on the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{4023e87c-34b1-4abd-9acc-ede5e4d68c7f-08_988_1561_312_244} The equation \(0.2 \cosh x - 0.4 x = 0\) has two roots, \(\alpha\) and \(\beta\) where \(\alpha < \beta\), in the interval \(0 < x < 3\). The secant method with \(x _ { 0 } = 1\) and \(x _ { 1 } = 2\) is to be used to find \(\beta\).

1. On the copy of the graph in the Printed Answer Booklet, show how the secant method works with these two values of \(x\) to obtain an improved approximation to \(\beta\). The spreadsheet output in the table below shows the result of applying the secant method with \(x _ { 0 } = 1\) and \(x _ { 1 } = 2\).

   	I	J	K	L	M
   2	\(r\)	\(\mathrm { x } _ { \mathrm { r } }\)	f(x)	\(\mathrm { X } _ { \mathrm { r } + 1 }\)	\(\mathrm { f } \left( \mathrm { x } _ { \mathrm { r } + 1 } \right)\)
   3	0	1	-0.0914	2	-0.0476
   4	1	2	-0.0476	3.08529	0.95784
   5	2	3.08529	0.95784	2.05134	-0.0298
   6	3	2.05134	-0.0298	2.08259	-0.0181
   7	4	2.08259	-0.0181	2.13042	0.00155
   8	5	2.13042	0.00155	2.12664	\(- 7 \mathrm { E } - 05\)

2. Write down a suitable cell formula for cell J4.
3. Write down a suitable cell formula for cell L4.
4. Write down the most accurate approximation to \(\beta\) which is displayed in the table.
5. Determine whether your answer to part (d) is correct to 5 decimal places. You should not calculate any more iterates.
6. It is decided to use the secant method with starting values \(x _ { 0 } = 1\) and \(\mathrm { x } _ { 1 } = \mathrm { a }\), where \(a > 1\), to find \(\alpha\). State a suitable value for \(a\).

2 The following spreadsheet printout shows the bisection method being applied to the equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = \mathrm { e } ^ { x } - x ^ { 2 } - 2\). Some values of \(\mathrm { f } ( x )\) are shown in columns B and D.

1. The formula in cell A 3 is \(= \mathrm { IF } ( \mathrm { F } 2 > 0\), A2, E2). State the purpose of this formula.
2. The formula in cell C 3 is \(= \mathrm { IF } ( \mathrm { F } 2 > 0 , \ldots , \ldots )\). What are the missing cell references?
3. In which row is the magnitude of the maximum possible error (mpe) less than \(5 \times 10 ^ { - 7 }\) for the first time?

3. $$\mathrm { f } ( x ) = x ^ { 2 } + \frac { 3 } { x } - 1 , \quad x < 0$$ The only real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ - 2 , - 1 ]\).

1. Taking - 1.5 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 2 decimal places.
2. Show that your answer to part (a) gives \(\alpha\) correct to 2 decimal places.
   
   tion 3continued -

5

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 12 } \ln \left( x ^ { 2 } + 5 \right) \mathrm { d } x\), giving your answer to three significant figures.
2. A curve has equation \(y = \ln \left( x ^ { 2 } + 5 \right)\).
   1. Show that this equation can be rewritten as \(x ^ { 2 } = \mathrm { e } ^ { y } - 5\).
   2. The region bounded by the curve, the lines \(y = 5\) and \(y = 10\) and the \(y\)-axis is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the exact value of the volume of the solid generated.
3. The graph with equation \(y = \ln \left( x ^ { 2 } + 5 \right)\) is stretched with scale factor 4 parallel to the \(x\)-axis, and then translated through \(\left[ \begin{array} { l } 0
   3 \end{array} \right]\) to give the graph with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

2 Find the real root of the equation \(\frac { 2 \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { 2 + \mathrm { e } ^ { x } } = 3\), giving your answer correct to 3 decimal places. Your working should show clearly that the equation has only one real root.

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.

7 The equation \(x ^ { 2 } = x ^ { 3 } + x - 3\) has a single solution, \(x = \alpha\)
7

1. By considering a suitable change of sign, show that \(\alpha\) lies between 1.5 and 1.6
   [0pt] [2 marks]
   7
2. Show that the equation \(x ^ { 2 } = x ^ { 3 } + x - 3\) can be rearranged into the form $$x ^ { 2 } = x - 1 + \frac { 3 } { x }$$ 7
3. Use the iterative formula $$x _ { n + 1 } = \sqrt { x _ { n } - 1 + \frac { 3 } { x _ { n } } }$$ with \(x _ { 1 } = 1.5\), to find \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to four decimal places.
   7
4. Hence, deduce an interval of width 0.001 in which \(\alpha\) lies.

1. Find a small positive value of \(x\) which is an approximate solution of the equation.

$$\cos x - 4 \sin x = x ^ { 2 }$$

9. The diagram shows the curve with equation \(y = 2 x - 3 \ln ( 2 x + 5 )\) and the normal to the curve at the point \(P ( - 2 , - 4 )\).

1. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) intersects the curve again at the point \(Q\) with \(x\)-coordinate \(q\).
2. Show that \(1 < q < 2\).
3. Show that \(q\) is a solution of the equation $$x = \frac { 12 } { 7 } \ln ( 2 x + 5 ) - 2 .$$
4. Use an iterative process based on the equation above with a starting value of 1.5 to find the value of \(q\) to 3 significant figures and justify the accuracy of your answer.