Sign Change with Function Evaluation

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = 6 \ln ( 2 x + 3 ) - \frac { 1 } { 2 } x ^ { 2 } + 4 \quad x > - \frac { 3 } { 2 }$$ The curve cuts the negative \(x\)-axis at the point \(P\), as shown in Figure 1.

1. Show that the \(x\) coordinate of \(P\) lies in the interval \([ - 1.25 , - 1.2 ]\) The curve cuts the positive \(x\)-axis at the point \(Q\), also shown in Figure 1.
   Using the iterative formula $$x _ { n + 1 } = \sqrt { 12 \ln \left( 2 x _ { n } + 3 \right) + 8 } \text { with } x _ { 1 } = 6$$
2. 1. find, to 4 decimal places, the value of \(x _ { 2 }\)
   2. find, by continued iteration, the \(x\) coordinate of \(Q\). Give your answer to 4 decimal places. The curve has a maximum turning point at \(M\), as shown in Figure 1.
3. Using calculus and showing each stage of your working, find the \(x\) coordinate of \(M\).

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 x + 1 \right) \mathrm { e } ^ { x ^ { 2 } }$$ The curve cuts the \(x\)-axis at points \(A\) and \(B\) as shown in Figure 2 .

1. Calculate the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\), giving your answers to 3 decimal places.
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The curve has a minimum turning point at the point \(P\) as shown in Figure 2.
3. Show that the \(x\) coordinate of \(P\) is the solution of $$x = - \frac { 3 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 2 \right) }$$
4. Use the iteration formula $$x _ { n + 1 } = - \frac { 3 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 2 \right) } , \quad \text { with } x _ { 0 } = - 2.4$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places. The \(x\) coordinate of \(P\) is \(\alpha\).
5. By choosing a suitable interval, prove that \(\alpha = - 2.43\) to 2 decimal places.

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 2 \cos \left( \frac { 1 } { 2 } x ^ { 2 } \right) + x ^ { 3 } - 3 x - 2$$ The curve crosses the \(x\)-axis at the point \(Q\) and has a minimum turning point at \(R\).

1. Show that the \(x\) coordinate of \(Q\) lies between 2.1 and 2.2
2. Show that the \(x\) coordinate of \(R\) is a solution of the equation $$x = \sqrt { 1 + \frac { 2 } { 3 } x \sin \left( \frac { 1 } { 2 } x ^ { 2 } \right) }$$ Using the iterative formula $$x _ { n + 1 } = \sqrt { 1 + \frac { 2 } { 3 } x _ { n } \sin \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } \right) } , \quad x _ { 0 } = 1.3$$
3. find the values of \(x _ { 1 }\) and \(x _ { 2 }\) to 3 decimal places.

8
\includegraphics[max width=\textwidth, alt={}, center]{e0e2a26b-d4d6-46ea-ac12-a882f3465e5e-3_588_915_954_614} The diagram shows part of each of the curves \(y = e ^ { \frac { 1 } { 5 } x }\) and \(y = \sqrt [ 3 ] { } ( 3 x + 8 )\). 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.
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 5 } { 3 } \ln ( 3 x + 8 )\).
3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(P\) correct to 2 decimal places.
4. Use integration, and your answer to part (iii), to find an approximate value of the area of the region \(R\).
   \includegraphics[max width=\textwidth, alt={}, center]{e0e2a26b-d4d6-46ea-ac12-a882f3465e5e-4_625_647_264_749} The function f is defined by \(\mathrm { f } ( x ) = \sqrt { } ( m x + 7 ) - 4\), where \(x \geqslant - \frac { 7 } { m }\) and \(m\) is a positive constant. The diagram shows the curve \(y = \mathrm { f } ( x )\).
5. A sequence of transformations maps the curve \(y = \sqrt { } x\) to the curve \(y = \mathrm { f } ( x )\). Give details of these transformations.
6. Explain how you can tell that f is a one-one function and find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
7. It is given that the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) do not meet. Explain how it can be deduced that neither curve meets the line \(y = x\), and hence determine the set of possible values of \(m\). [5]

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$f ( x ) = 4 \cos 2 x - 2 x + 1 \quad x > 0$$ and where \(x\) is measured in radians.
The curve crosses the \(x\)-axis at the point \(A\), as shown in figure 1 .
Given that \(x\)-coordinate of \(A\) is \(\alpha\)
a. show that \(\alpha\) lies between 0.7 and 0.8 Given that \(x\)-coordinates of \(B\) and \(C\) are \(\beta\) and \(\gamma\) respectively and they are two smallest values of \(x\) at which local maxima occur
b. find, using calculus, the value of \(\beta\) and the value of \(\gamma\), giving your answers to 3 significant figures.
c. taking \(x _ { 0 } = 0.7\) or 0.8 as a first approximation to \(\alpha\), apply the Newton-Raphson method once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Show, your method and give your answer to 2 significant figures.

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 .

9 This question is about the equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } - x - \frac { 1 } { 3 x - 2 }\).
Fig. 9.1 shows the curve \(y = f ( x )\).
Fig. 9.1
\includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-06_940_929_518_239}

1. Show, by calculation, that the equation \(\mathrm { f } ( x ) = 0\) has a root between \(x = 1\) and \(x = 2\).
2. Fig. 9.2 shows part of a spreadsheet being used to find a root of the equation. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 9.2}

   	A	B
   1	\(x\)	\(f ( x )\)
   2	1.5	3.1625
   3	1.25	0.619977679
   4	1.125	- 0.250466087
   5

   \end{table} Write down a suitable number to use as the next value of \(x\) in the spreadsheet.
3. Determine a root of the equation \(\mathrm { f } ( x ) = 0\). Give your answer correct to \(\mathbf { 1 }\) decimal place.
4. Fig. 9.3 shows a similar spreadsheet being used to search for another root of \(\mathrm { f } ( x ) = 0\). \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 9.3}

   	A	B
   1	x	f(x)
   2	0	0.5
   3	1	-1
   4	0.5	1.5625
   5	0.75	-4.4336
   6	0.6	4.5296
   7	0.7	-10.4599
   8	0.65	19.5285
   9	0.675	-40.4674
   10	0.6625	79.5301
   11	0.66875	-160.4687
   10

   \end{table}
   1. Explain why it looks from rows 2 and 3 of the spreadsheet as if there is a root between 0 and 1.
   2. Explain why this process will not find a root between 0 and 1 .

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 10 + \ln ( 3 x ) - \frac { 1 } { 2 } \mathrm { e } ^ { x } , 0.1 \leq x \leq 3.3 .$$ Given that \(\mathrm { f } ( k ) = 0\),

1. show, by calculation, that \(3.1 < k < 3.2\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The tangent to the graph at \(x = 1\) intersects the \(y\)-axis at the point \(P\).
3. 1. Find an equation of this tangent.
   2. Find the exact \(y\)-coordinate of \(P\), giving your answer in the form \(a + \ln b\).

9. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = x + 2 \ln ( \mathrm { e } - x )$$

1. 1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left( \frac { \mathrm { e } } { 2 - \mathrm { e } } \right) x + 2$$
   2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer.
2. The equation \(\mathrm { f } ( x ) = 0\) has one positive root, \(\alpha\).
   1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer.
   2. Show that the roots of \(\mathrm { f } ( x ) = 0\) satisfy the equation $$x = \mathrm { e } - \mathrm { e } ^ { - \frac { x } { 2 } }$$ [2 marks]
   3. Use the recurrence relation $$x _ { n + 1 } = \mathrm { e } - \mathrm { e } ^ { - \frac { x _ { n } } { 2 } }$$ with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\) giving your answers to three decimal places.
      [0pt] [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.
      [0pt] [2 marks] \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{75e73922-4324-441c-b942-c709a71d9025-22_1236_1566_1519_360}
      \end{figure} [BLANK PAGE]
      [0pt] [BLANK PAGE]
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      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]

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\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-03_579_901_959_623} The diagram shows part of each of the curves \(y = e ^ { \frac { 1 } { 5 } x }\) and \(y = \sqrt [ 3 ] { } ( 3 x + 8 )\). 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.
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 5 } { 3 } \ln ( 3 x + 8 )\).
3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(P\) correct to 2 decimal places.
4. Use integration, and your answer to part (iii), to find an approximate value of the area of the region \(R\).

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200 3 A curve is defined by the parametric equations $$x = t ^ { 3 } + 2 , \quad y = t ^ { 2 } - 1$$ 3

1. Find the gradient of the curve at the point where \(t = - 2\)
   [0pt] [4 marks]
   3
2. Find a Cartesian equation of the curve.
   4 The equation \(x ^ { 3 } - 3 x + 1 = 0\) has three real roots. 4
3. Show that one of the roots lies between - 2 and - 1
   4
4. Taking \(x _ { 1 } = - 2\) as the first approximation to one of the roots, use the Newton-Raphson method to find \(x _ { 2 }\), the second approximation.
   [0pt] [3 marks]
   4
5. Explain why the Newton-Raphson method fails in the case when the first approximation is \(x _ { 1 } = - 1\)
   [0pt] [1 mark]

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200 3 A curve is defined by the parametric equations $$x = t ^ { 3 } + 2 , \quad y = t ^ { 2 } - 1$$ 3

1. Find the gradient of the curve at the point where \(t = - 2\)
   [0pt] [4 marks]
   3
2. Find a Cartesian equation of the curve.
   4 The equation \(x ^ { 3 } - 3 x + 1 = 0\) has three real roots. 4
3. Show that one of the roots lies between - 2 and - 1
   4
4. Taking \(x _ { 1 } = - 2\) as the first approximation to one of the roots, use the Newton-Raphson method to find \(x _ { 2 }\), the second approximation.
   [0pt] [3 marks]
   4
5. Explain why the Newton-Raphson method fails in the case when the first approximation is \(x _ { 1 } = - 1\)
   [0pt] [1 mark]