Rearrange to iterative form

5

1. Given that \(2 \ln ( x + 1 ) + \ln x = \ln ( x + 9 )\), show that \(x = \sqrt { \frac { 9 } { x + 2 } }\).
2. It is given that the equation \(x = \sqrt { \frac { 9 } { x + 2 } }\) has a single root. Show by calculation that this root lies between 1.5 and 2.0.
3. Use an iterative formula, based on the equation in part (b), to find the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{b9556031-871d-4dd3-9523-e3438a41339f-3_655_685_262_730} The diagram shows the curve \(y = x \mathrm { e } ^ { 2 x }\) and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Show that the curve intersects the line \(y = 20\) at the point whose \(x\)-coordinate is the root of the equation $$x = \frac { 1 } { 2 } \ln \left( \frac { 20 } { x } \right)$$
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 20 } { x _ { n } } \right)$$ with initial value \(x _ { 1 } = 1.3\), to calculate the root correct to 2 decimal places, giving the result of each iteration to 4 decimal places.

6

1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 4 x - 2$$ where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.9\).
3. Show that this root also satisfies the equation $$x = \frac { 1 + 2 \tan x } { 4 \tan x }$$
4. Use the iterative formula \(x _ { n + 1 } = \frac { 1 + 2 \tan x _ { n } } { 4 \tan x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. By sketching a suitable pair of graphs, show that the equation $$3 \mathrm { e } ^ { x } = 8 - 2 x$$ has only one root.
2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.8\).
3. Show that this root also satisfies the equation $$x = \ln \left( \frac { 8 - 2 x } { 3 } \right)$$
4. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 8 - 2 x _ { n } } { 3 } \right)\) to determine this root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

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.

10

1. It is given that \(2 \tan 2 x + 5 \tan ^ { 2 } x = 0\). Denoting \(\tan x\) by \(t\), form an equation in \(t\) and hence show that either \(t = 0\) or \(t = \sqrt [ 3 ] { } ( t + 0.8 )\).
2. It is given that there is exactly one real value of \(t\) satisfying the equation \(t = \sqrt [ 3 ] { } ( t + 0.8 )\). Verify by calculation that this value lies between 1.2 and 1.3 .
3. Use the iterative formula \(t _ { n + 1 } = \sqrt [ 3 ] { } \left( t _ { n } + 0.8 \right)\) to find the value of \(t\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
4. Using the values of \(t\) found in previous parts of the question, solve the equation $$2 \tan 2 x + 5 \tan ^ { 2 } x = 0$$ for \(- \pi \leqslant x \leqslant \pi\).

6 \includegraphics[max width=\textwidth, alt={}, center]{87392b1c-3683-45b4-8d55-36760b5f0cc1-10_547_531_260_806} The diagram shows the curve \(y = x ^ { 4 } - 2 x ^ { 3 } - 7 x - 6\). The curve intersects the \(x\)-axis at the points \(( a , 0 )\) and \(( b , 0 )\), where \(a < b\). It is given that \(b\) is an integer.

1. Find the value of \(b\).
2. Hence show that \(a\) satisfies the equation \(a = - \frac { 1 } { 3 } \left( 2 + a ^ { 2 } + a ^ { 3 } \right)\).
3. 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.

3 The equation \(x ^ { 5 } - 3 x ^ { 3 } + x ^ { 2 } - 4 = 0\) has one positive root.

1. Verify by calculation that this root lies between 1 and 2 .
2. Show that the equation can be rearranged in the form $$\left. x = \sqrt [ 3 ] { ( } 3 x + \frac { 4 } { x ^ { 2 } } - 1 \right)$$
3. Use an iterative formula based on this rearrangement to determine the positive root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. By sketching a suitable pair of graphs, show that the equation $$2 - x = \ln x$$ has only one root.
2. Verify by calculation that this root lies between 1.4 and 1.7.
3. Show that this root also satisfies the equation $$x = \frac { 1 } { 3 } ( 4 + x - 2 \ln x )$$
4. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( 4 + x _ { n } - 2 \ln x _ { n } \right)$$ with initial value \(x _ { 1 } = 1.5\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5

1. By sketching a suitable pair of graphs, show that the equation $$\sec x = 3 - \frac { 1 } { 2 } x ^ { 2 }$$ 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 1 and 1.4.
3. Show that this root also satisfies the equation $$x = \cos ^ { - 1 } \left( \frac { 2 } { 6 - x ^ { 2 } } \right)$$
4. Use an iterative formula based on the equation in part (iii) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_561_732_255_705} The diagram shows the curve \(y = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 4 x - 16\), which crosses the \(x\)-axis at the points \(( \alpha , 0 )\) and \(( \beta , 0 )\) where \(\alpha < \beta\). It is given that \(\alpha\) is an integer.

1. Find the value of \(\alpha\).
2. Show that \(\beta\) satisfies the equation \(x = \sqrt [ 3 ] { } ( 8 - 2 x )\).
3. Use an iteration process based on the equation in part (ii) to find the value of \(\beta\) correct to 2 decimal places. Show the result of each iteration to 4 decimal places.

5 The equation of a curve is \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 x + 1 }\) and the point \(P\) on the curve has \(y\)-coordinate 10 .

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 2 } \ln ( 40 x + 10 )\).
2. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 40 x _ { n } + 10 \right)\) with \(x _ { 1 } = 2.3\) to find the \(x\)-coordinate of \(P\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
3. Find the gradient of the curve at \(P\), giving the answer correct to 3 significant figures.

6

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) in the interval \(0 < x < \frac { 1 } { 2 } \pi\) that is a root of the equation $$\cot x = x$$
2. Verify by calculation that this root lies between 0.8 and 0.9 radians.
3. Show that this value of \(x\) is also a root of the equation $$x = \tan ^ { - 1 } \left( \frac { 1 } { x } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { x _ { n } } \right)$$ to determine this root correct to 2 decimal places, showing the result of each iteration.

5

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) that is a root of the equation $$\frac { 1 } { x } = \ln x$$
2. Verify by calculation that this root lies between 1 and 2 .
3. Show that this root also satisfies the equation $$x = \mathrm { e } ^ { \frac { 1 } { x } }$$
4. Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \frac { 1 } { x _ { n } } }$$ with initial value \(x _ { 1 } = 1.8\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. Verify by calculation that the cubic equation $$x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3 = 0$$ has a root that lies between \(x = 0.7\) and \(x = 0.8\).
2. Show that this root also satisfies an equation of the form $$x = \frac { a x ^ { 2 } + 3 } { x ^ { 2 } + b }$$ where the values of \(a\) and \(b\) are to be found.
3. With these values of \(a\) and \(b\), use the iterative formula $$x _ { n + 1 } = \frac { a x _ { n } ^ { 2 } + 3 } { x _ { n } ^ { 2 } + b }$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 \includegraphics[max width=\textwidth, alt={}, center]{faf83d93-40b6-4557-bfd5-f94c67470dfa-2_449_639_388_753} The diagram shows the curve \(y = x ^ { 4 } + 2 x - 9\). The curve cuts the positive \(x\)-axis at the point \(P\).

1. Verify by calculation that the \(x\)-coordinate of \(P\) lies between 1.5 and 1.6.
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt [ 3 ] { \left( \frac { 9 } { x } - 2 \right) }$$
3. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { \left( \frac { 9 } { x _ { n } } - 2 \right) }$$ to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 \includegraphics[max width=\textwidth, alt={}, center]{a3e778cb-9f95-4750-ba49-a57ee22af018-2_449_639_388_753} The diagram shows the curve \(y = x ^ { 4 } + 2 x - 9\). The curve cuts the positive \(x\)-axis at the point \(P\).

1. Verify by calculation that the \(x\)-coordinate of \(P\) lies between 1.5 and 1.6.
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt [ 3 ] { \left( \frac { 9 } { x } - 2 \right) }$$
3. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { \left( \frac { 9 } { x _ { n } } - 2 \right) }$$ to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{c703565b-8aa8-424b-9684-6592d4effdf8-3_597_931_260_607} The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 4 } - 3 x ^ { 3 } + 3 x ^ { 2 } - 25 x + 48 .$$ The diagram shows the curve \(y = \mathrm { p } ( x )\) which crosses the \(x\)-axis at ( \(\alpha , 0\) ) and ( 3,0 ).

1. Divide \(\mathrm { p } ( x )\) by a suitable linear factor and hence show that \(\alpha\) is a root of the equation \(x = \sqrt [ 3 ] { } ( 16 - 3 x )\).
2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 16 - 3 x _ { n } \right)\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4 \includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-06_652_789_260_676} The diagram shows the curve with equation $$y = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 12 x - 32$$ The curve crosses the \(x\)-axis at points with coordinates \(( \alpha , 0 )\) and \(( \beta , 0 )\).

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of $$x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 12 x - 32$$
2. Show that \(\beta\) satisfies an equation of the form \(x = \sqrt [ 3 ] { } ( p + q x )\), and state the values of \(p\) and \(q\). [3]
3. Use an iterative formula based on the equation in part (ii) to find the value of \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 The curve with equation $$y = 5 \mathrm { e } ^ { 2 x } - 8 x ^ { 2 } - 20$$ crosses the \(x\)-axis at only one point. This point has coordinates \(( p , 0 )\).

1. Show that \(p\) satisfies the equation \(x = \frac { 1 } { 2 } \ln \left( 1.6 x ^ { 2 } + 4 \right)\).
2. Show by calculation that \(0.75 < p < 0.85\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 5 significant figures. Give the result of each iteration to 7 significant figures.
4. Find the gradient of the curve at the point \(( p , 0 )\).

1. A curve has equation \(y = \mathrm { f } ( x )\) where

$$\mathrm { f } ( x ) = x ^ { 4 } - 5 x ^ { 2 } + 4 x - 7 \quad x \in \mathbb { R }$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval [2,3]
2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt [ 3 ] { \frac { 5 x ^ { 2 } - 4 x + 7 } { x } }$$ The iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { \frac { 5 x _ { n } ^ { 2 } - 4 x _ { n } + 7 } { x _ { n } } }$$ is used to find \(\alpha\)
3. Starting with \(x _ { 1 } = 2\) and using the iterative formula,
   1. find, to 4 decimal places, the value of \(x _ { 2 }\)
   2. find, to 4 decimal places, the value of \(\alpha\)

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = x \left( x ^ { 2 } - 4 \right) e ^ { - \frac { 1 } { 2 } x }$$

1. Find \(f ^ { \prime } ( x )\). The line \(l\) is the normal to the curve at \(O\) and meets the curve again at the point \(P\). The point \(P\) lies in the 3rd quadrant, as shown in Figure 3.
2. Show that the \(x\) coordinate of \(P\) is a solution of the equation $$x = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x } }$$
3. Using the iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x _ { n } } } \quad \text { with } x _ { 1 } = - 2$$ find, to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the \(x\) coordinate of \(P\).

5. \begin{figure}[h] \end{figure} The profit made by a company, \(\pounds P\) million, \(t\) years after the company started trading, is modelled by the equation $$P = \frac { 4 t - 1 } { 10 } + \frac { 3 } { 4 } \ln \left[ \frac { t + 1 } { ( 2 t + 1 ) ^ { 2 } } \right]$$ The graph of \(P\) against \(t\) is shown in Figure 2. According to the model,

1. show that exactly one year after it started trading, the company had made a loss of approximately £ 830000 A manager of the company wants to know the value of \(t\) for which \(P = 0\)
2. Show that this value of \(t\) occurs in the interval [6,7]
3. Show that the equation \(P = 0\) can be expressed in the form $$t = \frac { 1 } { 4 } + \frac { 15 } { 8 } \ln \left[ \frac { ( 2 t + 1 ) ^ { 2 } } { t + 1 } \right]$$
4. Using the iteration formula $$t _ { n + 1 } = \frac { 1 } { 4 } + \frac { 15 } { 8 } \ln \left[ \frac { \left( 2 t _ { n } + 1 \right) ^ { 2 } } { t _ { n } + 1 } \right] \text { with } t _ { 1 } = 6$$ find the value of \(t _ { 2 }\) and the value of \(t _ { 6 }\), giving your answers to 3 decimal places.
5. Hence find, according to the model, how many months it takes in total, from when the company started trading, for it to make a profit.
   (2)

2. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { \left( \frac { 4 ( 3 - x ) } { ( 3 + x ) } \right) } \quad x \neq - 3$$ The equation \(x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12 = 0\) has a single root which is between 1 and 2
2. Use the iteration formula $$x _ { n + 1 } = \sqrt { \left( \frac { 4 \left( 3 - x _ { n } \right) } { \left( 3 + x _ { n } \right) } \right) } \quad n \geqslant 0$$ with \(x _ { 0 } = 1\) to find, to 2 decimal places, the value of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) The root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\).
3. By choosing a suitable interval, prove that \(\alpha = 1.272\) to 3 decimal places.
   

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5. $$f ( x ) = - x ^ { 3 } + 4 x ^ { 2 } - 6$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root between \(x = 1\) and \(x = 2\)
2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as $$x = \sqrt { \left( \frac { 6 } { 4 - x } \right) }$$
3. Starting with \(x _ { 1 } = 1.5\) use the iteration \(x _ { n + 1 } = \sqrt { \left( \frac { 6 } { 4 - x _ { n } } \right) }\) to calculate the values of \(x _ { 2 }\), \(x _ { 3 }\) and \(x _ { 4 }\) giving all your answers to 4 decimal places.
4. Using a suitable interval, show that 1.572 is a root of \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.