Derive stationary point equation

5 \includegraphics[max width=\textwidth, alt={}, center]{8bdd1285-9e39-465a-8c09-bbe410504f9d-06_442_698_260_721} The diagram shows part of the curve with equation \(y = x ^ { 3 } \cos 2 x\). The curve has a maximum at the point \(M\).

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \sqrt [ 3 ] { 1.5 x ^ { 2 } \cot 2 x }\).
2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(M\) lies between 0.59 and 0.60.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{2d6fc4c5-70ec-4cd8-9b48-59d5ce0e39b7-08_575_618_262_762} The diagram shows the curve with equation \(y = \frac { 3 x + 2 } { \ln x }\). The curve has a minimum point \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { 3 x + 2 } { 3 \ln x }\). [3]
2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(M\) lies between 3 and 4.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 5 significant figures. Give the result of each iteration to 7 significant figures.

5 A curve has equation \(\mathrm { y } = \frac { 1 + \mathrm { e } ^ { 2 \mathrm { x } } } { 1 + 3 \mathrm { x } }\). The curve has exactly one stationary point \(P\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) and hence show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 6 } + \frac { 1 } { 2 } \mathrm { e } ^ { - 2 x }\).
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.35 and 0.45 .
3. Use an iterative formula based on the equation in part (a) to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-10_451_647_258_699} The diagram shows the curve with equation \(\mathrm { y } = \sqrt { \sin 2 \mathrm { x } + \sin ^ { 2 } 2 \mathrm { x } }\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\). The shaded region is bounded by the curve and the straight lines \(x = \frac { 1 } { 6 } \pi\) and \(y = 0\).

4 The curve with equation \(y = x \mathrm { e } ^ { 2 x } + 5 \mathrm { e } ^ { - x }\) has a minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { 1 } { 3 } \ln 5 - \frac { 1 } { 3 } \ln ( 1 + 2 x )\).
2. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Use an initial value of 0.35 and give the result of each iteration to 5 significant figures.

3 The curve with equation $$y = 5 x - 2 \tan 2 x$$ has exactly one stationary point in the interval \(0 \leqslant x < \frac { 1 } { 4 } \pi\).
Find the coordinates of this stationary point, giving each coordinate correct to 3 significant figures.

5 The curve with equation \(y = x \ln ( 4 x + 1 ) - 3 x\) has one stationary point \(P\).

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \frac { 2 x + 0.75 } { \ln ( 4 x + 1 ) } - 0.25$$
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 1.8 and 1.9.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{d4bec1a9-2d24-4cf8-9991-9ab61ddbc865-08_430_990_260_539} Th id ag am sto cn \(\mathrm { y } = \frac { \sin 2 \mathrm { x } } { \mathrm { x } + 2 }\) fo \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Tb \(x\)-co \(\dot { \mathrm { d } } \mathbf { a }\) te 6 th max mm \(\dot { \mathrm { p } } n M\) is d t ed y \(\alpha\).

1. Fid \(\frac { \mathrm { dy } } { \mathrm { dx } }\) ad th t \(\alpha\) satisfies th eq tin \(\tan 2 x = 2 x + 4\) [4]
2. Stw alch atin \(\mathbf { b }\) t \(\alpha\) lies b tweerfd nd
3. Use th iterati fo mu a \(x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( 2 x _ { n } + 4 \right.\) to id b \& le \(6 \alpha\) co rect tod cimal p aces. Gie th resu to each teratio od cimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{48ab71ff-c37b-4e0b-b031-d99b0cf517a8-3_421_976_251_580} The diagram shows the curve \(y = \frac { \sin 2 x } { x + 2 }\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The \(x\)-coordinate of the maximum point \(M\) is denoted by \(\alpha\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\alpha\) satisfies the equation \(\tan 2 x = 2 x + 4\).
2. Show by calculation that \(\alpha\) lies between 0.6 and 0.7 .
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( 2 x _ { n } + 4 \right)\) to find the value of \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

4 \includegraphics[max width=\textwidth, alt={}, center]{873a104f-e2e2-49bb-b943-583769728fbb-06_355_839_260_653} The diagram shows the curve with equation \(y = \frac { 5 \ln x } { 2 x + 1 }\). The curve crosses the \(x\)-axis at the point \(P\) and has a maximum point \(M\).

1. Find the gradient of the curve at the point \(P\).
2. Show that the \(x\)-coordinate of the point \(M\) satisfies the equation \(x = \frac { x + 0.5 } { \ln x }\).
3. Use an iterative formula based on the equation in part (ii) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Show the result of each iteration to 6 significant figures.

4 \includegraphics[max width=\textwidth, alt={}, center]{20de5ba6-9426-4431-99af-6e8e62607f3e-2_513_895_1055_625} The diagram shows the curve \(y = \frac { \sin x } { x }\) for \(0 < x \leqslant 2 \pi\), and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation $$x = \tan x$$
2. The iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } \right) + \pi$$ can be used to determine the \(x\)-coordinate of \(M\). Use this formula to determine the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 The curve \(y = \frac { \ln x } { x + 1 }\) has one stationary point.

1. Show that the \(x\)-coordinate of this point satisfies the equation $$x = \frac { x + 1 } { \ln x }$$ and that this \(x\)-coordinate lies between 3 and 4 .
2. Use the iterative formula $$x _ { n + 1 } = \frac { x _ { n } + 1 } { \ln x _ { n } }$$ to determine the \(x\)-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 The curve with equation \(y = x ^ { 2 } \cos \frac { 1 } { 2 } x\) has a stationary point at \(x = p\) in the interval \(0 < x < \pi\).

1. Show that \(p\) satisfies the equation \(\tan \frac { 1 } { 2 } p = \frac { 4 } { p }\).
2. Verify by calculation that \(p\) lies between 2 and 2.5.
3. Use the iterative formula \(p _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 4 } { p _ { n } } \right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

10 \includegraphics[max width=\textwidth, alt={}, center]{83a6d80b-dc74-4936-ac32-858a517a843c-18_353_675_260_735} The diagram shows the curve \(y = x ^ { 2 } \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The curve has a maximum point at \(M\) where \(x = p\).

1. Show that \(p\) satisfies the equation \(p = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { p } \right)\).
2. Use the iterative formula \(p _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { p _ { n } } \right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
3. Find, showing all necessary working, the exact area of the region bounded by the curve and the \(x\)-axis.

10 \includegraphics[max width=\textwidth, alt={}, center]{9e3b4c96-0989-4ffb-bd74-0e73b79ca45a-3_430_807_1375_667} The diagram shows the curve \(y = x \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The point \(M\) is a maximum point.

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(1 = 2 x \tan 2 x\).
2. The equation in part (i) can be rearranged in the form \(x = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x } \right)\). Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x _ { n } } \right) ,$$ with initial value \(x _ { 1 } = 0.4\), to calculate the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
3. Use integration by parts to find the exact area of the region enclosed between the curve and the \(x\)-axis from 0 to \(\frac { 1 } { 4 } \pi\).

5 The curve with equation \(y = \mathrm { e } ^ { - 2 x } \ln ( x - 1 )\) has a stationary point when \(x = p\).

1. Show that \(p\) satisfies the equation \(x = 1 + \exp \left( \frac { 1 } { 2 ( x - 1 ) } \right)\), where \(\exp ( x )\) denotes \(\mathrm { e } ^ { x }\).
2. Verify by calculation that \(p\) lies between 2.2 and 2.6.
3. Use an iterative formula based on the equation in part (i) to determine \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{6694ccc1-c8b1-42a7-8b21-829a89af74c9-08_732_807_258_667} The diagram shows the curve with equation \(y = \frac { 8 + x ^ { 3 } } { 2 - 5 x }\). The maximum point is denoted by \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine the gradient of the curve at the point where the curve crosses the \(x\)-axis.
2. Show that the \(x\)-coordinate of the point \(M\) satisfies the equation \(x = \sqrt { } \left( 0.6 x + 4 x ^ { - 1 } \right)\).
3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{729aa2f6-2b62-445f-a2aa-a63b45cb6b64-3_604_971_262_587} The diagram shows the curve \(y = x ^ { 2 } \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Show by differentiation that the \(x\)-coordinate of \(M\) satisfies the equation $$\tan x = \frac { 2 } { x }$$
2. Verify by calculation that this equation has a root (in radians) between 1 and 1.2.
3. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { x _ { n } } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{72d50061-ead5-466a-96fc-2203438d1407-3_296_675_945_735} The diagram shows part of the curve \(y = \frac { x ^ { 2 } } { 1 + \mathrm { e } ^ { 3 x } }\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(m\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m\) satisfies the equation \(x = \frac { 2 } { 3 } \left( 1 + \mathrm { e } ^ { - 3 x } \right)\).
2. Show by calculation that \(m\) lies between 0.7 and 0.8 .
3. Use an iterative formula based on the equation in part (i) to find \(m\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{a257f49d-5c8f-4c23-be78-46619b746fde-10_353_689_262_726} The diagram shows the curve \(y = \frac { \tan ^ { - 1 } x } { \sqrt { x } }\) and its maximum point \(M\) where \(x = a\).

1. Show that \(a\) satisfies the equation $$a = \tan \left( \frac { 2 a } { 1 + a ^ { 2 } } \right)$$
2. Verify by calculation that \(a\) lies between 1.3 and 1.5.
3. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5 \includegraphics[max width=\textwidth, alt={}, center]{72042f09-3495-42e9-bee9-96ec5ac0bf0c-06_352_643_274_744} The diagram shows the part of the curve \(y = x ^ { 2 } \cos 3 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\), and its maximum point \(M\), where \(x = a\).

1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 3 } \tan ^ { - 1 } \left( \frac { 2 } { 3 a } \right)\).
2. Use an iterative formula based on the equation in (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-10_620_517_260_774} The diagram shows the curve \(\mathrm { y } = \mathrm { xe } ^ { 2 \mathrm { x } } - 5 \mathrm { x }\) and its minimum point \(M\), where \(x = \alpha\).

1. Show that \(\alpha\) satisfies the equation \(\alpha = \frac { 1 } { 2 } \ln \left( \frac { 5 } { 1 + 2 \alpha } \right)\).
2. Verify by calculation that \(\alpha\) lies between 0.4 and 0.5.
3. Use an iterative formula based on the equation in part (a) to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

10 \includegraphics[max width=\textwidth, alt={}, center]{77a45360-8e1d-4f4f-9830-075d832a14cf-18_549_933_260_605} The diagram shows the curve \(y = \sqrt { x } \cos x\), for \(0 \leqslant x \leqslant \frac { 3 } { 2 } \pi\), and its minimum point \(M\), where \(x = a\). The shaded region between the curve and the \(x\)-axis is denoted by \(R\).

1. Show that \(a\) satisfies the equation \(\tan a = \frac { 1 } { 2 a }\).
2. The sequence of values given by the iterative formula \(a _ { n + 1 } = \pi + \tan ^ { - 1 } \left( \frac { 1 } { 2 a _ { n } } \right)\), with initial value \(x _ { 1 } = 3\), converges to \(a\). Use this formula to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
3. Find the volume of the solid obtained when the region \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 The curve with equation \(y = \frac { x ^ { 3 } } { \mathrm { e } ^ { x } - 1 }\) has a stationary point at \(x = p\), where \(p > 0\).

1. Show that \(p = 3 \left( 1 - \mathrm { e } ^ { - p } \right)\).
2. Verify by calculation that \(p\) lies between 2.5 and 3 .
3. Use an iterative formula based on the equation in part (a) to determine \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6. $$\mathrm { f } ( x ) = x \cos \left( \frac { x } { 3 } \right) \quad x > 0$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\)
2. Show that the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\) can be written as $$x = k \arctan \left( \frac { k } { x } \right)$$ where \(k\) is an integer to be found.
3. Starting with \(x _ { 1 } = 2.5\) use the iteration formula $$x _ { n + 1 } = k \arctan \left( \frac { k } { x _ { n } } \right)$$ with the value of \(k\) found in part (b), to calculate the values of \(x _ { 2 }\) and \(x _ { 6 }\) giving your answers to 3 decimal places.
4. Using a suitable interval and a suitable function that should be stated, show that a root of \(\mathrm { f } ^ { \prime } ( x ) = 0\) is 2.581 correct to 3 decimal places.
   In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. The curve \(C\) has equation

$$y = x ^ { 2 } \cos \left( \frac { 1 } { 2 } x \right) \quad 0 < x \leqslant \pi$$ The curve has a stationary point at the point \(P\).

1. Show, using calculus, that the \(x\) coordinate of \(P\) is a solution of the equation $$x = 2 \arctan \left( \frac { 4 } { x } \right)$$ Using the iteration formula $$x _ { n + 1 } = 2 \arctan \left( \frac { 4 } { x _ { n } } \right) \quad x _ { 1 } = 2$$
2. find the value of \(x _ { 2 }\) and the value of \(x _ { 6 }\), giving your answers to 3 decimal places.