Find coordinate from gradient condition

6 A curve has equation \(y = x ^ { 3 } \mathrm { e } ^ { 0.2 x }\) where \(x \geqslant 0\). At the point \(P\) on the curve, the gradient of the curve is 15 .

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \sqrt { \frac { 75 \mathrm { e } ^ { - 0.2 x } } { 15 + x } }\).
2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(P\) lies between 1.7 and 1.8.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-08_615_469_260_799} The diagram shows part of the curve with equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 3 } } { \mathrm { x } + 2 }\). At the point \(P\), the gradient of the curve is 6 .

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \sqrt [ 3 ] { 12 x + 12 }\).
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 3.8 and 4.0 .
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Show the result of each iteration to 5 significant figures.

6 The equation of a curve is \(y = \frac { 3 x ^ { 2 } } { x ^ { 2 } + 4 }\). At the point on the curve with positive \(x\)-coordinate \(p\), the gradient of the curve is \(\frac { 1 } { 2 }\).

1. Show that \(p = \sqrt { } \left( \frac { 48 p - 16 } { p ^ { 2 } + 8 } \right)\).
2. Show by calculation that \(2 < p < 3\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 The equation of a curve is \(y = 6 x \mathrm { e } ^ { \frac { 1 } { 3 } x }\). At the point on the curve with \(x\)-coordinate \(p\), the gradient of the curve is 40 .

1. Show that \(p = 3 \ln \left( \frac { 20 } { p + 3 } \right)\).
2. Show by calculation that \(3.3 < p < 3.5\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-2_435_597_1516_776} The diagram shows the curves \(y = \mathrm { e } ^ { 2 x - 3 }\) and \(y = 2 \ln x\). When \(x = a\) the tangents to the curves are parallel.

1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 2 } ( 3 - \ln a )\).
2. Verify by calculation that this equation has a root between 1 and 2 .
3. Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)\) to calculate \(a\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{98ee8d3e-9aba-46a2-aa9c-b1e2093f393e-10_702_597_258_772} The diagram shows the curves \(y = 4 \cos \frac { 1 } { 2 } x\) and \(y = \frac { 1 } { 4 - x }\), for \(0 \leqslant x < 4\). When \(x = a\), the tangents to the curves are perpendicular.

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

9 \includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-3_481_483_1434_831} The diagram shows the curves \(y = x \cos x\) and \(y = \frac { k } { x }\), where \(k\) is a constant, for \(0 < x \leqslant \frac { 1 } { 2 } \pi\). The curves touch at the point where \(x = a\).

1. Show that \(a\) satisfies the equation \(\tan a = \frac { 2 } { a }\).
2. Use the iterative formula \(a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { a _ { n } } \right)\) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
3. Hence find the value of \(k\) correct to 2 decimal places.

5 The equation of a curve is \(y = x \ln ( 8 - x )\). The gradient of the curve is equal to 1 at only one point, when \(x = a\).

1. Show that \(a\) satisfies the equation \(x = 8 - \frac { 8 } { \ln ( 8 - x ) }\).
2. Verify by calculation that \(a\) lies between 2.9 and 3.1.
3. Use an iterative formula based on the equation in part (i) 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]{e82fee05-0c55-4fe2-b781-e5e82186c153-2_608_999_1430_571} The diagram shows the curve \(y = ( x - 4 ) \mathrm { e } ^ { \frac { 1 } { 2 } x }\). The curve has a gradient of 3 at the point \(P\).

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = 2 + 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x }$$
2. Verify that the equation in part (i) has a root between \(x = 3.1\) and \(x = 3.3\).
3. Use the iterative formula \(x _ { n + 1 } = 2 + 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{ebfaafca-3279-44f8-a910-0756ffa25c70-10_542_789_260_676} The diagram shows the curve $$y = x ^ { 2 } + 3 x + 1 + 5 \cos \frac { 1 } { 2 } x .$$ The curve crosses the \(y\)-axis at the point \(P\) and the gradient of the curve at \(P\) is \(m\). The point \(Q\) on the curve has \(x\)-coordinate \(q\) and the gradient of the curve at \(Q\) is \(- m\).

1. Find the value of \(m\) and hence show that \(q\) satisfies the equation $$x = a \sin \frac { 1 } { 2 } x + b ,$$ where the values of the constants \(a\) and \(b\) are to be determined.
2. Show by calculation that \(- 4.5 < q < - 4.0\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{da5162f3-b5d5-417f-9b6c-5ae0024f22d9-10_542_789_260_676} The diagram shows the curve $$y = x ^ { 2 } + 3 x + 1 + 5 \cos \frac { 1 } { 2 } x .$$ The curve crosses the \(y\)-axis at the point \(P\) and the gradient of the curve at \(P\) is \(m\). The point \(Q\) on the curve has \(x\)-coordinate \(q\) and the gradient of the curve at \(Q\) is \(- m\).

1. Find the value of \(m\) and hence show that \(q\) satisfies the equation $$x = a \sin \frac { 1 } { 2 } x + b ,$$ where the values of the constants \(a\) and \(b\) are to be determined.
2. Show by calculation that \(- 4.5 < q < - 4.0\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

9 \includegraphics[max width=\textwidth, alt={}, center]{3149080d-ad1a-4d2e-8e20-eb9977ced619-14_558_686_260_726} The diagram shows the curves \(y = \cos x\) and \(y = \frac { k } { 1 + x }\), where \(k\) is a constant, for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The curves touch at the point where \(x = p\).

1. Show that \(p\) satisfies the equation \(\tan p = \frac { 1 } { 1 + p }\).
2. Use the iterative formula \(p _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + p _ { n } } \right)\) to determine the value of \(p\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
3. Hence find the value of \(k\) correct to 2 decimal places.

7 The equation of a curve is \(y = \frac { x } { \cos ^ { 2 } x }\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\). At the point where \(x = a\), the tangent to the curve has gradient equal to 12 .

1. Show that \(a = \cos ^ { - 1 } \left( \sqrt [ 3 ] { \frac { \cos a + 2 a \sin a } { 12 } } \right)\).
2. Verify by calculation that \(a\) lies between 0.9 and 1 .
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.

4 The gradient of the curve \(y = \left( 2 x ^ { 2 } + 9 \right) ^ { \frac { 5 } { 2 } }\) at the point \(P\) is 100 .

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = 10 \left( 2 x ^ { 2 } + 9 \right) ^ { - \frac { 3 } { 2 } }\).
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.3 and 0.4 .
3. Use an iterative formula, based on the equation in part (i), to find the \(x\)-coordinate of \(P\) correct to 4 decimal places. You should show the result of each iteration.