1.09b Sign change methods: understand failure cases

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.

7 \includegraphics[max width=\textwidth, alt={}, center]{388d7076-636c-417d-84cb-e6e2a3e9a6a0-10_465_785_260_680} The diagram shows the curve with parametric equations $$x = 4 t + \mathrm { e } ^ { 2 t } , \quad y = 6 t \sin 2 t$$ for \(0 \leqslant t \leqslant 1\). The point \(P\) on the curve has parameter \(p\) and \(y\)-coordinate 3 .

1. Show that \(p = \frac { 1 } { 2 \sin 2 p }\).
2. Show by calculation that the value of \(p\) lies between 0.5 and 0.6 .
3. Use an iterative formula, based on the equation in part (a), to find the value of \(p\) correct to 3 significant figures. Use an initial value of 0.55 and give the result of each iteration to 5 significant figures.
4. Find the gradient of the curve at \(P\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 \includegraphics[max width=\textwidth, alt={}, center]{61df367d-741f-4906-8ab9-2f32e8711aa6-10_465_785_260_680} The diagram shows the curve with parametric equations $$x = 4 t + \mathrm { e } ^ { 2 t } , \quad y = 6 t \sin 2 t$$ for \(0 \leqslant t \leqslant 1\). The point \(P\) on the curve has parameter \(p\) and \(y\)-coordinate 3 .

1. Show that \(p = \frac { 1 } { 2 \sin 2 p }\).
2. Show by calculation that the value of \(p\) lies between 0.5 and 0.6 .
3. Use an iterative formula, based on the equation in part (a), to find the value of \(p\) correct to 3 significant figures. Use an initial value of 0.55 and give the result of each iteration to 5 significant figures.
4. Find the gradient of the curve at \(P\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5

1. By sketching the graphs of $$y = | 5 - 2 x | \quad \text { and } \quad y = 3 \ln x$$ on the same diagram, show that the equation \(| 5 - 2 x | = 3 \ln x\) has exactly two roots.
2. Show that the value of the larger root satisfies the equation \(x = 2.5 + 1.5 \ln x\).
3. Show by calculation that the value of the larger root lies between 4.5 and 5.0.
4. Use an iterative formula, based on the equation in part (b), to find the value of the larger root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

4

1. \includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-05_753_944_278_630} The diagram shows the graph of \(y = 3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x }\).
   On the diagram, sketch the graph of \(y = | 5 x - 4 |\), and show that the equation \(3 - e ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) has exactly two real roots. It is given that the two roots of \(3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
2. Show by calculation that \(\alpha\) lies between 0.36 and 0.37 .
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 7 - \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to find \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

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.

5

1. By sketching a suitable pair of graphs, show that the equation $$\sec x = 3 - x$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 1.0 and 1.2.
3. Show that this root also satisfies the equation $$x = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } } \right)$$ with initial value \(x _ { 1 } = 1.1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

8 The constant \(a\), where \(a > 1\), is such that \(\int _ { 1 } ^ { a } \left( x + \frac { 1 } { x } \right) \mathrm { d } x = 6\).

1. Find an equation satisfied by \(a\), and show that it can be written in the form $$a = \sqrt { } ( 13 - 2 \ln a )$$
2. Verify, by calculation, that the equation \(a = \sqrt { } ( 13 - 2 \ln a )\) has a root between 3 and 3.5.
3. Use the iterative formula $$a _ { n + 1 } = \sqrt { } \left( 13 - 2 \ln a _ { n } \right)$$ with \(a _ { 1 } = 3.2\), to calculate the value of \(a\) 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 $$\cot x = 4 x - 2$$ 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.

4

1. By sketching a suitable pair of graphs, show that the equation $$3 \ln x = 15 - x ^ { 3 }$$ has exactly one real root.
2. Show by calculation that the root lies between 2.0 and 2.5.
3. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 15 - 3 \ln x _ { n } \right)\) to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-3_380_641_258_751} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The angle \(B O C\) is \(x\) radians. The area of the shaded segment is a quarter of the area of the semicircle.

1. Show that \(x\) satisfies the equation $$x = \frac { 3 } { 4 } \pi - \sin x$$
2. This equation has one root. Verify by calculation that the root lies between 1.3 and 1.5.
3. Use the iterative formula $$x _ { n + 1 } = \frac { 3 } { 4 } \pi - \sin x _ { n }$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5 \includegraphics[max width=\textwidth, alt={}, center]{d1377d66-73c8-4d97-9cae-d784b41fb0a8-2_519_800_1359_669} The diagram shows a circle with centre \(O\) and radius \(r\). The tangents to the circle at the points \(A\) and \(B\) meet at \(T\), and the angle \(A O B\) is \(2 x\) radians. The shaded region is bounded by the tangents \(A T\) and \(B T\), and by the minor \(\operatorname { arc } A B\). The perimeter of the shaded region is equal to the circumference of the circle.

1. Show that \(x\) satisfies the equation $$\tan x = \pi - x .$$
2. This equation has one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Verify by calculation that this root lies between 1 and 1.3.
3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \pi - x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4 The equation \(x ^ { 3 } - x - 3 = 0\) has one real root, \(\alpha\).

1. Show that \(\alpha\) lies between 1 and 2 . Two iterative formulae derived from this equation are as follows: $$\begin{aligned} & x _ { n + 1 } = x _ { n } ^ { 3 } - 3 \\ & x _ { n + 1 } = \left( x _ { n } + 3 \right) ^ { \frac { 1 } { 3 } } \end{aligned}$$ Each formula is used with initial value \(x _ { 1 } = 1.5\).
2. Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5

1. Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) - \tan x = 2$$ can be written in the form $$\tan ^ { 2 } x + 2 \tan x - 1 = 0$$
2. Hence solve the equation $$\tan \left( 45 ^ { \circ } + x \right) - \tan x = 2$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

9 The constant \(a\) is such that \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x = 6\).

1. Show that \(a\) satisfies the equation $$x = 2 + \mathrm { e } ^ { - \frac { 1 } { 2 } x } .$$
2. By sketching a suitable pair of graphs, show that this equation has only one root.
3. Verify by calculation that this root lies between 2 and 2.5.
4. Use an iterative formula based on the equation in part (i) to calculate the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4

1. By sketching suitable graphs, show that the equation $$4 x ^ { 2 } - 1 = \cot x$$ has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 0.6 and 1 .
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \sqrt { } \left( 1 + \cot x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{0900b607-6136-4bf7-a42e-6824d1a21e43-3_451_451_255_845} The diagram shows part of the curve \(y = 8 x + \frac { 1 } { 2 } \mathrm { e } ^ { x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = a\), where \(a\) is positive. The area of \(R\) is equal to \(\frac { 1 } { 2 }\).

1. Find an equation satisfied by \(a\), and show that the equation can be written in the form $$a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)$$
2. Verify by calculation that the equation \(a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)\) has a root between 0.2 and 0.3.
3. Use the iterative formula \(a _ { n + 1 } = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a _ { n } } } { 8 } \right)\) to determine this root 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]{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.

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.

6 \includegraphics[max width=\textwidth, alt={}, center]{293e1e27-77e9-4b19-a152-96d71b75346e-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.

4

1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
2. Verify by calculation that \(4.5 < \alpha < 5.0\).
3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) 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 \(\ln x = 3 x - x ^ { 2 }\) has one real root.
2. Verify by calculation that the root lies between 2 and 2.8.
3. Use the iterative formula \(x _ { n + 1 } = \sqrt { 3 x _ { n } - \ln x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4 Find \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\). Give your answer in a simplified exact form.