1.07q Product and quotient rules: differentiation

6 Find the exact coordinates of the point on the curve \(y = x \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) at which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\).

8 \includegraphics[max width=\textwidth, alt={}, center]{dde12c57-5129-43ae-b385-9a8f21f51e49-3_566_787_255_680} The diagram shows the curve \(y = x \sin x\), for \(0 \leqslant x \leqslant \pi\). The point \(Q \left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi \right)\) lies on the curve.

1. Show that the normal to the curve at \(Q\) passes through the point \(( \pi , 0 )\).
2. Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin x - x \cos x )\).
3. Hence evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).

8 \includegraphics[max width=\textwidth, alt={}, center]{2aceb797-097c-499b-99b6-cce9f287cb51-3_566_787_255_680} The diagram shows the curve \(y = x \sin x\), for \(0 \leqslant x \leqslant \pi\). The point \(Q \left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi \right)\) lies on the curve.

1. Show that the normal to the curve at \(Q\) passes through the point \(( \pi , 0 )\).
2. Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin x - x \cos x )\).
3. Hence evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).

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.

2 The curve with equation \(y = \frac { \sin 2 x } { \mathrm { e } ^ { 2 x } }\) has one stationary point in the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Find the exact \(x\)-coordinate of this point.

2 The curve \(y = \frac { \mathrm { e } ^ { 3 x - 1 } } { 2 x }\) has one stationary point. Find the coordinates of this stationary point.

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]{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.

2 A curve has equation $$y = \frac { 3 x + 1 } { x - 5 }$$ Find the coordinates of the points on the curve at which the gradient is - 4 .

5 Find the \(x\)-coordinates of the stationary points of the following curves:

1. \(y = 4 x \mathrm { e } ^ { - 3 x }\);
2. \(y = \frac { 4 x ^ { 2 } } { x + 1 }\).

7 The equation of a curve is \(y = \frac { \sin 2 x } { \cos x + 1 }\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \left( \cos ^ { 2 } x + \cos x - 1 \right) } { \cos x + 1 }\).
2. Find the \(x\)-coordinate of each stationary point of the curve in the interval \(- \pi < x < \pi\). Give each answer correct to 3 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-2_556_844_1731_648} The diagram shows the curve \(y = \frac { 4 \ln x } { x ^ { 2 } + 1 }\) and its stationary point \(M\). The \(x\)-coordinate of \(M\) is \(m\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m = \mathrm { e } ^ { 0.5 \left( 1 + m ^ { - 2 } \right) }\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(m\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

3 A curve has equation \(y = \frac { 3 + 2 \ln x } { 1 + \ln x }\). Find the exact gradient of the curve at the point for which \(y = 4\).

5 Find the exact coordinates of the stationary point of the curve with equation \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } ( 2 x + 5 )\).

4 The curve with equation \(y = \mathrm { e } ^ { 2 x } ( \sin x + 3 \cos x )\) has a stationary point in the interval \(0 \leqslant x \leqslant \pi\).

1. Find the \(x\)-coordinate of this point, giving your answer correct to 2 decimal places.
2. Determine whether the stationary point is a maximum or a minimum.

8 \includegraphics[max width=\textwidth, alt={}, center]{8c26235b-c78c-40d8-9e8e-213dc1311186-12_437_686_274_719} The diagram shows the curve \(y = x ^ { 3 } \ln x\), for \(x > 0\), and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { 1 } { 2 }\). [5]

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.

3 The equation of a curve is \(y = \sin x \sin 2 x\). The curve has a stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this point, giving your answer correct to 3 significant figures.

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

$$f ( x ) = ( x - 4 ) ( 2 x + 1 ) ^ { 2 }$$ The curve touches the \(x\)-axis at the point \(P\) and crosses the \(x\)-axis at the point \(Q\).

1. State the coordinates of the point \(P\).
2. Find \(f ^ { \prime } ( x )\).
3. Hence show that the equation of the tangent to the curve at the point where \(x = \frac { 5 } { 2 }\) can be expressed in the form \(y = k\), where \(k\) is a constant to be found. The curve with equation \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, passes through the origin \(O\).
4. State the possible values of \(a\).
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8. The curve \(C\) has equation $$y = ( x - 2 ) ( x - 4 ) ^ { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 32$$ The line \(l _ { 1 }\) is the tangent to \(C\) at the point where \(x = 6\)
2. Find the equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The line \(l _ { 2 }\) is the tangent to \(C\) at the point where \(x = \alpha\) Given that \(l _ { 1 }\) and \(l _ { 2 }\) are parallel and distinct,
3. find the value of \(\alpha\)

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 2 )$$ The point \(P\) lies on \(C\) and has \(x\) coordinate \(\frac { 1 } { 2 }\) The line \(l\), as shown on Figure 6, is the tangent to \(C\) at \(P\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Use part (a) to find an equation for \(l\) in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. The finite region \(R\), shown shaded in Figure 6, is bounded by the line \(l\), the curve \(C\) and the \(x\)-axis. The line \(l\) meets the curve again at the point \(( 2,0 )\)
3. Use integration to find the exact area of the shaded region \(R\).

8. The curve \(C _ { 1 }\) has equation $$y = x ^ { 2 } ( x + 2 )$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Sketch \(C _ { 1 }\), showing the coordinates of the points where \(C _ { 1 }\) meets the \(x\)-axis.
3. Find the gradient of \(C _ { 1 }\) at each point where \(C _ { 1 }\) meets the \(x\)-axis. The curve \(C _ { 2 }\) has equation $$y = ( x - k ) ^ { 2 } ( x - k + 2 )$$ where \(k\) is a constant and \(k > 2\)
4. Sketch \(C _ { 2 }\), showing the coordinates of the points where \(C _ { 2 }\) meets the \(x\) and \(y\) axes.

10. The curve \(C\) has equation $$y = ( x + 1 ) ( x + 3 ) ^ { 2 }$$

1. Sketch \(C\), showing the coordinates of the points at which \(C\) meets the axes.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 14 x + 15\). The point \(A\), with \(x\)-coordinate - 5 , lies on \(C\).
3. Find the equation of the tangent to \(C\) at \(A\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(B\) also lies on \(C\). The tangents to \(C\) at \(A\) and \(B\) are parallel.
4. Find the \(x\)-coordinate of \(B\).

10. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\), where $$f ( x ) = ( 2 x - 5 ) ^ { 2 } ( x + 3 )$$

1. Given that
   1. the curve with equation \(y = \mathrm { f } ( x ) - k , x \in \mathbb { R }\), passes through the origin, find the value of the constant \(k\),
   2. the curve with equation \(y = \mathrm { f } ( x + c ) , x \in \mathbb { R }\), has a minimum point at the origin, find the value of the constant \(c\).
2. Show that \(\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 16 x - 35\) Points \(A\) and \(B\) are distinct points that lie on the curve \(y = \mathrm { f } ( x )\).
   The gradient of the curve at \(A\) is equal to the gradient of the curve at \(B\).
   Given that point \(A\) has \(x\) coordinate 3
3. find the \(x\) coordinate of point \(B\).
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1. The height of a river above a fixed point on the riverbed was monitored over a 7-day period.

The height of the river, \(H\) metres, \(t\) days after monitoring began, was given by $$H = \frac { \sqrt { t } } { 20 } \left( 20 + 6 t - t ^ { 2 } \right) + 17 \quad 0 \leqslant t \leqslant 7$$ Given that \(H\) has a stationary value at \(t = \alpha\)

1. use calculus to show that \(\alpha\) satisfies the equation $$5 \alpha ^ { 2 } - 18 \alpha - 20 = 0$$
2. Hence find the value of \(\alpha\), giving your answer to 3 decimal places.
3. Use further calculus to prove that \(H\) is a maximum at this value of \(\alpha\).