1.07n Stationary points: find maxima, minima using derivatives

5

1. A curve has equation \(y = \mathrm { e } ^ { 2 x } - 10 \mathrm { e } ^ { x } + 12 x\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      (2 marks)
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
      (1 mark)
2. The points \(P\) and \(Q\) are the stationary points of the curve.
   1. Show that the \(x\)-coordinates of \(P\) and \(Q\) are given by the solutions of the equation $$\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } + 6 = 0$$ (1 mark)
   2. By using the substitution \(z = \mathrm { e } ^ { x }\), or otherwise, show that the \(x\)-coordinates of \(P\) and \(Q\) are \(\ln 2\) and \(\ln 3\).
   3. Find the \(y\)-coordinates of \(P\) and \(Q\), giving each of your answers in the form \(m + 12 \ln n\), where \(m\) and \(n\) are integers.
   4. Using the answer to part (a)(ii), determine the nature of each stationary point.

3 A curve is defined for \(0 \leqslant x \leqslant \frac { \pi } { 4 }\) by the equation \(y = x \cos 2 x\), and is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-3_757_878_402_559}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The point \(A\), where \(x = \alpha\), on the curve is a stationary point.
   1. Show that \(1 - 2 \alpha \tan 2 \alpha = 0\).
   2. Show that \(0.4 < \alpha < 0.5\).
   3. Show that the equation \(1 - 2 x \tan 2 x = 0\) can be rearranged to become \(x = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x } \right)\).
   4. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x _ { n } } \right)\) with \(x _ { 1 } = 0.4\) to find \(x _ { 3 }\), giving your answer to two significant figures.
3. Use integration by parts to find \(\int _ { 0 } ^ { 0.5 } x \cos 2 x \mathrm {~d} x\), giving your answer to three significant figures.

9

1. Given that \(x = \frac { \sin y } { \cos y }\), use the quotient rule to show that $$\frac { \mathrm { d } x } { \mathrm {~d} y } = \sec ^ { 2 } y$$ (3 marks)
2. Given that \(\tan y = x - 1\), use a trigonometrical identity to show that $$\sec ^ { 2 } y = x ^ { 2 } - 2 x + 2$$
3. Show that, if \(y = \tan ^ { - 1 } ( x - 1 )\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x ^ { 2 } - 2 x + 2 }$$ (l mark)
4. A curve has equation \(y = \tan ^ { - 1 } ( x - 1 ) - \ln x\).
   1. Find the value of the \(x\)-coordinate of each of the stationary points of the curve.
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   3. Hence show that the curve has a minimum point which lies on the \(x\)-axis.

7

1. By writing \(\sec x = ( \cos x ) ^ { - 1 }\), use the chain rule to show that, if \(y = \sec x\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x$$
2. The function f is defined by $$\mathrm { f } ( x ) = 2 \tan x - 3 \sec x , \text { for } 0 < x < \frac { \pi } { 2 }$$ Find the value of the \(y\)-coordinate of the stationary point of the graph of \(y = \mathrm { f } ( x )\), giving your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are integers.
   [0pt] [6 marks]

4. The curve with equation \(y = x ^ { \frac { 5 } { 2 } } \ln \frac { x } { 4 } , x > 0\) crosses the \(x\)-axis at the point \(P\).

1. Write down the coordinates of \(P\). The normal to the curve at \(P\) crosses the \(y\)-axis at the point \(Q\).
2. Find the area of triangle \(O P Q\) where \(O\) is the origin. The curve has a stationary point at \(R\).
3. Find the \(x\)-coordinate of \(R\) in exact form.

8. The curve \(C\) has the equation \(y = \sqrt { x } + \mathrm { e } ^ { 1 - 4 x } , x \geq 0\).

1. Find an equation for the normal to the curve at the point \(\left( \frac { 1 } { 4 } , \frac { 3 } { 2 } \right)\). The curve \(C\) has a stationary point with \(x\)-coordinate \(\alpha\) where \(0.5 < \alpha < 1\).
2. Show that \(\alpha\) is a solution of the equation $$x = \frac { 1 } { 4 } [ 1 + \ln ( 8 \sqrt { x } ) ]$$
3. Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 4 } \left[ 1 + \ln \left( 8 \sqrt { x _ { n } } \right) \right]$$ with \(x _ { 0 } = 1\) to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving the value of \(x _ { 4 }\) to 3 decimal places.
4. Show that your value for \(x _ { 4 }\) is the value of \(\alpha\) correct to 3 decimal places.
5. Another attempt to find \(\alpha\) is made using the iteration formula $$x _ { n + 1 } = \frac { 1 } { 64 } \mathrm { e } ^ { 8 x _ { n } - 2 }$$ with \(x _ { 0 } = 1\). Describe the outcome of this attempt.

6. A curve has the equation \(y = \mathrm { e } ^ { 3 x } \cos 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { 3 x } ( 5 \cos 2 x - 12 \sin 2 x )\). The curve has a stationary point in the interval \([ 0,1 ]\).
3. Find the \(x\)-coordinate of the stationary point to 3 significant figures.
4. Determine whether the stationary point is a maximum or minimum point and justify your answer.

3. $$f ( x ) = x ^ { 2 } + 5 x - 2 \sec x , \quad x \in \mathbb { R } , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } .$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [1,1.5]. A more accurate estimate of this root is to be found using iterations of the form $$x _ { n + 1 } = \arccos \mathrm { g } \left( x _ { n } \right) .$$
2. Find a suitable form for \(\mathrm { g } ( x )\) and use this formula with \(x _ { 0 } = 1.25\) to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 3 decimal places. The curve \(y = \mathrm { f } ( x )\) has a stationary point at \(P\).
3. Show that the \(x\)-coordinate of \(P\) is 1.0535 correct to 5 significant figures.

5. (a) Show that \(( 2 x + 3 )\) is a factor of \(\left( 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\).
(b) Hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$ (c) Find the coordinates of the stationary points of the curve with equation $$y = \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$

6. (a) Use the derivative of \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$ The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
(b) Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
(c) Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).

8. A curve has the equation \(y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , \quad x \neq 0\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   [0pt]
2. Show that the curve has a stationary point in the interval [1.3,1.4]. The point \(A\) on the curve has \(x\)-coordinate 2 .
3. Show that the tangent to the curve at \(A\) passes through the origin. The tangent to the curve at \(A\) intersects the curve again at the point \(B\).
   The \(x\)-coordinate of \(B\) is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with \(x _ { 0 } = - 1\).
4. Find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 7 significant figures and hence state the \(x\)-coordinate of \(B\) to 5 significant figures.

4. The curve \(C\) has the equation \(y = x ^ { 2 } - 5 x + 2 \ln \frac { x } { 3 } , x > 0\).

1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3 x + 5 y + 21 = 0$$
2. Find the \(x\)-coordinates of the stationary points of \(C\).

8. $$f ( x ) = 2 x + \sin x - 3 \cos x$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [0.7, 0.8].
2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where it crosses the \(y\)-axis.
3. Find the values of the constants \(a , b\) and \(c\), where \(b > 0\) and \(0 < c < \frac { \pi } { 2 }\), such that $$f ^ { \prime } ( x ) = a + b \cos ( x - c )$$
4. Hence find the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\), giving your answers to 2 decimal places.

7. The curve \(C\) has equation \(y = \frac { x } { 4 + x ^ { 2 } }\).

1. Use calculus to find the coordinates of the turning points of \(C\). Using the result \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 x \left( x ^ { 2 } - 12 \right) } { \left( 4 + x ^ { 2 } \right) ^ { 3 } }\), or otherwise,
2. determine the nature of each of the turning points.
3. Sketch the curve \(C\).

5. \begin{figure}[h] \end{figure} Figure 1 shows a graph of \(y = x \sqrt { } \sin x , 0 < x < \pi\). The maximum point on the curve is \(A\).

1. Show that the \(x\)-coordinate of the point \(A\) satisfies the equation \(2 \tan x + x = 0\). The finite region enclosed by the curve and the \(x\)-axis is shaded as shown in Fig. 1.
   A solid body \(S\) is generated by rotating this region through \(2 \pi\) radians about the \(x\)-axis.
2. Find the exact value of the volume of \(S\).
   (7)

7. (a) Prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( a ^ { x } \right) = a ^ { x } \ln a .$$ A curve has the equation \(y = 4 ^ { x } - 2 ^ { x - 1 } + 1\).
(b) Show that the tangent to the curve at the point where it crosses the \(y\)-axis has the equation $$3 x \ln 2 - 2 y + 3 = 0 .$$ (c) Find the exact coordinates of the stationary point of the curve.
7. continued

7 A particle of mass 0.8 kg is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to a fixed point \(O\). The particle is held at rest at \(O\) and then released. When the extension of the string is \(x \mathrm {~m}\), the particle is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy show that \(v ^ { 2 } = 39.2 + 19.6 x - 12.5 x ^ { 2 }\).
2. Hence find
   1. the maximum extension of the string,
   2. the maximum speed of the particle,
   3. the maximum magnitude of the acceleration of the particle.

6. A smooth wire, with ends \(A\) and \(B\), is in the shape of a semicircle of radius \(r\). The line \(A B\) is horizontal and the midpoint of \(A B\) is \(O\). The wire is fixed in a vertical plane. A small ring \(R\) of mass \(2 m\) is threaded on the wire and is attached to two light inextensible strings. One string passes through a small smooth ring fixed at \(A\) and is attached to a particle of mass \(\sqrt { 6 } m\). The other string passes through a small smooth ring fixed at \(B\) and is attached to a second particle of mass \(\sqrt { 6 } \mathrm {~m}\). The particles hang freely under gravity, as shown in Figure 3. The angle between the radius \(O R\) and the downward vertical is \(2 \theta\), where \(- \frac { \pi } { 4 } < \theta < \frac { \pi } { 4 }\)

1. Show that the potential energy of the system is $$2 m g r ( 2 \sqrt { 3 } \cos \theta - \cos 2 \theta ) + \text { constant }$$
2. Find the values of \(\theta\) for which the system is in equilibrium.
3. Determine the stability of the position of equilibrium for which \(\theta > 0\)

3 A uniform rigid rod AB of length \(2 a\) and mass \(m\) is smoothly hinged to a fixed point at A so that it can rotate freely in a vertical plane. A light elastic string of modulus \(\lambda\) and natural length \(a\) connects the midpoint of AB to a fixed point C which is vertically above A with \(\mathrm { AC } = a\). The rod makes an angle \(2 \theta\) with the upward vertical, where \(\frac { 1 } { 3 } \pi \leqslant 2 \theta \leqslant \pi\). This is shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Find the potential energy, \(V\), of the system relative to A in terms of \(m , \lambda , a\) and \(\theta\). Show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 a \cos \theta ( 2 \lambda \sin \theta - 2 m g \sin \theta - \lambda ) .$$ Assume now that the system is set up so that the result (*) continues to hold when \(\pi < 2 \theta \leqslant \frac { 5 } { 3 } \pi\).
2. In the case \(\lambda < 2 m g\), show that there is a stable position of equilibrium at \(\theta = \frac { 1 } { 2 } \pi\). Show that there are no other positions of equilibrium in this case.
3. In the case \(\lambda > 2 m g\), find the positions of equilibrium for \(\frac { 1 } { 3 } \pi \leqslant 2 \theta \leqslant \frac { 5 } { 3 } \pi\) and determine for each whether the equilibrium is stable or unstable, justifying your conclusions.

2 A uniform rod AB of length 0.5 m and mass 0.5 kg is freely hinged at A so that it can rotate in a vertical plane. Attached at B are two identical light elastic strings BC and BD each of natural length 0.5 m and stiffness \(2 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). The ends C and D are fixed at the same horizontal level as A and with \(\mathrm { AC } = \mathrm { CD } = 0.5 \mathrm {~m}\). The system is shown in Fig. 2.1 with the angle \(\mathrm { BAC } = \theta\). You may assume that \(\frac { 1 } { 3 } \pi \leqslant \theta \leqslant \frac { 5 } { 3 } \pi\) so that both strings are taut. \begin{figure}[h] \end{figure}

1. Show that the length of BC in metres is \(\sin \frac { 1 } { 2 } \theta\).
2. Find the potential energy, \(V \mathrm {~J}\), of the system relative to AD in terms of \(\theta\). Hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 1.5 \sin \theta - 1.225 \cos \theta - \frac { 0.5 \sin \theta } { \sqrt { 1.25 - \cos \theta } } - 0.5 \cos \frac { 1 } { 2 } \theta .$$
3. Fig. 2.2 shows a graph of the function \(\mathrm { f } ( \theta ) = 1.5 \sin \theta - 1.225 \cos \theta - \frac { 0.5 \sin \theta } { \sqrt { 1.25 - \cos \theta } } - 0.5 \cos \frac { 1 } { 2 } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bc637a95-b469-493b-8fd4-d3b12049878b-2_453_1264_2021_397} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} Use the graph both to estimate, correct to 1 decimal place, the values of \(\theta\) for which the system is in equilibrium and also to determine their stability.

5 A curve has equation \(y = x ^ { 3 } - 12 x\).
The point \(A\) on the curve has coordinates ( \(2 , - 16\) ).
The point \(B\) on the curve has \(x\)-coordinate \(2 + h\).

1. Show that the gradient of the line \(A B\) is \(6 h + h ^ { 2 }\).
2. Explain how the result of part (a) can be used to show that \(A\) is a stationary point on the curve.
   \includegraphics[max width=\textwidth, alt={}]{763d89e4-861a-4754-a93c-d0902987673f-06_1894_1709_813_153}

2

1. 1. Sketch on the axes below the graphs of \(y = \sinh x\) and \(y = \cosh x\).
   2. Use your graphs to explain why the equation $$( k + \sinh x ) \cosh x = 0$$ where \(k\) is a constant, has exactly one solution.
2. A curve \(C\) has equation \(y = 6 \sinh x + \cosh ^ { 2 } x\). Show that \(C\) has only one stationary point and show that its \(y\)-coordinate is an integer. \includegraphics[max width=\textwidth, alt={}, center]{53d742f4-923b-478c-8ae6-ada6c0bb4a7e-2_560_704_1416_171} \includegraphics[max width=\textwidth, alt={}, center]{53d742f4-923b-478c-8ae6-ada6c0bb4a7e-2_560_711_1416_964}

1 A family of functions is defined as $$f ( x ) = a x + \frac { x ^ { 2 } } { 1 + x } , \quad x \neq - 1$$ where the parameter \(a\) is a real number. You may find it helpful to use a slider (for \(a\) ) to investigate the family of curves \(y = f ( x )\). \begin{enumerate}[label=(\alph*)] \item \begin{enumerate}[label=(\roman*)] \item On the axes in the Printed Answer Booklet, sketch the curve \(y = f ( x )\) in each of the following cases.

• \(a = - 2\)
• \(a = - 1\)
• \(a = 0\)
• State a feature which is common to the curve in all three cases, \(a = - 2\), \(a = - 1\) and \(a = 0\).
• State a feature of the curve for the cases \(a = - 2 , a = - 1\) that is not a feature of the curve in the case \(a = 0\).
• 1. Determine the equation of the oblique asymptote to the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) in terms of \(a\).
  2. For \(b \neq - 1,0,1\) let \(A\) be the point with coordinates ( \(- b , \mathrm { f } ( - b )\) ) and let \(B\) be the point with coordinates ( \(b , \mathrm { f } ( b )\) ).

Show that the \(y\)-coordinate of the point at which the chord to the curve \(y = f ( x )\) between \(A\) and \(B\) meets the \(y\)-axis is independent of \(a\).

With \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), determine the range of values of \(a\) for which

Find its coordinates and fully justify that it is a cusp.

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a non-zero constant.

The point \(P \left( c p , \frac { c } { p } \right)\), where \(p \neq 0\), lies on \(H\).

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$p ^ { 3 } x - p y + c \left( 1 - p ^ { 4 } \right) = 0$$ The normal to \(H\) at the point \(P\) meets \(H\) again at the point \(Q\).
2. Find the coordinates of the midpoint of \(P Q\) in terms of \(c\) and \(p\), simplifying your answer where possible.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the rectangular hyperbola \(H\) with equation $$x y = c ^ { 2 } \quad x > 0$$ where \(c\) is a positive constant.
The point \(P \left( c t , \frac { c } { t } \right)\) lies on \(H\).
The line \(l\) is the tangent to \(H\) at the point \(P\).
The line \(l\) crosses the \(x\)-axis at the point \(A\) and crosses the \(y\)-axis at the point \(B\).
The region \(R\), shown shaded in Figure 2, is bounded by the \(x\)-axis, the \(y\)-axis and the line \(l\). Given that the length \(O B\) is twice the length of \(O A\), where \(O\) is the origin, and that the area of \(R\) is 32 , find the exact coordinates of the point \(P\).