1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates

1. (a) Sketch the polar curve \(C\), with equation

$$r = 3 + \sqrt { 5 } \cos \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ On your sketch clearly label the pole, the initial line and the value of \(r\) at the point where the curve intersects the initial line. The tangent to \(C\) at the point \(A\), where \(0 < \theta < \frac { \pi } { 2 }\), is parallel to the initial line.
(b) Use calculus to show that at \(A\) $$\cos \theta = \frac { 1 } { \sqrt { 5 } }$$ (c) Hence determine the value of \(r\) at \(A\).

1

1. Differentiate the following with respect to \(x\).
   1. \(\frac { 1 } { ( 3 x - 4 ) ^ { 2 } }\)
   2. \(\frac { \ln ( x + 2 ) } { x }\)
   3. Find \(\int \mathrm { e } ^ { ( 2 x + 3 ) } \mathrm { d } x\).

2. (a) Differentiate with respect to \(x\)

1. \(3 \sin ^ { 2 } x + \sec 2 x\),
2. \(\{ x + \ln ( 2 x ) \} ^ { 3 }\). Given that \(y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } , x \neq 1\),
   (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { ( x - 1 ) ^ { 3 } }\).

9

1. Given that \(y = x ^ { - 2 } \ln x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 \ln x } { x ^ { 3 } }\).
2. Using integration by parts, find \(\int x ^ { - 2 } \ln x \mathrm {~d} x\).
3. The sketch shows the graph of \(y = x ^ { - 2 } \ln x\). \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-007_593_1034_696_543}
   1. Using the answer to part (a), find, in terms of e, the \(x\)-coordinate of the stationary point \(A\).
   2. The region \(R\) is bounded by the curve, the \(x\)-axis and the line \(x = 5\). Using your answer to part (b), show that the area of \(R\) is $$\frac { 1 } { 5 } ( 4 - \ln 5 )$$

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \tan 3 x\).
   (2 marks)
2. Given that \(y = \frac { 3 x + 1 } { 2 x + 1 }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }\).
   (3 marks)

1

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 1.5 } ^ { 5.5 } \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 ) \mathrm { d } x\), giving your answer to three decimal places.
   [0pt] [4 marks]
2. Find the exact value of the gradient of the curve \(y = \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 )\) at the point on the curve where \(x = 2\).
   [0pt] [4 marks]

6

1. Sketch, on the axes below, the curve with equation \(y = \sin ^ { - 1 } ( 3 x )\), where \(y\) is in radians. State the exact values of the coordinates of the end points of the graph.
2. Given that \(x = \frac { 1 } { 3 } \sin y\), write down \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) and hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\). \section*{Answer space for question 6}
   1. \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-14_839_1451_813_324}

5 Find an equation of the tangent to the curve $$y = ( x - 2 ) ^ { 4 }$$ at the point where \(x = 0\)

17 A particle is moving such that its position vector, \(\mathbf { r }\) metres, at time \(t\) seconds, is given by $$\mathbf { r } = \mathrm { e } ^ { t } \cos t \mathbf { i } + \mathrm { e } ^ { t } \sin t \mathbf { j }$$ Show that the magnitude of the acceleration of the particle, \(a \mathrm {~ms} ^ { - 2 }\), is given by $$a = 2 \mathrm { e } ^ { t }$$ Fully justify your answer. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-27_2490_1728_217_141}

12 A curve, \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 12 x + 12 } { x ^ { 2 } + 4 x - 4 }\) The line \(y = k\) intersects the curve, \(C _ { 1 }\) 12

1. 1. Show that \(( k + 3 ) ( k - 1 ) \geq 0\) [0pt] [5 marks]
      12
      1. (ii) Hence find the coordinates of the stationary point of \(C _ { 1 }\) that is a maximum point.
         [0pt] [4 marks] 12
   2. Show that the curve \(C _ { 2 }\) whose equation is \(y = \frac { 1 } { \mathrm { f } ( x ) }\), has no vertical asymptotes.
      [0pt] [2 marks]
      12
   3. State the equation of the line that is a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\).
      [0pt] [1 mark]

1. The vertical height, \(h \mathrm {~m}\), above horizontal ground, of a passenger on a fairground ride, \(t\) seconds after the ride starts, where \(t \leqslant 5\), is modelled by the differential equation

$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} h } { \mathrm {~d} t } + 2 h = t ^ { 3 }$$

1. Given that \(t = \mathrm { e } ^ { x }\), show that
   1. \(t \frac { \mathrm {~d} h } { \mathrm {~d} t } = \frac { \mathrm { d } h } { \mathrm {~d} x }\)
   2. \(t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } h } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } h } { \mathrm {~d} x }\)
2. Hence show that the transformation \(t = \mathrm { e } ^ { x }\) transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } h } { \mathrm {~d} x ^ { 2 } } - 3 \frac { \mathrm {~d} h } { \mathrm {~d} x } + 2 h = \mathrm { e } ^ { 3 x }$$
3. Hence show that $$h = A t + B t ^ { 2 } + \frac { 1 } { 2 } t ^ { 3 }$$ where \(A\) and \(B\) are constants. Given that when \(t = 1 , h = 2.5\) and when \(t = 2 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = - 1\)
4. determine the height of the passenger above the ground 5 seconds after the start of the ride.

5. \begin{figure}[h] \end{figure} Figure 3 shows a container in the shape of a hollow, inverted, right circular cone.
The height of the container is 30 cm and the radius is 12 cm , as shown in Figure 3.
The container is initially empty when water starts flowing into it.
When the height of water is \(h \mathrm {~cm}\), the surface of the water has radius \(r \mathrm {~cm}\) and the volume of water is \(V \mathrm {~cm} ^ { 3 }\)

1. Show that $$V = \frac { 4 \pi h ^ { 3 } } { 75 }$$ [The volume \(V\) of a right circular cone with vertical height \(h\) and base radius \(r\) is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) ] Given that water flows into the container at a constant rate of \(2 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
2. find, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), the rate at which \(h\) is changing, exactly 1.5 minutes after water starts flowing into the container.

9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}

1. Copy and complete the diagram to show all the forces acting on the particle.
2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.

7

1. Differentiate \(3 \ln \left( x ^ { 2 } + 1 \right)\).
2. Find \(\int \frac { x ^ { 2 } } { 3 - 4 x ^ { 3 } } \mathrm {~d} x\).

9 A curve has equation \(y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x }\). Find the exact coordinates of the stationary points of the curve.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-4_620_896_959_333} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

12 Boyle's Law states that when a gas is kept at a constant temperature, its pressure \(P\) pascals is inversely proportional to its volume \(V \mathrm {~m} ^ { 3 }\). When the volume of a certain gas is \(80 \mathrm {~m} ^ { 3 }\), its pressure is 5 pascals and the rate at which the volume is increasing is \(10 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the pressure is decreasing at this volume.

5 \includegraphics[max width=\textwidth, alt={}, center]{48b63de9-f022-4881-a187-f08e3c7d9f1a-3_570_734_219_667} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
2. Given that \(P = 20\), show that \(A = \left( \frac { 10 } { 1 + x } \right) ^ { 2 }\).
3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{48b63de9-f022-4881-a187-f08e3c7d9f1a-4_620_894_1064_342} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled \(C\) has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\). In the second model the equation is \(y = f \cos ( \lambda x ) + g\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

7 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).

1. Determine \(f ^ { \prime \prime } ( 1 )\).
2. Differentiate ( \(*\) ) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
4. Deduce, to 3 decimal places, an approximation for f(1.1).

3 Show that the equation of the tangent to the curve \(y = \ln \left( x ^ { 2 } + 3 \right)\) at the point \(( 1 , \ln 4 )\) is $$2 y - x = \ln ( 16 ) - 1$$

a) Differentiate each of the following functions with respect to \(x\). i) \(x ^ { 5 } \ln x\) ii) \(\frac { \mathrm { e } ^ { 3 x } } { x ^ { 3 } - 1 }\) iii) \(( \tan x + 7 x ) ^ { \frac { 1 } { 2 } }\) b) A function is defined implicitly by $$3 y + 4 x y ^ { 2 } - 5 x ^ { 3 } = 8$$ Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
□
1
The function \(f ( x )\) is defined by $$f ( x ) = \frac { \sqrt { x ^ { 2 } - 1 } } { x }$$ with domain \(x \geqslant 1\).
a) Find an expression for \(f ^ { - 1 } ( x )\). State the domain for \(f ^ { - 1 }\) and sketch both \(f ( x )\) and \(f ^ { - 1 } ( x )\) on the same diagram.
b) Explain why the function \(f f ( x )\) cannot be formed.
□
1
A chord \(A B\) subtends an angle \(\theta\) radians at the centre of a circle. The chord divides the circle into two segments whose areas are in the ratio \(1 : 2\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-5_572_576_1197_749}
a) Show that \(\sin \theta = \theta - \frac { 2 \pi } { 3 }\).
b) i) Show that \(\theta\) lies between \(2 \cdot 6\) and \(2 \cdot 7\).
ii) Starting with \(\theta _ { 0 } = 2 \cdot 6\), use the Newton-Raphson Method to find the value of \(\theta\) correct to three decimal places. \section*{TURN OVER} Wildflowers grow on the grass verge by the side of a motorway. The area populated by wildflowers at time \(t\) years is \(A \mathrm {~m} ^ { 2 }\). The rate of increase of \(A\) is directly proportional to \(A\).
a) Write down a differential equation that is satisfied by \(A\).
b) At time \(t = 0\), the area populated by wildflowers is \(0.2 \mathrm {~m} ^ { 2 }\). One year later, the area has increased to \(1.48 \mathrm {~m} ^ { 2 }\). Find an expression for \(A\) in terms of \(t\) in the form \(p q ^ { t }\), where \(p\) and \(q\) are rational numbers to be determined.

Differentiate the following functions with respect to \(x\). a) \(x ^ { 3 } \ln ( 5 x )\) b) \(( x + \cos 3 x ) ^ { 4 }\)

\includegraphics{figure_2} A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is \(h\) cm, the volume of water \(V\) cm\(^3\) is given by $$V = 4\pi h(h + 4), \quad 0 \leq h \leq 25$$ Water flows into the vase at a constant rate of \(80\pi\) cm\(^3\)s\(^{-1}\) Find the rate of change of the depth of the water, in cm s\(^{-1}\), when \(h = 6\) \hfill [5]