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

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\cot x^2\) [2]
2. \(x^2 e^{-x}\) [3]
3. \(\frac{\sin x}{3 + 2\cos x}\) [4]

Fig. 9 shows the curve \(y = \frac{x^2}{3x - 1}\). P is a turning point, and the curve has a vertical asymptote \(x = a\). \includegraphics{figure_1}

1. Write down the value of \(a\). [1]
2. Show that \(\frac{dy}{dx} = \frac{x(3x - 2)}{(3x - 1)^2}\) [3]
3. Find the exact coordinates of the turning point P. Calculate the gradient of the curve when \(x = 0.6\) and \(x = 0.8\), and hence verify that P is a minimum point. [7]
4. Using the substitution \(u = 3x - 1\), show that \(\int \frac{x^2}{3x - 1} dx = \frac{1}{27} \int \left( u + 2 + \frac{1}{u} \right) du\). Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = \frac{2}{3}\) and \(x = 1\). [7]

Differentiate \(\sqrt{1 + 6x^2}\). [4]

\includegraphics{figure_9} Fig. 9 shows a hemispherical bowl, of radius 10 cm, filled with water to a depth of \(x\) cm. It can be shown that the volume of water, \(V\) cm\(^3\), is given by $$V = \pi(10x^2 - \frac{1}{3}x^3).$$ Water is poured into a leaking hemispherical bowl of radius 10 cm. Initially, the bowl is empty. After \(t\) seconds, the volume of water is changing at a rate, in cm\(^3\) s\(^{-1}\), given by the equation $$\frac{dV}{dt} = k(20 - x),$$ where \(k\) is a constant.

1. Find \(\frac{dV}{dx}\), and hence show that \(\pi x \frac{dx}{dt} = k\). [4]
2. Solve this differential equation, and hence show that the bowl fills completely after \(T\) seconds, where \(T = \frac{50\pi}{k}\). [5]

Once the bowl is full, the supply of water to the bowl is switched off, and water then leaks out at a rate of \(k\) cm\(^3\) s\(^{-1}\).

1. Show that, \(t\) seconds later, \(\pi(20 - x) \frac{dx}{dt} = -k\). [3]
2. Solve this differential equation. Hence show that the bowl empties in \(3T\) seconds. [6]

Water is leaking from a container. After \(t\) seconds, the depth of water in the container is \(x\) cm, and the volume of water is \(V\) cm\(^3\), where \(V = \frac{1}{3}x^3\). The rate at which water is lost is proportional to \(x\), so that \(\frac{dV}{dt} = -kx\), where \(k\) is a constant.

1. Show that \(x \frac{dx}{dt} = -k\). [3]

Initially, the depth of water in the container is 10 cm.

1. Show by integration that \(x = \sqrt{100 - 2kt}\). [4]
2. Given that the container empties after 50 seconds, find \(k\). [2]

Once the container is empty, water is poured into it at a constant rate of 1 cm\(^3\) per second. The container continues to lose water as before.

1. Show that, \(t\) seconds after starting to pour the water in, \(\frac{dx}{dt} = \frac{1-x}{x^2}\). [2]
2. Show that \(\frac{1}{1-x} - x - 1 = \frac{x^2}{1-x}\). Hence solve the differential equation in part (iv) to show that $$t = \ln\left(\frac{1}{1-x}\right) - \frac{1}{2}x^2 - x.$$ [6]
3. Show that the depth cannot reach 1 cm. [1]

The motion of a particle is modelled by the differential equation $$v \frac{dv}{dt} + 4x = 0,$$ where \(x\) is its displacement from a fixed point, and \(v\) is its velocity. Initially \(x = 1\) and \(v = 4\).

1. Solve the differential equation to show that \(v^2 = 20 - 4x^2\). [4]

Now consider motion for which \(x = \cos 2t + 2 \sin 2t\), where \(x\) is the displacement from a fixed point at time \(t\).

1. Verify that, when \(t = 0\), \(x = 1\). Use the fact that \(v = \frac{dx}{dt}\) to verify that when \(t = 0\), \(v = 4\). [4]
2. Express \(x\) in the form \(R \cos(2t - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and obtain the corresponding expression for \(v\). Hence or otherwise verify that, for this motion too, \(v^2 = 20 - 4x^2\). [7]
3. Use your answers to part (iii) to find the maximum value of \(x\), and the earliest time at which \(x\) reaches this maximum value. [3]

\includegraphics{figure_2} Figure 2 shows a hemispherical bowl of radius 5 cm. The bowl is filled with water but the water leaks from a hole at the base of the bowl. At time \(t\) minutes, the depth of water is \(h\) cm and the volume of water in the bowl is \(V\) cm³, where $$V = \frac{1}{3}\pi h^2(15 - h).$$ In a model it is assumed that the rate at which the volume of water in the bowl decreases is proportional to \(V\).

1. Show that $$\frac{dh}{dt} = -\frac{kh(15-h)}{3(10-h)},$$ where \(k\) is a positive constant. [5]
2. Express \(\frac{3(10-h)}{h(15-h)}\) in partial fractions. [3]

Given that when \(t = 0\), \(h = 5\),

1. show that $$h^2(15-h) = 250e^{-kt}.$$ [6]

Given also that when \(t = 2\), \(h = 4\),

1. find the value of \(k\) to 3 significant figures. [3]

\includegraphics{figure_2} Figure 2 shows a vertical cross-section of a vase. The inside of the vase is in the shape of a right-circular cone with the angle between the sides in the cross-section being \(60°\). When the depth of water in the vase is \(h\) cm, the volume of water in the vase is \(V\) cm\(^3\).

1. Show that \(V = \frac{1}{9}\pi h^3\). [3]

The vase is initially empty and water is poured in at a constant rate of 120 cm\(^3\) s\(^{-1}\).

1. Find, to 2 decimal places, the rate at which \(h\) is increasing
   1. when \(h = 6\),
   2. after water has been poured in for 8 seconds. [7]

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\cot x^2\) [2]
2. \(\frac{\sin x}{3 + 2\cos x}\) [4]

\includegraphics{figure_3} Fig. 9 shows a hemispherical bowl, of radius 10 cm, filled with water to a depth of \(x\) cm. It can be shown that the volume of water, \(V\) cm\(^3\), is given by $$V = \pi(10x^2 - \frac{1}{3}x^3).$$ Water is poured into a leaking hemispherical bowl of radius 10 cm. Initially, the bowl is empty. After \(t\) seconds, the volume of water is changing at a rate, in cm\(^3\) s\(^{-1}\), given by the equation $$\frac{dV}{dt} = k(20 - x),$$ where \(k\) is a constant.

1. Find \(\frac{dV}{dx}\), and hence show that \(\pi x \frac{dx}{dt} = k\). [4]
2. Solve this differential equation, and hence show that the bowl fills completely after \(T\) seconds, where \(T = \frac{50\pi}{k}\). [5]

Once the bowl is full, the supply of water to the bowl is switched off, and water then leaks out at a rate of \(kx\) cm\(^3\) s\(^{-1}\).

1. Show that, \(t\) seconds later, \(\pi(20 - x) \frac{dx}{dt} = -k\). [3]
2. Solve this differential equation. Hence show that the bowl empties in 37 seconds. [6]

Some years ago an island was populated by red squirrels and there were no grey squirrels. Then grey squirrels were introduced. The population \(x\), in thousands, of red squirrels is modelled by the equation $$x = \frac{a}{1 + kt},$$ where \(t\) is the time in years, and \(a\) and \(k\) are constants. When \(t = 0\), \(x = 2.5\).

1. Show that \(\frac{dx}{dt} = -\frac{kx^2}{a}\). [3]
2. Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after one year, calculate \(a\) and \(k\). [3]
3. What is the long-term population of red squirrels predicted by this model? [1]

The population \(y\), in thousands, of grey squirrels is modelled by the differential equation $$\frac{dy}{dt} = 2y - y^2.$$ When \(t = 0\), \(y = 1\).

1. Express \(\frac{1}{2y - y^2}\) in partial fractions. [4]
2. Hence show by integration that \(\ln\left(\frac{y}{2-y}\right) = 2t\). Show that \(y = \frac{2}{1 + e^{-2t}}\). [7]
3. What is the long-term population of grey squirrels predicted by this model? [1]

A spaceship is moving in a straight line in deep space and needs to increase its speed. This is done by ejecting fuel backwards from the spaceship at a constant speed \(c\) relative to the spaceship. When the speed of the spaceship is \(v\), its mass is \(m\).

1. Show that, while the spaceship is ejecting fuel, $$\frac{dv}{dm} = -\frac{c}{m}.$$ [5]

The initial mass of the spaceship is \(m_0\) and at time \(t\) the mass of the spaceship is given by \(m = m_0(1 - kt)\), where \(k\) is a positive constant.

1. Find the acceleration of the spaceship at time \(t\). [4]

In this question you must show detailed reasoning. \includegraphics{figure_5} The diagram shows the curve with equation \(y = \frac{2x - 3}{4x^2 + 1}\). The tangent to the curve at the point \(P\) has gradient 2.

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$4x^3 + 3x - 3 = 0.$$ [5]
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.5 and 1. [2]
3. Show that the iteration $$x_{n+1} = \frac{3 - 4x_n^3}{3}$$ cannot converge to the \(x\)-coordinate of \(P\) whatever starting value is used. [2]
4. Use the Newton-Raphson method, with initial value 0.5, to determine the coordinates of \(P\) correct to 5 decimal places. [5]

A piece of wire of length 66 cm is bent to form the five sides of a pentagon. The pentagon consists of three sides of a rectangle and two sides of an equilateral triangle. The sides of the rectangle measure \(x\) cm and \(y\) cm and the sides of the triangle measure \(x\) cm, as shown in the diagram below. \includegraphics{figure_10}

1. 1. You are given that \(\sin 60° = \frac{\sqrt{3}}{2}\) Explain why the area of the triangle is \(\frac{\sqrt{3}}{4}x^2\) [1 mark]
   2. Show that the area enclosed by the wire, \(A\) cm\(^2\), can be expressed by the formula $$A = 33x - \frac{1}{4}(6 - \sqrt{3})x^2$$ [3 marks]
2. Use calculus to find the value of \(x\) for which the wire encloses the maximum area. Give your answer in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are integers. Fully justify your answer. [7 marks]

A singer has a social media account with a number of followers. The singer releases a new song and the number of followers grows exponentially. The number of followers, \(F\), may be modelled by the formula $$F = ae^{kt}$$ where \(t\) is the number of days since the song was released and \(a\) and \(k\) are constants. • Two days after the song is released the account has 2050 followers. • Five days after the song is released the account has 9200 followers. On the graph below ln \(F\) has been plotted against \(t\) for these two pieces of data. A line has been drawn passing through these two data points. \includegraphics{figure_2}

1. 1. Show that \(\ln F = \ln a + kt\) [2 marks]
   2. Using the graph, estimate the value of the constant \(a\) and the value of the constant \(k\) [4 marks]
2. 1. Show that \(\frac{dF}{dt} = kF\) [2 marks]
   2. Using the model, estimate the rate at which the number of followers is increasing 5 days after the song is released. [2 marks]
3. The singer claims that 30 days after the song is released, the account will have more than a billion followers. Comment on the singer's claim. [1 mark]

The volume of a spherical bubble is increasing at a constant rate. Show that the rate of increase of the radius, \(r\), of the bubble is inversely proportional to \(r^2\) Volume of a sphere = \(\frac{4}{3}\pi r^3\) [4 marks]

Use the chain rule to find \(\frac{dy}{dx}\) when \(y = (x^3 + 5x)^7\) [2 marks]

A curve has equation \(y = \frac{2x + 3}{4x^2 + 7}\)

1. 1. Find \(\frac{dy}{dx}\) [2 marks]
   2. Hence show that \(y\) is increasing when \(4x^2 + 12x - 7 < 0\) [4 marks]
2. Find the values of \(x\) for which \(y\) is increasing. [2 marks]

A function \(f\) is defined by \(f(x) = \frac{x}{\sqrt{2x - 2}}\)

1. State the maximum possible domain of \(f\). [2 marks]
2. Use the quotient rule to show that \(f'(x) = \frac{x - 2}{(2x - 2)^{\frac{3}{2}}}\). [3 marks]
3. Show that the graph of \(y = f(x)\) has exactly one point of inflection. [7 marks]
4. Write down the values of \(x\) for which the graph of \(y = f(x)\) is convex. [1 mark]

Water is poured into an empty cone at a constant rate of 8 cm³/s After \(t\) seconds the depth of the water in the inverted cone is \(h\) cm, as shown in the diagram below. \includegraphics{figure_8} When the depth of the water in the inverted cone is \(h\) cm, the volume, \(V\) cm³, is given by $$V = \frac{\pi h^3}{12}$$

1. Show that when \(t = 3\) $$\frac{dV}{dh} = 6 \sqrt[3]{6\pi}$$ [4 marks]
2. Hence, find the rate at which the depth is increasing when \(t = 3\) Give your answer to three significant figures. [3 marks]

\includegraphics{figure_3} The curve \(C_1\), shown in Figure 3, has equation \(y = 4x^2 - 6x + 4\). The point \(P\left(\frac{1}{2}, 2\right)\) lies on \(C_1\) The curve \(C_2\), also shown in Figure 3, has equation \(y = \frac{1}{2}x + \ln(2x)\). The normal to \(C_1\) at the point \(P\) meets \(C_2\) at the point \(Q\). Find the exact coordinates of \(Q\). (Solutions based entirely on graphical or numerical methods are not acceptable.) [8]