1.07n Stationary points: find maxima, minima using derivatives

Fig. 11 shows the curve \(y = x^3 - 3x^2 - x + 3\). \includegraphics{figure_11}

1. Use calculus to find \(\int_{-1}^{3} (x^3 - 3x^2 - x + 3) dx\) and state what this represents. [6]
2. Find the \(x\)-coordinates of the turning points of the curve \(y = x^3 - 3x^2 - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x^3 - 3x^2 - x + 3\) is a decreasing function. [5]

The curve \(C\) has equation \(y = \text{f}(x)\), where $$\text{f}(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$ The point \(P\) is a stationary point on \(C\).

1. Calculate the \(x\)-coordinate of \(P\). [4]
2. Show that the \(y\)-coordinate of \(P\) may be expressed in the form \(k - k \ln k\), where \(k\) is a constant to be found. [2]

The point \(Q\) on \(C\) has \(x\)-coordinate 1.

1. Find an equation for the normal to \(C\) at \(Q\). [4]

The normal to \(C\) at \(Q\) meets \(C\) again at the point \(R\).

1. Show that the \(x\)-coordinate of \(R\)
   1. satisfies the equation \(6 \ln x + x + \frac{2}{x} - 3 = 0\),
   2. lies between 0.13 and 0.14. [4]

A curve has equation \(y = \frac{x}{2 + 3\ln x}\). Find \(\frac{dy}{dx}\). Hence find the exact coordinates of the stationary point of the curve.

A curve is defined by the equation \((x + y)^2 = 4x\). The point \((1, 1)\) lies on this curve. By differentiating implicitly, show that \(\frac{dy}{dx} = \frac{2}{x + y} - 1\). Hence verify that the curve has a stationary point at \((1, 1)\). [4]

A curve has equation \(x^2 + 2y^2 = 4x\).

1. By differentiating implicitly, find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [3]
2. Hence find the exact coordinates of the stationary points of the curve. [You need not determine their nature.] [3]

The function f(x) is defined as \(f(x) = \frac{\ln x}{x}\). The graph of the function is shown in Fig. 6. \includegraphics{figure_6}

1. Give the coordinates of the point, P, where the curve crosses the \(x\)-axis. [1]
2. Use calculus to find the coordinates of the stationary point, Q, and show that it is a maximum. [6]

A curve has the equation \(y = (2x + 3)e^{-x}\).

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies in the interval \([-2, -1]\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3e^{x_n}}{e^{x_n} - 2}$$ with \(x_0 = -1\), to find \(x_1\), \(x_2\), \(x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [3]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]

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

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

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

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies between \(-2\) and \(-1\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3\mathrm{e}^{x_n}}{\mathrm{e}^{x_n} - 2},$$ with \(x_0 = -1\), to find \(x_1, x_2, x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [2]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]

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]

Show that the curve \(y = x^2 \ln x\) has a stationary point when \(x = \frac{1}{\sqrt{e}}\). [6]

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

1. Show that \(\frac{dy}{dx} = \frac{2x(x + 1)}{(2x + 1)^2}\). [4]
2. Find the coordinates of the stationary points of the curve. You need not determine their nature. [4]

A curve has equation \(y = \frac{x}{2 + 3\ln x}\). Find \(\frac{dy}{dx}\). Hence find the exact coordinates of the stationary point of the curve. [7]

\includegraphics{figure_1} The curve \(C\) has equation \(y = f(x)\), \(x \in \mathbb{R}\). Figure 1 shows the part of \(C\) for which \(0 \leq x \leq 2\). Given that $$\frac{dy}{dx} = e^x - 2x^2,$$ and that \(C\) has a single maximum, at \(x = k\),

1. show that \(1.48 < k < 1.49\). [3]

Given also that the point \((0, 5)\) lies on \(C\),

1. find \(f(x)\). [4]

The finite region \(R\) is bounded by \(C\), the coordinate axes and the line \(x = 2\).

1. Use integration to find the exact area of \(R\). [4]

The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation $$\frac{dn}{dt} = e^{0.5t} - 5, \quad t \geq 0.$$

1. Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue. [3]
2. Given that there are 20 people in the queue when the Post Office opens, solve the differential equation. [4]
3. Explain why this model would not be appropriate for large values of \(t\). [1]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(AC\), each of mass \(2m\) and length \(2L\), are freely jointed at \(A\). The mid-points of the rods are \(D\) and \(E\) respectively. A light inextensible string of length \(s\) is fixed to \(E\) and passes round small, smooth light pulleys at \(D\) and \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs vertically. The points \(A\), \(B\) and \(C\) lie in the same vertical plane with \(B\) and \(C\) on a smooth horizontal surface. The angles \(PAB\) and \(PAC\) are each equal to \(\theta\) (\(\theta > 0\)), as shown in Fig. 2.

1. Find the length of \(AP\) in terms of \(s\), \(L\) and \(\theta\). [2]
2. Show that the potential energy \(V\) of the system is given by $$V = 2mgL(3\cos\theta + \sin\theta) + \text{constant}.$$ [4]
3. Hence find the value of \(\theta\) for which the system is in equilibrium. [4]
4. Determine whether this position of equilibrium is stable or unstable. [4]

\includegraphics{figure_1} A smooth wire \(PMQ\) is in the shape of a semicircle with centre \(O\) and radius \(a\). The wire is fixed in a vertical plane with \(PQ\) horizontal and the mid-point \(M\) of the wire vertically below \(O\). A smooth bead \(B\) of mass \(m\) is threaded on the wire and is attached to one end of a light elastic string. The string has modulus of elasticity \(4mg\) and natural length \(\frac{3}{4}a\). The other end of the string is attached to a fixed point \(P\) which is a distance \(a\) vertically above \(O\), as shown in Fig. 1.

1. Show that, when \(\angle BFO = \theta\), the potential energy of the system is $$\frac{1}{16}mga(8 \cos \theta - 5)^2 - 2mga \cos^2\theta + \text{constant}.$$ [6]
2. Hence find the values of \(\theta\) for which the system is in equilibrium. [6]
3. Determine the nature of the equilibrium at each of these positions. [5]

\includegraphics{figure_1} A smooth wire with ends \(A\) and \(B\) is in the shape of a semi-circle of radius \(a\). The mid-point of \(AB\) is \(O\) and is fixed in a vertical plane and hangs below \(AB\) which is horizontal. A small ring \(R\), of mass \(m\sqrt{2}\), is threaded on the wire and is attached to two light inextensible strings. The other end of each string is attached to a particle of mass \(\frac{3m}{2}\). The particles hang vertically under gravity, as shown in Figure 1.

1. Show that, when the radius \(OR\) makes an angle \(2\theta\) with the vertical, the potential energy, \(V\), of the system is given by $$V = \sqrt{2}mga(3 \cos \theta - \cos 2\theta) + \text{constant}.$$ [7]
2. Find the values of \(\theta\) for which the system is in equilibrium. [6]
3. Determine the stability of the position of equilibrium for which \(\theta > 0\). [4]

A non-uniform rod \(BC\) has mass \(m\) and length \(3l\). The centre of mass of the rod is at distance \(l\) from \(B\). The rod can turn freely about a fixed smooth horizontal axis through \(B\). One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\frac{mg}{6}\), is attached to \(C\). The other end of the string is attached to a point \(P\) which is at a height \(3l\) vertically above \(B\).

1. Show that, while the string is stretched, the potential energy of the system is $$mgl(\cos^2 \theta - \cos \theta) + \text{constant},$$ where \(\theta\) is the angle between the string and the downward vertical and \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\). [6]
2. Find the values of \(\theta\) for which the system is in equilibrium with the string stretched. [6]

A curve \(C\) has equation \(y = \frac{1}{x(x + 2)}\).

1. Write down the equations of all the asymptotes of \(C\). [2 marks]
2. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is \(-1\).
   1. Find the \(y\)-coordinate of the stationary point. [1 mark]
   2. Sketch the curve \(C\). [2 marks]
3. Solve the inequality $$\frac{1}{x(x + 2)} \leqslant \frac{1}{8}$$ [5 marks]

The equation of a curve is $$y = \frac{kx}{(x-1)^2},$$ where \(k\) is a positive constant.

1. Write down the equations of the asymptotes of the curve. [2]
2. Show that \(y \geq -\frac{1}{4}k\). [4]
3. Show that the \(x\)-coordinate of the stationary point of the curve is independent of \(k\), and sketch the curve. [4]