1.07p Points of inflection: using second derivative

A pencil holder is in the shape of an open circular cylinder of radius \(r\) cm and height \(h\) cm. The surface area of the cylinder (including the base) is 250 cm\(^2\).

1. Show that the volume, V cm\(^3\), of the cylinder is given by \(V = 125r - \frac{\pi r^3}{2}\). [4]
2. Use calculus to find the value of \(r\) for which \(V\) has a stationary value. [3]
3. Prove that the value of \(r\) you found in part (b) gives a maximum value for \(V\). [2]
4. Calculate, to the nearest cm\(^3\), the maximum volume of the pencil holder. [2]

\includegraphics{figure_3} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 4. Given that the capacity of a carton has to be 1030 cm\(^3\),

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm\(^2\), of a carton is given by $$A = 4x^2 + \frac{3090}{x}.$$ [3]

The manufacturer needs to minimise the surface area of a carton.

1. Use calculus to find the value of \(x\) for which \(A\) is a minimum. [5]
2. Calculate the minimum value of \(A\). [2]
3. Prove that this value of \(A\) is a minimum. [2]

The curve \(y = (1 - x)(x^2 + 4x + k)\) has a stationary point when \(x = -3\).

1. Find the value of the constant \(k\). [7]
2. Determine whether the stationary point is a maximum or minimum point. [2]
3. Given that \(y = 9x - 9\) is the equation of the tangent to the curve at the point \(A\), find the coordinates of \(A\). [5]

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

1. Verify that the curve has a stationary point when \(x = 1\). [5]
2. Determine the nature of this stationary point. [2]
3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\). [2]

\includegraphics{figure_2} A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side \(x\) cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 2.

1. Show that the volume, \(V\) cm\(^3\), of the tray is given by $$V = 4x(x^2 - 45x + 500)$$ [3]
2. State the range of possible values of \(x\). [1]
3. Find the value of \(x\) for which \(V\) is a maximum. [4]
4. Hence find the maximum value of \(V\). [2]
5. Justify that the value of \(V\) you found in part (d) is a maximum. [2]

\includegraphics{figure_3} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 3. Given that the capacity of a carton has to be 1030 cm\(^3\),

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm\(^2\), of a carton is given by \(A = 4x^2 + \frac{3090}{x}\). [3]

The manufacturer needs to minimise the surface area of a carton.

1. Use calculus to find the value of \(x\) for which \(A\) is a minimum. [5]
2. Calculate the minimum value of \(A\). [2]
3. Prove that this value of \(A\) is a minimum. [2]

\includegraphics{figure_3} A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side \(x\) cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 3.

1. Show that the volume, \(V\) cm\(^3\), of the tray is given by \(V = 4x(x^2 - 45x + 500)\). [3]
2. State the range of possible values of \(x\). [1]
3. Find the value of \(x\) for which \(V\) is a maximum. [4]
4. Hence find the maximum value of \(V\). [2]
5. Justify that the value of \(V\) you found in part (d) is a maximum. [2]

On a journey, the average speed of a car is \(v\) m s\(^{-1}\). For \(v \geq 5\), the cost per kilometre, \(C\) pence, of the journey is modelled by \(C = \frac{160}{v} + \frac{v^2}{100}\). Using this model,

1. show, by calculus, that there is a value of \(v\) for which \(C\) has a stationary value, and find this value of \(v\). [5]
2. Justify that this value of \(v\) gives a minimum value of \(C\). [2]
3. Find the minimum value of \(C\) and hence find the minimum cost of a 250 km car journey. [3]

For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),

1. find \(\frac{dy}{dx}\), [2]
2. find the coordinates of each of the stationary points, [5]
3. determine the nature of each stationary point. [3]

The point \(A\), on the curve \(C\), has \(x\)-coordinate \(1\).

1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

$$f(x) = 2 - x + 3x^{\frac{1}{2}}, \quad x > 0.$$

1. Find \(f'(x)\) and \(f''(x)\). [3]
2. Find the coordinates of the turning point of the curve \(y = f(x)\). [4]
3. Determine whether the turning point is a maximum or minimum point. [2]

\includegraphics{figure_3} Two uniform rods \(AB\) and \(BC\), each of length \(a\) and mass \(m\), are rigidly joined together so that \(AB\) is perpendicular to \(BC\). The rod \(AB\) is freely hinged to a fixed point at \(A\). The rods can rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda mg\) is attached to \(B\). The other end of the string is attached to a fixed point \(D\) vertically above \(A\), where \(AD = a\). The string \(BD\) makes an angle \(\theta\) radians with the downward vertical (see diagram).

1. Taking \(D\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = \frac{1}{2}mga(\sin 2\theta - 3\cos 2\theta) + \frac{1}{2}\lambda mga(2\cos \theta - 1)^2 - 2mga.$$ [5]
2. Given that \(\theta = \frac{1}{3}\pi\) is a position of equilibrium, find the exact value of \(\lambda\). [4]
3. Find \(\frac{d^2V}{d\theta^2}\) and hence determine whether the position of equilibrium at \(\theta = \frac{1}{3}\pi\) is stable or unstable. [4]

Find the coordinates of the stationary points of the curve with equation $$x^3 + y^3 - 3xy = 48$$ and determine their nature. [14]

The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_4}

1. Determine the coordinates of \(B\). [6]
2. Hence determine the exact value of \(\tan \alpha\). [3]

A curve has equation \(y = 2\ln(k - 3x) + x^2 - 3x\), where \(k\) is a positive constant.

1. Given that the curve has a point of inflection where \(x = 1\), show that \(k = 6\). [5] It is also given that the curve intersects the \(x\)-axis at exactly one point.
2. Show by calculation that the \(x\)-coordinate of this point lies between 0.5 and 1.5. [2]
3. Use the Newton-Raphson method, with initial value \(x_0 = 1\), to find the \(x\)-coordinate of the point where the curve intersects the \(x\)-axis, giving your answer correct to 5 decimal places. Show the result of each iteration to 6 decimal places. [3]
4. By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places. [1]

\includegraphics{figure_8} The diagram shows the curve \(M\) with equation \(y = xe^{-2x}\).

1. Show that \(M\) has a point of inflection at the point \(P\) where \(x = 1\). [5]

The line \(L\) passes through the origin \(O\) and the point \(P\). The shaded region \(R\) is enclosed by the curve \(M\) and the line \(L\).

1. Show that the area of \(R\) is given by $$\frac{1}{4}(a + be^{-2}),$$ where \(a\) and \(b\) are integers to be determined. [6]

A curve has equation $$y = \frac{a}{\sqrt{x}} + bx^2 + \frac{c}{x^3} \quad \text{for } x > 0$$ where \(a\), \(b\) and \(c\) are positive constants. The curve has a single turning point. Use the second derivative of \(y\) to determine the nature of this turning point. You do not need to find the coordinates of the turning point. Fully justify your answer. [7 marks]

A curve has equation $$y = x^3 - 6x + \frac{9}{x}$$

1. Show that the \(x\) coordinates of the stationary points of the curve satisfy the equation $$x^4 - 2x^2 - 3 = 0$$ [3 marks]
2. Deduce that the curve has exactly two stationary points. [3 marks]
3. Find the coordinates and nature of the two stationary points. Fully justify your answer. [4 marks]
4. Write down the equation of a line which is a tangent to the curve in two places. [1 mark]

The function f is defined by $$f(x) = x^2 + 2 \cos x \text{ for } -\pi \leq x \leq \pi$$ Determine whether the curve with equation \(y = f(x)\) has a point of inflection at the point where \(x = 0\) Fully justify your answer. [4 marks]

A curve has equation \(y = 2x \cos 3x + (3x^2 - 4) \sin 3x\)

1. Find \(\frac{dy}{dx}\), giving your answer in the form \((mx^2 + n) \cos 3x\), where \(m\) and \(n\) are integers. [4 marks]
2. Show that the \(x\)-coordinates of the points of inflection of the curve satisfy the equation $$\cot 3x = \frac{9x^2 - 10}{6x}$$ [4 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]

The diagram shows part of the graph of \(y = e^{-x^2}\) \includegraphics{figure_7} The graph is formed from two convex sections, where the gradient is increasing, and one concave section, where the gradient is decreasing.

1. Find the values of \(x\) for which the graph is concave. [4 marks]
2. The finite region bounded by the \(x\)-axis and the lines \(x = 0.1\) and \(x = 0.5\) is shaded. \includegraphics{figure_7b} Use the trapezium rule, with 4 strips, to find an estimate for \(\int_{0.1}^{0.5} e^{-x^2} dx\) Give your estimate to four decimal places. [3 marks]
3. Explain with reference to your answer in part (a), why the answer you found in part (b) is an underestimate. [2 marks]
4. By considering the area of a rectangle, and using your answer to part (b), prove that the shaded area is 0.4 correct to 1 decimal place. [3 marks]

The equation of a curve is $$y = 6x^4 + 8x^3 - 21x^2 + 12x - 6.$$

1. In this question you must show detailed reasoning. Determine
   [12]
2. On the axes in the Printed Answer Booklet, sketch the curve whose equation is $$y = 6x^4 + 8x^3 - 21x^2 + 12x - 6.$$ [3]

A curve C has equation \(f(x) = 5x^3 + 2x^2 - 3x\).

1. Find the \(x\)-coordinate of the point of inflection. State, with a reason, whether the point of inflection is stationary or non-stationary. [5]
2. Determine the range of values of \(x\) for which C is concave. [2]