1.07o Increasing/decreasing: functions using sign of dy/dx

$$f(x) = x^3 + x - 3.$$ The equation \(f(x) = 0\) has a root, \(\alpha\), between 1 and 2.

1. By considering \(f'(x)\), show that \(\alpha\) is the only real root of the equation \(f(x) = 0\). [3]
2. Taking 1.2 as your first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(f(x)\) to obtain a second approximation to \(\alpha\). Give your answer to 3 significant figures. [2]
3. Prove that your answer to part (b) gives the value of \(\alpha\) correct to 3 significant figures. [2]

The curve \(C\) has the equation $$y = 3 - x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}, \quad x > 0.$$

1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis. [4]
2. Find the exact coordinates of the stationary point of \(C\). [5]
3. Determine the nature of the stationary point. [2]
4. Sketch the curve \(C\). [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]

Differentiate \(2x^3 + 9x^2 - 24x\). Hence find the set of values of \(x\) for which the function \(f(x) = 2x^3 + 9x^2 - 24x\) is increasing. [4]

Use calculus to find the set of values of \(x\) for which \(\text{f}(x) = 12x - x^3\) is an increasing function. [3]

\includegraphics{figure_11} Fig. 11 shows a sketch of the curve with equation \(y = x - \frac{4}{x^2}\).

1. Find \(\frac{dy}{dx}\) and show that \(\frac{d^2y}{dx^2} = -\frac{24}{x^4}\). [3]
2. Hence find the coordinates of the stationary point on the curve. Verify that the stationary point is a maximum. [5]
3. Find the equation of the normal to the curve when \(x = -1\). Give your answer in the form \(ax + by + c = 0\). [5]

The curve \(C\) has the equation $$y = 3 - x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}, \quad x > 0.$$

1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis. [4]
2. Find the exact coordinates of the stationary point of \(C\). [5]
3. Determine the nature of the stationary point. [2]
4. Sketch the curve \(C\). [2]

Differentiate \(2x^3 + 9x^2 - 24x\). Hence find the set of values of \(x\) for which the function \(f(x) = 2x^3 + 9x^2 - 24x\) is increasing. [4]

Find the set of values of \(x\) for which \(x^2 - 7x\) is a decreasing function. [3]

The gradient of a curve is given by \(\frac{dy}{dx} = x^2 - 6x\). Find the set of values of \(x\) for which \(y\) is an increasing function of \(x\). [3]

The function f, defined for \(x \in \mathbb{R}, x > 0\), is such that $$f'(x) = x^2 - 2 + \frac{1}{x^2}$$

1. Find the value of f''(x) at \(x = 4\). [3]
2. Given that f(3) = 0, find f(x). [4]
3. Prove that f is an increasing function. [3]

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 $$y = x^3 - 6x + \frac{9}{x}$$

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]

A continuous curve has equation \(y = f(x)\) The curve passes through the points \(A(2, 1)\), \(B(4, 5)\) and \(C(6, 1)\) It is given that \(f'(4) = 0\) Jasmin made two statements about the nature of the curve \(y = f(x)\) at the point \(B\): Statement 1: There is a turning point at \(B\) Statement 2: There is a maximum point at \(B\)

1. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is correct and Statement 2 is correct. [1 mark]
2. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is correct and Statement 2 is not correct. [1 mark]
3. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is not correct and Statement 2 is not correct. [1 mark]

It is given that for the continuous function \(g\) • \(g'(1) = 0\) • \(g'(4) = 0\) • \(g''(x) = 2x - 5\)

1. Determine the nature of each of the turning points of \(g\) Fully justify your answer. [3 marks]
2. Find the set of values of \(x\) for which \(g\) is an increasing function. [2 marks]

Chris claims that, "for any given value of \(x\), the gradient of the curve \(y = 2x^3 + 6x^2 - 12x + 3\) is always greater than the gradient of the curve \(y = 1 + 60x - 6x^2\)". Show that Chris is wrong by finding all the values of \(x\) for which his claim is not true. [7 marks]

A curve has gradient function $$\frac{dy}{dx} = 3x^2 - 12x + c$$ The curve has a turning point at \((-1, 1)\)

1. Find the coordinates of the other turning point of the curve. Fully justify your answer. [6 marks]
2. Find the set of values of \(x\) for which \(y\) is increasing. [2 marks]

Prove that the graph of the curve with equation $$y = x^3 + 15x - \frac{18}{x}$$ has no stationary points. [5 marks]

Prove that the function \(f(x) = x^3 - 3x^2 + 15x - 1\) is an increasing function. [6 marks]

An open-topped fish tank is to be made for an aquarium. It will have a square horizontal base, rectangular vertical sides and a volume of 60 m\(^3\) The materials cost:

• £15 per m\(^2\) for the base
• £8 per m\(^2\) for the sides.

1. Modelling the sides and base of the fish tank as laminae, use calculus to find the height of the tank for which the overall cost of the materials has its minimum value. Fully justify your answer. [8 marks]
2. 1. In reality, the thickness of the base and sides of the tank is 2.5 cm Briefly explain how you would refine your modelling to take account of the thickness of the sides and base of the tank. [1 mark]
   2. How would your refinement affect your answer to part (a)? [1 mark]