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

Which one of these functions is decreasing for all real values of \(x\)? Circle your answer. \(f(x) = e^x\) \quad \(f(x) = -e^{1-x}\) \quad \(f(x) = -e^{x-1}\) \quad \(f(x) = -e^{-x}\) [1 mark]

A cylinder is to be cut out of the circular face of a solid hemisphere. The cylinder and the hemisphere have the same axis of symmetry. The cylinder has height \(h\) and the hemisphere has a radius of \(R\). \includegraphics{figure_9}

1. Show that the volume, \(V\), of the cylinder is given by $$V = \pi R^2 h - \pi h^3$$ [3 marks]
2. Find the maximum volume of the cylinder in terms of \(R\). Fully justify your answer. [7 marks]

A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds. The cost of the edging strip is £1.80 per metre and can be purchased for any length. One of the flowerbeds has a radius of 5 metres and an angle at the centre of 0.7 radians as shown in the diagram below. \includegraphics{figure_5}

1. 1. Find the area of this flowerbed. [2 marks]
   2. Find the cost of the edging strip required for this flowerbed. [3 marks]
2. A flowerbed is to be made with an area of 20 m²
   1. Show that the cost, £\(C\), of the edging strip required for this flowerbed is given by $$C = \frac{18}{5}\left(\frac{20}{r} + r\right)$$ where \(r\) is the radius measured in metres. [3 marks]
   2. Hence, show that the minimum cost of the edging strip for this flowerbed occurs when \(r \approx 4.5\) Fully justify your answer. [5 marks]

A function f is defined for all real values of \(x\) as $$f(x) = x^4 + 5x^3$$ The function has exactly two stationary points when \(x = 0\) and \(x = -\frac{15}{4}\)

1. 1. Find \(f''(x)\) [2 marks]
   2. Determine the nature of the stationary points. Fully justify your answer. [4 marks]
2. State the range of values of \(x\) for which $$f(x) = x^4 + 5x^3$$ is an increasing function. [1 mark]
3. A second function g is defined for all real values of \(x\) as $$g(x) = x^4 - 5x^3$$
   1. State the single transformation which maps f onto g. [1 mark]
   2. State the range of values of \(x\) for which g is an increasing function. [1 mark]

\includegraphics{figure_4} Figure 4 shows the plan view of the design for a swimming pool. The shape of this pool \(ABCDEA\) consists of a rectangular section \(ABDE\) joined to a semicircular section \(BCD\) as shown in Figure 4. Given that \(AE = 2x\) metres, \(ED = y\) metres and the area of the pool is \(250\text{m}^2\),

1. show that the perimeter, \(P\) metres, of the pool is given by $$P = 2x + \frac{250}{x} + \frac{\pi x}{2}$$ [4]
2. Explain why \(0 < x < \sqrt{\frac{500}{\pi}}\) [2]
3. Find the minimum perimeter of the pool, giving your answer to \(3\) significant figures. [4]

\includegraphics{figure_2} Figure 2 shows a solid cuboid \(ABCDEFGH\). \(AB = x\) cm, \(BC = 2x\) cm, \(AE = h\) cm The total surface area of the cuboid is 180 cm\(^2\). The volume of the cuboid is \(V\) cm\(^3\).

1. Show that \(V = 60x - \frac{4x^3}{3}\) [4]

Given that \(x\) can vary,

1. use calculus to find, to 3 significant figures, the value of \(x\) for which \(V\) is a maximum. Justify that this value of \(x\) gives a maximum value of \(V\). [5]
2. Find the maximum value of \(V\), giving your answer to the nearest cm\(^3\). [2]

Prove that \(f(x) = x^3 - 6x^2 + 13x - 7\) is an increasing function. [4]

The diagram below shows a sketch of the curve \(y = f(x)\), where \(f(x) = 10x + 3x^2 - x^3\). The curve intersects the \(x\)-axis at the origin \(O\) and at the points \(A(-2, 0)\), \(B(5, 0)\). The tangent to the curve at the point \(C(2, 24)\) intersects the \(y\)-axis at the point \(D\). \includegraphics{figure_11}

1. Find the coordinates of \(D\). [5]
2. Find the area of the shaded region. [6]
3. Determine the range of values of \(x\) for which \(f(x)\) is an increasing function. [4]

A curve \(C\) has equation \(f(x) = 3x^3 - 5x^2 + x - 6\).

1. Find the coordinates of the stationary points of \(C\) and determine their nature. [8]
2. Without solving the equations, determine the number of distinct real roots for each of the following:
   1. \(3x^3 - 5x^2 + x + 1 = 0\),
   2. \(6x^3 - 10x^2 + 2x + 1 = 0\). [4]

A curve C has equation \(y = -x^3 + 12x - 20\).

1. Find the coordinates of the stationary points of C and determine their nature. [7]
2. Determine the range of values of \(x\) for which the curve is decreasing. Give your answer in set notation. [3]

Find the range of values of \(x\) for which the function $$f(x) = x^3 - 5x^2 - 8x + 13$$ is an increasing function. [5]

The diagram below shows a sketch of the graph of \(y = f'(x)\) for the interval \([x_1, x_5]\). \includegraphics{figure_13}

1. Find the interval on which \(f(x)\) is both decreasing and convex. Give reasons for your answer. [2]
2. Write down the \(x\)-coordinate of a point of inflection of the graph of \(y = f(x)\). [1]

A curve has equation $$y = 2x^3 - 2x^2 - 2x + 8$$

1. Find \(\frac{dy}{dx}\) [2]
2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation. [4]

\includegraphics{figure_10} The figure above shows solid right prism of height \(h\) cm. The cross section of the prism is a circular sector of radius \(r\) cm, subtending an angle of 2 radians at the centre.

1. Given that the volume of the prism is 1000 cm\(^3\), show clearly that $$S = 2r^2 + \frac{4000}{r},$$ where \(S\) cm\(^2\) is the total surface area of the prism. [5]
2. Hence determine the value of \(r\) and the value of \(h\) which make \(S\) least, fully justifying your answer. [7]

In this question you must show detailed reasoning. Find the values of \(x\) for which the gradient of the curve \(y = \frac{2}{3}x^3 + \frac{5}{2}x^2 - 3x + 7\) is positive. Give your answer in set notation. [5]

Let \(y = (x - 1)\left(\frac{2}{x^2} + t\right)\) define \(y\) as a function of \(x\) (\(x > 0\)), for each value of the real parameter \(t\).

1. When \(t = 0\),
   1. determine the set of values of \(x\) for which \(y\) is positive and an increasing function, [3]
   2. locate the stationary point of \(y\), and determine its nature. [2]
2. It is given that \(t = 2\) and \(y = -2\).
   1. Show that \(x\) satisfies \(f(x) = 0\), where \(f(x) = x^3 + x - 1\). [1]
   2. Prove that \(f\) has no stationary points. [2]
   3. Use the Newton-Raphson method, with \(x_0 = 1\), to find \(x\) correct to 4 significant figures. [4]

Let \(y = (2x - 3)e^{-2x}\).

1. Find \(\frac{dy}{dx}\), giving your answer in the form \(e^{-2x}(ax + b)\), where \(a\) and \(b\) are integers. [3]
2. Determine the set of values of \(x\) for which \(y\) is increasing. [2]