1.07n Stationary points: find maxima, minima using derivatives

The curve \(C\) has equation $$y = \frac{x^2 + ax - 1}{x + 1},$$ where \(a\) is constant and \(a > 1\).

1. Find the equations of the asymptotes of \(C\). [3]
2. Show that \(C\) intersects the \(x\)-axis twice. [1]
3. Justifying your answer, find the number of stationary points on \(C\). [2]
4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis. [3]

The curve \(C\) has equation $$y = \frac{5x^2 + 5x + 1}{x^2 + x + 1}.$$

1. Find the equation of the asymptote of \(C\). [2]
2. Show that, for all real values of \(x\), \(-\frac{1}{5} \leqslant y < 5\). [4]
3. Find the coordinates of any stationary points of \(C\). [2]
4. Sketch \(C\), stating the coordinates of any intersections with the \(y\)-axis. [2]

A particle starts from rest and moves in a straight line. The velocity of the particle at time \(t\) s after the start is \(v\) m s\(^{-1}\), where $$v = -0.01t^3 + 0.22t^2 - 0.4t.$$

1. Find the two positive values of \(t\) for which the particle is instantaneously at rest. [2]
2. Find the time at which the acceleration of the particle is greatest. [3]
3. Find the distance travelled by the particle while its velocity is positive. [4]

A particle moves in a straight line starting from a point \(O\) with initial velocity \(1\) m s\(^{-1}\). The acceleration of the particle at time \(t\) s after leaving \(O\) is \(a\) m s\(^{-2}\), where $$a = 1.2t^{\frac{1}{2}} - 0.6t.$$

1. At time \(T\) s after leaving \(O\) the particle reaches its maximum velocity. Find the value of \(T\). [2]
2. Find the velocity of the particle when its acceleration is maximum (you do not need to verify that the acceleration is a maximum rather than a minimum). [6]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A curve has equation $$y = 256x^4 - 304x - 35 + \frac{27}{x^2} \quad x \neq 0$$

1. Find \(\frac{dy}{dx}\) [3]
2. Hence find the coordinates of the stationary points of the curve. [5]

\includegraphics{figure_3} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the stage is \(2x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2x\) metres. The perimeter of the stage is 80 m.

1. Show that the area, \(A\) m², of the stage is given by $$A = 80x - \left(2 + \frac{\pi}{2}\right)x^2.$$ [4]
2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value. [4]
3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\). [2]
4. Calculate, to the nearest m², the maximum area of the stage. [2]

Find the coordinates of the stationary point on the curve with equation \(y = 2x^2 - 12x\). [4]

The curve \(C\) has equation \(y = 2x^3 - 5x^2 - 4x + 2\).

1. Find \(\frac{dy}{dx}\). [2]
2. Using the result from part (a), find the coordinates of the turning points of \(C\). [4]
3. Find \(\frac{d^2y}{dx^2}\). [2]
4. Hence, or otherwise, determine the nature of the turning points of \(C\). [2]

\includegraphics{figure_4} Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm. The total surface area of the brick is 600 cm².

1. Show that the volume, \(V\) cm³, of the brick is given by \(V = 200x - \frac{4x^3}{3}\). [4]

Given that \(x\) can vary,

1. use calculus to find the maximum value of \(V\), giving your answer to the nearest cm³. [5]
2. Justify that the value of \(V\) you have found is a maximum. [2]

\includegraphics{figure_4} Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle \(x\) metres by \(y\) metres. The height of the tank is \(x\) metres. The capacity of the tank is 100 m³.

1. Show that the area \(A\) m² of the sheet metal used to make the tank is given by $$A = \frac{300}{x} + 2x^2.$$ [4]
2. Use calculus to find the value of \(x\) for which \(A\) is stationary. [4]
3. Prove that this value of \(x\) gives a minimum value of \(A\). [2]
4. Calculate the minimum area of sheet metal needed to make the tank. [2]

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]

\includegraphics{figure_5} Figure 2 shows part of the curve with equation $$y = x^3 - 6x^2 + 9x.$$ The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).

1. Show that the equation of the curve may be written as $$y = x(x - 3)^2,$$ and hence write down the coordinates of \(A\). [2]
2. Find the coordinates of \(B\). [5]

The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

1. Find the area of \(R\). [5]

Given that \(f(x) = 15 - 7x - 2x^2\),

1. find the coordinates of all points at which the graph of \(y = f(x)\) crosses the coordinate axes. [3]
2. Sketch the graph of \(y = f(x)\). [2]
3. Calculate the coordinates of the stationary point of \(f(x)\). [3]

\includegraphics{figure_9} 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]

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]

The curve \(C\) has equation \(y = f(x)\), where $$f(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$ 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 particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \text{ ms}^{-1}\) in the positive \(x\)-direction, where \(v = 3t^2 - 4t + 3\). When \(t = 0\), \(P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) is moving with minimum velocity. [8]

A cuboid has a volume of \(8 \text{m}^3\). The base of the cuboid is square with sides of length \(x\) metres. The surface area of the cuboid is \(A \text{m}^2\).

1. Show that \(A = 2x^2 + \frac{32}{x}\). [3]
2. Find \(\frac{dA}{dx}\). [3]
3. Find the value of \(x\) which gives the smallest surface area of the cuboid, justifying your answer. [4]

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]

The curve with equation \(y = 2x^3 - 8x^{\frac{1}{3}}\) has a minimum at the point \(A\).

1. Find \(\frac{dy}{dx}\). [3]
2. Find the \(x\)-coordinate of \(A\). [3]

The point \(B\) on the curve has \(x\)-coordinate 2.

1. Find an equation for the tangent to the curve at \(B\) in the form \(y = mx + c\). [6]

A curve has the equation $$y = x^3 + ax^2 - 15x + b,$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \((-1, 12)\),

1. find the values of \(a\) and \(b\), [6]
2. find the coordinates of the other stationary point of the curve. [3]