A particle $\displaystyle P$ moves in a straight line. At time $\displaystyle t$ seconds, the velocity, $\displaystyle v$$\displaystyle \mathrm{m}/\mathrm{s}$, of $\displaystyle P$ is given by

(a) Show that $\displaystyle P$ never comes to rest.

(b) Find the acceleration of $\displaystyle P$, in $\displaystyle \mathrm{m}/\mathrm{s}^{2}$, when $\displaystyle t = 4$

Using algebra

(c) find the distance, in m, that $\displaystyle P$ travels in the interval $\displaystyle 0 \leq t \leq 5$

Figure 2 shows a hollow hemisphere with radius 20 cm

The hemisphere contains liquid, which is dripping out of a small hole at the lowest point $\displaystyle A$ at a constant rate of $\displaystyle k \mathrm{cm}^3/\mathrm{s}$

At time $\displaystyle t$ seconds after the liquid starts to drip from the hemisphere, the height of the liquid is $\displaystyle h$ cm above $\displaystyle A$

The volume $\displaystyle V \mathrm{cm}^3$ of liquid in the hemisphere is given by

When $\displaystyle h = 12$, the height of the liquid is decreasing at a rate of $\displaystyle \frac{1}{60} \mathrm{cm}/\mathrm{s}$

Find the value of $\displaystyle k$

Give your answer in terms of $\displaystyle \pi$

Show that

Figure 2 shows a hollow right circular cone fixed with its axis of symmetry vertical.

The vertical angle of the cone is $\displaystyle 60^\circ$.

Initially the cone is empty.

At time $\displaystyle t=0$ liquid starts to fill the cone at a constant rate of $\displaystyle 0.2\,\mathrm{cm}^3/\mathrm{s}$.

At time $\displaystyle t$ seconds after the liquid starts to fill the cone, the height of the liquid is $\displaystyle h$ cm above $\displaystyle X$.

(a) Show that

The surface area of the liquid, shown shaded in Figure 2, is increasing at a constant rate of $\displaystyle p\,\mathrm{cm}^2/\mathrm{s}$ when $\displaystyle t=6$.

(b) Find, to 3 significant figures, the value of $\displaystyle p$.

A particle $\displaystyle P$ is moving along a straight line, from a fixed origin $\displaystyle O$

At time $\displaystyle t$ seconds ($\displaystyle t \geq 0$), the velocity, $\displaystyle v$$\displaystyle \mathrm{m}/\mathrm{s}$, of $\displaystyle P$ is given by

At time $\displaystyle t$ seconds the acceleration of $\displaystyle P$ is $\displaystyle a$$\displaystyle \mathrm{m}/\mathrm{s}^{2}$

(a) Find the value of $\displaystyle t$ for which $\displaystyle a = 0$

$\displaystyle P$ is instantaneously at rest at time $\displaystyle T_1$ seconds and at time $\displaystyle T_2$ seconds where $\displaystyle T_2 > T_1$

(b) Find the value of $\displaystyle T_1$ and the value of $\displaystyle T_2$

(c) Find the exact distance, in metres, that $\displaystyle P$ travels between the times $\displaystyle T_1$ and $\displaystyle T_2$

Show your working clearly.

Figure 3 shows a hollow right circular cone with radius 3 metres and height 4 metres above the vertex $\displaystyle A$

The cone is fixed with its axis of symmetry vertical.
The cone is initially empty.

Water pours into the cone at a constant rate of $\displaystyle 9\pi \mathrm{m}^3/\mathrm{s}$

At time $\displaystyle t$ seconds after the water starts to pour into the cone, the height of the water is $\displaystyle h$ metres above $\displaystyle A$

Find, in $\displaystyle \mathrm{m}/\mathrm{s}$, the rate at which the height of the water is increasing at the instant when $\displaystyle h = 2$

Figure 1 shows a solid right circular cylinder with radius $\displaystyle r$ cm and height $\displaystyle h$ cm

The total surface area of the cylinder is $\displaystyle 700\pi$$\displaystyle \mathrm{cm}^{2}$

The volume of the cylinder is $\displaystyle V$$\displaystyle \mathrm{cm}^{3}$

(a) Show that $\displaystyle V = \pi r(350 - r^2)$

Given that $\displaystyle r$ can vary and using calculus,

(b) find, in cm to 3 significant figures, the value of $\displaystyle r$ for which $\displaystyle V$ is a maximum.
Justify that this value of $\displaystyle r$ gives a maximum value of $\displaystyle V$

(c) Find, to 3 significant figures, the height $\displaystyle h$ cm for which $\displaystyle V$ is a maximum.

(a) Show that $\displaystyle 2\mathrm{e}^{-4t} \mathrm{sin} 2t = -\frac{\mathrm{d}y}{\mathrm{d}t} - 4y$

Given that $\displaystyle \frac{\mathrm{d}^{2}y}{\mathrm{d}t^2} + M \frac{\mathrm{d}y}{\mathrm{d}t} + Ny = 0$ where $\displaystyle M$ and $\displaystyle N$ are integers

(b) find the value of $\displaystyle M$ and the value of $\displaystyle N$

A particle $\displaystyle P$ is moving along a straight line. At time $\displaystyle t$ seconds ($\displaystyle t \geq 0$), its velocity, $\displaystyle v$$\displaystyle \mathrm{m}/\mathrm{s}$, is given by $\displaystyle v = 4t^2 - 6t + 5$ (a) Find the minimum speed of $\displaystyle P$

The acceleration of $\displaystyle P$ at time $\displaystyle T$ seconds is $\displaystyle 18 \mathrm{m}/\mathrm{s}^2$ (b) Find the value of $\displaystyle T$

When $\displaystyle t = 0$, $\displaystyle P$ is at the point $\displaystyle X$, and when $\displaystyle t = 3$, $\displaystyle P$ is at the point $\displaystyle Y$ (c) Find the distance $\displaystyle XY$

A solid right circular cylinder has base radius $\displaystyle r$ cm and height $\displaystyle h$ cm, as shown in Figure 1.

The cylinder has a total surface area of $\displaystyle 322\pi\,\mathrm{cm}^2$ and a volume of $\displaystyle V\,\mathrm{cm}^3$.

(a) Show that

Given that $\displaystyle r$ can vary,

(b) use calculus to find, to 3 significant figures, the value of $\displaystyle r$ for which $\displaystyle V$ is a maximum, justifying that this value of $\displaystyle r$ gives a maximum value of $\displaystyle V$.

(c) Find, to 3 significant figures, the maximum value of $\displaystyle V$.

Figure 3 shows a solid cuboid with sides of length $\displaystyle x$ cm, $\displaystyle 2x$ cm and $\displaystyle y$ cm

The volume of the cuboid is $\displaystyle 72 \mathrm{cm}^3$

The total surface area of the cuboid is $\displaystyle A$$\displaystyle \mathrm{cm}^2$

(a) Show that $\displaystyle A = \frac{216}{x} + 4x^2$

(b) Use calculus to find the minimum value of $\displaystyle A$
Justify that your value is a minimum value of $\displaystyle A$

Use calculus to find the gradient of the curve with equation

at the point on the curve where $\displaystyle x = 1$

Give your answer to 3 significant figures.

The surface area of a sphere with radius $\displaystyle r$ cm is increasing at a constant rate of $\displaystyle 50\pi$$\displaystyle \mathrm{cm}^{2}$/s

Find, in $\displaystyle \mathrm{cm}^{3}$, the exact volume of the sphere at the instant when the rate of increase

of $\displaystyle r$ is $\displaystyle \frac{5}{12}$ cm/s

A particle $\displaystyle P$ is moving along the $\displaystyle x$-axis.

At time $\displaystyle t$ seconds ($\displaystyle t \geq 0$) the acceleration, $\displaystyle a$$\displaystyle \mathrm{m}/\mathrm{s}^{2}$, of $\displaystyle P$ is given by $\displaystyle a = 3t - 4$

When $\displaystyle t = 0$, $\displaystyle P$ is at rest.

(a) Find the velocity of $\displaystyle P$ when $\displaystyle t = 4$

At time $\displaystyle T$ seconds, $\displaystyle T > 0$, $\displaystyle P$ is instantaneously at rest.

(b) Find the value of $\displaystyle T$

When $\displaystyle t = 0$, $\displaystyle P$ is at the point with coordinates $\displaystyle (-10, 0)$

(c) Find the displacement of $\displaystyle P$ from the origin when $\displaystyle t = 3$

Figure 1 shows a sketch of part of the curve $\displaystyle C$ with equation

The point $\displaystyle P$ lies on $\displaystyle C$ and has coordinates $\displaystyle (4, a)$

(a) Show that $\displaystyle a = 6$

The line $\displaystyle L$ is the normal to $\displaystyle C$ at the point $\displaystyle P$

(b) Show that an equation of $\displaystyle L$ is $\displaystyle 5y + 4x - 46 = 0$

The finite region $\displaystyle R$ is bounded by the curve $\displaystyle C$, the line $\displaystyle L$, the $\displaystyle x$-axis and the line with equation $\displaystyle x = 1$

(c) Use calculus to find the exact area of $\displaystyle R$

A particle $\displaystyle P$ is moving along a straight line.

At time $\displaystyle t$ seconds ($\displaystyle t \geq 0$), the velocity, $\displaystyle v$$\displaystyle \mathrm{m}/\mathrm{s}$, of $\displaystyle P$ is given by

(a) Find the values of $\displaystyle t$ when $\displaystyle P$ is instantaneously at rest.

At time $\displaystyle t$ seconds the acceleration of $\displaystyle P$ is $\displaystyle a$$\displaystyle \mathrm{m}/\mathrm{s}^{2}$

(b) Find the range of values of $\displaystyle t$ for which $\displaystyle a > 0$

(c) Find the distance, in m, that $\displaystyle P$ travels in the interval $\displaystyle 1 \leq t \leq 4$

Figure 2 shows a shape $\displaystyle ABCDEOFG$

$\displaystyle ABCO$ is a quarter circle with radius $\displaystyle r$ cm

$\displaystyle CDEO$ and $\displaystyle AOFG$ are congruent rectangles of length $\displaystyle r$ cm and width $\displaystyle x$ cm

The total area of the shape is $\displaystyle 100\mathrm{cm}^2$

The perimeter of the shape is $\displaystyle P$ cm

(b) Use calculus to find the value of $\displaystyle r$ for which $\displaystyle P$ is a minimum, justifying that this value of $\displaystyle r$ gives a minimum value of $\displaystyle P$

(c) Find the minimum value of $\displaystyle P$

A particle $\displaystyle P$ is moving in a straight line. The displacement $\displaystyle s$ of $\displaystyle P$, in metres, at time $\displaystyle t$ seconds, $\displaystyle t \geq 0$, is given by

At time $\displaystyle t = 0$, $\displaystyle P$ is at the point $\displaystyle A$ and at time $\displaystyle t = \frac{\pi}{6}$, $\displaystyle P$ is at the point $\displaystyle B$

(a) Find the exact distance $\displaystyle AB$

(b) Find the exact velocity of $\displaystyle P$ when $\displaystyle t = \frac{\pi}{3}$

Figure 4 shows a solid right triangular prism $\displaystyle ABCDEF$

The cross section of the prism is an isosceles triangle.

• $\displaystyle \angle DEC = \angle AFB = 90^\circ$

• $\displaystyle AB = DC = x$ cm

• $\displaystyle AD = BC = FE = y$ cm

• $\displaystyle AF = BF = DE = CE$

The triangular faces of the prism are vertical and the edges $\displaystyle AD$, $\displaystyle BC$ and $\displaystyle FE$ are horizontal.

The volume of the prism is $\displaystyle 3.6 \mathrm{cm}^3$

The total external surface area of the prism is $\displaystyle S \mathrm{cm}^2$

(a) Show that $\displaystyle S$ satisfies the equation

Given that $\displaystyle x$ can vary,

(b) use calculus, to find to 3 significant figures, the value of $\displaystyle x$ for which $\displaystyle S$ is a minimum.

Justify that this value of $\displaystyle x$ gives a minimum value of $\displaystyle S$

(c) Hence find, to 2 significant figures, the minimum value of $\displaystyle S$

The curve $\displaystyle C$ has equation $\displaystyle y = \mathrm{e}^{3x}(2x-1)^4$

Using calculus, find the exact value of the gradient of the tangent to $\displaystyle C$ when $\displaystyle x = 1$

The height of liquid in a vessel $\displaystyle P$ is $\displaystyle h$

The volume, $\displaystyle V$, of the liquid in $\displaystyle P$ is given by $\displaystyle V = 6h^3$

Liquid is leaking from $\displaystyle P$ at a constant rate of $\displaystyle 36 \mathrm{cm}^3/\mathrm{s}$

Find the exact rate of change, in $\displaystyle \mathrm{cm}/\mathrm{s}$, of $\displaystyle h$ when $\displaystyle V = 384 \mathrm{cm}^3$

The force $\displaystyle F$ newtons between two magnetic poles is given by the formula

where $\displaystyle r$ is the distance, in centimetres, between the poles.

The distance between the two poles is increasing at a constant rate of 0.7 cm/s

Find the rate of change of $\displaystyle F$, in newtons/s to 3 significant figures, when the distance between the poles is 2.8 cm

Show that $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{sin} 2x + (A + Bx) \mathrm{cos} 2x}{\sqrt{3 + 2x}}$ where $\displaystyle A$ and $\displaystyle B$ are integers to be found.

A particle $\displaystyle P$ is moving along the $\displaystyle x$-axis. At time $\displaystyle t$ seconds, $\displaystyle t \geq 0$, the velocity, $\displaystyle v \mathrm{m}/\mathrm{s}$, of $\displaystyle P$ is given by $\displaystyle v = 2t^2 - 16t + 30$

(a) Find the acceleration, in $\displaystyle \mathrm{m}/\mathrm{s}^2$, of $\displaystyle P$ when $\displaystyle t = 5$

$\displaystyle P$ comes to instantaneous rest at the points $\displaystyle M$ and $\displaystyle N$ at times $\displaystyle t_1$ seconds and $\displaystyle t_2$ seconds where $\displaystyle t_2 > t_1$

(b) Find the exact distance $\displaystyle MN$

A solid cuboid has width $\displaystyle x$ cm, length $\displaystyle 4x$ cm and height $\displaystyle h$ cm.
The volume of the cuboid is $\displaystyle 75 \mathrm{cm}^3$ and the surface area of the cuboid is $\displaystyle S \mathrm{cm}^2$

(a) Show that $\displaystyle S = 8x^2 + \frac{375}{2x}$

Given that $\displaystyle x$ can vary, using calculus,

(b) (i) find to 3 significant figures, the value of $\displaystyle x$ for which $\displaystyle S$ is a minimum,

(ii) justify that this value of $\displaystyle x$ gives a minimum value of $\displaystyle S$

(c) Find, to 3 significant figures, the minimum value of $\displaystyle S$

The equation of a curve is $\displaystyle y = \sqrt{\frac{\mathrm{e}^{4x}}{2x-3}}$

When $\displaystyle x$ is increased to $\displaystyle (x + \delta x)$, $\displaystyle y$ increases to $\displaystyle (y + \delta y)$ where $\displaystyle \delta x$ and $\displaystyle \delta y$ are small.

(a) Show that $\displaystyle \delta y \approx \frac{\mathrm{e}^{2x}(4x-7)}{(2x-3)^{\frac{3}{2}}} \delta x$

Given that $\displaystyle x = 2.5$

(b) find an estimate, to 2 significant figures, of the value of $\displaystyle \delta y$ when the value of $\displaystyle x$ increases by 0.2%

(a) Find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$

Give your answer in the form $\displaystyle \frac{Ae^{3x+1}(Bx-A)}{Cx^3}$ where $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are prime numbers to be found.

The value of $\displaystyle x$ increases by 2%

(b) Use your answer to part (a) to find an estimate, in terms of $\displaystyle x$, for the percentage change in $\displaystyle y$

Give your answer in the form $\displaystyle (Px - Q)$ where $\displaystyle P$ and $\displaystyle Q$ are integers.

Figure 4 shows a container in the shape of a right circular cone.

The container is fixed with its axis of symmetry vertical.

The vertical angle of the container is $\displaystyle 60^\circ$ as shown in the diagram.

At time $\displaystyle t$ seconds, $\displaystyle t > 0$, the height of oil in the container is $\displaystyle h$ cm and the volume of oil in the container is $\displaystyle V \mathrm{cm}^3$

(a) Show that $\displaystyle V = \frac{1}{9} \pi h^3$

At time $\displaystyle t$ seconds the surface area of oil in the container is $\displaystyle A \mathrm{cm}^2$, as shown in Figure 4

Oil is dripping out of the bottom of the container at a constant rate of $\displaystyle 4 \mathrm{cm}^3/\mathrm{s}$.

(b) Find the exact rate of change, in $\displaystyle \mathrm{cm}^2/\mathrm{s}$, of the surface area of oil in the container when $\displaystyle h = 24$

The curve with equation $\displaystyle y = mx^2 + 64\sqrt{x} + 39$ has a stationary point with coordinates $\displaystyle (4, n)$ where $\displaystyle m$ and $\displaystyle n$ are integers.

Using calculus

(a) find the value of $\displaystyle m$ and the value of $\displaystyle n$

(b) determine the nature of the stationary point.

Figure 3 shows a right triangular prism $\displaystyle ABCDEF$. A cross section $\displaystyle ABC$ of the prism is a triangle in which $\displaystyle AB = AC = r\mathrm{cm}$ and $\displaystyle \angle CAB = \frac{\pi}{3}$ radians.

In the prism

(a) Show that the volume of the prism is $\displaystyle \frac{5\sqrt{3}}{4} r^2 \mathrm{cm}^3$

The volume of the prism is increasing in such a way that the size of $\displaystyle \angle CAB$ and the size of $\displaystyle \angle DEF$ remain constant and the length of $\displaystyle AE$, the length of $\displaystyle BF$ and the length of $\displaystyle CD$ remain constant.

The lengths of $\displaystyle AB$, $\displaystyle AC$, $\displaystyle ED$ and $\displaystyle EF$ are each increasing at a constant rate of $\displaystyle 0.2\mathrm{cm}/\mathrm{s}$

(b) Find the exact rate of increase, in $\displaystyle \mathrm{cm}^3/\mathrm{s}$, of the volume of the prism when the area of the rectangular face $\displaystyle BCDF$ is $\displaystyle 60\mathrm{cm}^2$

A particle $\displaystyle P$ moves along the $\displaystyle x$-axis.

At time $\displaystyle t$ seconds ($\displaystyle t \geq 0$) the acceleration, $\displaystyle a$$\displaystyle \mathrm{m}/\mathrm{s}^{2}$, of $\displaystyle P$ is given by $\displaystyle a = 6t - 16$

When $\displaystyle t = 0$, $\displaystyle P$ is at the origin and is moving with velocity 12 $\displaystyle \mathrm{m}/\mathrm{s}$.

(a) Find an expression in terms of $\displaystyle t$ for

(i) the velocity of $\displaystyle P$ at time $\displaystyle t$ seconds

(ii) the displacement of $\displaystyle P$ at time $\displaystyle t$ seconds.

(b) Hence find the time at which $\displaystyle P$ first returns to the origin.

The volume of oil in a container is $\displaystyle V$$\displaystyle \mathrm{cm}^{3}$ when the height of the oil is $\displaystyle h$ cm. Oil is pouring into the container at a constant rate of $\displaystyle 12$$\displaystyle \mathrm{cm}^{3}$/s. Given that $\displaystyle V = 3h^3$

find the exact rate, in cm/s, at which the height of the oil is increasing when $\displaystyle V = 1536$$\displaystyle \mathrm{cm}^{3}$

Two numbers $\displaystyle x$ and $\displaystyle y$ are such that $\displaystyle 3x - y = 4$

(a) Show that $\displaystyle S = 5x^3 + 9x^2 - 24x + 16$

Given that $\displaystyle x$ can vary,

(b) use calculus to find the value of $\displaystyle x$ for which $\displaystyle S$ is a minimum, justifying that this value of $\displaystyle x$ gives a minimum value of $\displaystyle S$

(c) Find the minimum value of $\displaystyle S$

Figure 1 shows the sector $\displaystyle OAB$ of a circle with centre $\displaystyle O$.

The radius of the circle is $\displaystyle r$ cm and the angle $\displaystyle AOB$ is $\displaystyle \theta$ radians.

The area of the sector is $\displaystyle 675 \mathrm{cm}^2$

(a) Show that the perimeter of the sector, $\displaystyle P$ cm, is given by

Given that $\displaystyle r$ can vary,

(b) find, using calculus, the minimum value of $\displaystyle P$

Give your answer in the form $\displaystyle a\sqrt{b}$ where $\displaystyle a$ is an integer and $\displaystyle b$ is a prime number.

(c) Justify that the value of $\displaystyle P$ you found in (b) is a minimum.

A particle $\displaystyle P$ is moving along the $\displaystyle x$-axis.

At time $\displaystyle t$ seconds, $\displaystyle t \geq 0$, the velocity, $\displaystyle v$$\displaystyle \mathrm{m}/\mathrm{s}$, of $\displaystyle P$ is given by

(a) Find the acceleration of $\displaystyle P$ when $\displaystyle t = 5$

The particle comes to instantaneous rest at the points $\displaystyle A$ and $\displaystyle B$ at times $\displaystyle t_1$ seconds and $\displaystyle t_2$ seconds respectively, where $\displaystyle t_1 < t_2$

(b) Find the value of $\displaystyle t_1$ and the value of $\displaystyle t_2$

(c) Use calculus to find the distance $\displaystyle AB$

The point with coordinates $\displaystyle (4,-104)$ lies on the curve $\displaystyle C$ with equation $\displaystyle y=f(x)$.

Given that

(a) (i) show that $\displaystyle C$ passes through the origin,

(ii) show that $\displaystyle C$ has a maximum at the point on the curve where $\displaystyle x=0.5$.

The curve $\displaystyle C$ has another turning point at $\displaystyle A$ and another turning point at $\displaystyle B$. Given that the $\displaystyle x$ coordinate of $\displaystyle A$ is negative,

(b) (i) find the coordinates of $\displaystyle A$ and the coordinates of $\displaystyle B$,

(ii) determine the nature of these turning points.

The volume of a sphere is $\displaystyle 500\,\mathrm{cm}^3$.

(a) Calculate the radius, in $\displaystyle \mathrm{cm}$ to 3 significant figures, of the sphere.

The surface area of the sphere is increased by $\displaystyle 20\,\mathrm{cm}^2$.

(b) Using calculus, find an estimate for the increase in the radius, in $\displaystyle \mathrm{cm}$ to 2 significant figures, of the sphere.

A particle $\displaystyle P$ is moving along a straight line which passes through the fixed point $\displaystyle O$. At time $\displaystyle t$ seconds, $\displaystyle t\geq0$, the velocity, $\displaystyle v\,\mathrm{m\,s^{-1}}$, of $\displaystyle P$ is given by

At time $\displaystyle t$ seconds the acceleration of $\displaystyle P$ is $\displaystyle a\,\mathrm{m\,s^{-2}}$.

(a) Find an expression for $\displaystyle a$ in terms of $\displaystyle t$.

The displacement of $\displaystyle P$ from $\displaystyle O$ is $\displaystyle 7\,\mathrm{m}$ when $\displaystyle t=2$.

(b) Find the exact displacement of $\displaystyle P$ from $\displaystyle O$ when $\displaystyle t=4$.

Differentiate with respect to $\displaystyle x$:

(a) $$\mathrm{e}^{4x}(6x+2)^{\frac{3}{2}}.$$

Give your answer in the form

where $\displaystyle A$ and $\displaystyle B$ are integers.

(b) $$\frac{\mathrm{sin}\,3x}{(2x-4)^3}.$$

A particle $\displaystyle P$ moves along the $\displaystyle x$-axis. At time $\displaystyle t$ seconds, the displacement, $\displaystyle x$ metres, of $\displaystyle P$ from the origin $\displaystyle O$ is given by

(a) Find the velocity, in $\displaystyle \mathrm{m\,s^{-1}}$, of $\displaystyle P$ when $\displaystyle t=3$.

(b) Find the value of $\displaystyle t$ for which $\displaystyle P$ is instantaneously at rest.

(c) Find the acceleration, in $\displaystyle \mathrm{m\,s^{-2}}$, of $\displaystyle P$ when $\displaystyle t=2$.

Figure 2 shows a waste paper basket in the shape of a right prism with 5 faces and a cross section that is a trapezium. The top, $\displaystyle EFGH$, of the waste paper basket is open.

The base of the prism $\displaystyle ABCD$ is a rectangle with

The cross sections $\displaystyle HGBA$ and $\displaystyle EFCD$ are such that

The top, $\displaystyle EFGH$, is such that $\displaystyle EH=FG=h\,\mathrm{cm}$.

The volume of the waste paper basket is $\displaystyle 2250\,\mathrm{cm}^3$. The total surface area of the 5 faces is $\displaystyle S\,\mathrm{cm}^2$.

(a) Show that

Given that $\displaystyle x$ can vary,

(b) use calculus to find, to 3 significant figures, the value of $\displaystyle x$ for which $\displaystyle S$ is a minimum. Justify that this value gives a minimum value of $\displaystyle S$.

(c) Find, to 3 significant figures, the minimum value of $\displaystyle S$.

A curve $\displaystyle C$ has equation

where $\displaystyle a$ is a constant and $\displaystyle x\neq\frac{6}{a}$.

(a) Find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$.

The curve crosses the $\displaystyle y$-axis at the point $\displaystyle A$.

The normal to $\displaystyle C$ at the point $\displaystyle A$ is the line $\displaystyle l$ with equation

Show that

(b) (i) $\displaystyle a=3$,

(ii) the equation of $\displaystyle C$ is

(c) Using the axes on the opposite page, sketch $\displaystyle C$, showing clearly the asymptotes with their equations and the coordinates of the points where $\displaystyle C$ crosses the coordinate axes.

The line $\displaystyle l$ meets $\displaystyle C$ again at the point $\displaystyle D$.

(d) Find the $\displaystyle x$ coordinate of $\displaystyle D$.

Give your answer as an improper fraction.

When poured from a pipe, concrete is formed into the shape of a cuboid with a square base of side $\displaystyle x$ and height $\displaystyle 3x$.

The volume of the cuboid increases at a constant rate of $\displaystyle 8\,\mathrm{m}^3\mathrm{s}^{-1}$.

Find the rate of increase, in $\displaystyle \mathrm{m\,s^{-1}}$, of $\displaystyle x$ when $\displaystyle x=2$ metres.

Figure 3 shows a solid metal right circular cylinder of radius $\displaystyle r\,\mathrm{cm}$ and height $\displaystyle h\,\mathrm{cm}$.

The total surface area of the cylinder is $\displaystyle 600\,\mathrm{cm}^2$. The volume of the cylinder is $\displaystyle V\,\mathrm{cm}^3$.

(a) Show that

Given that $\displaystyle r$ can vary,

(b) (i) use calculus to show that the exact value of $\displaystyle r$ for which $\displaystyle V$ is a maximum is

(ii) justify that this value of $\displaystyle r$ gives a maximum value of $\displaystyle V$.

The cylinder is melted down and reformed into a sphere of radius $\displaystyle p\,\mathrm{cm}$.

(c) Find, to one decimal place, the greatest possible value of $\displaystyle p$.

The curve $\displaystyle C$ has equation

(a) Using calculus, find the coordinates of the stationary points on $\displaystyle C$.

(b) Show that

(c) Hence, or otherwise, determine the nature of each of these stationary points.

Find an equation of the normal to the curve with equation

at the point on the curve with coordinates $\displaystyle (1,-1)$.

Figure 3 shows a metal solid $\displaystyle S$. The solid is a right triangular prism.

The cross section of $\displaystyle S$ is an equilateral triangle with sides of length $\displaystyle x\,\mathrm{cm}$. The length of $\displaystyle S$ is $\displaystyle 4x\,\mathrm{cm}$.

The prism is being heated so that the cross-sectional area is increasing at a constant rate of $\displaystyle 0.03\,\mathrm{cm}^2\mathrm{s}^{-1}$.

(a) Find, giving your answer to 3 significant figures, $\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}$ when $\displaystyle x=2$.

(b) Find the rate of increase, in $\displaystyle \mathrm{cm}^3\mathrm{s}^{-1}$, of the volume of $\displaystyle S$ when $\displaystyle x=2$.

Differentiate with respect to $\displaystyle x$:

Given that

(a) show that

(b) Find the value of $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$ when $\displaystyle x=0$.

(c) Find an equation of the normal to the curve with equation $\displaystyle 2xy+5y=\mathrm{e}^x$ at the point where $\displaystyle x=0$. Give your answer in the form

where $\displaystyle p$, $\displaystyle q$ and $\displaystyle r$ are integers.

Figure 4 shows a right circular cone with base radius $\displaystyle r\,\mathrm{cm}$ and slant height $\displaystyle l\,\mathrm{cm}$. It also shows a sector of a circle with radius $\displaystyle R\,\mathrm{cm}$ and arc length $\displaystyle L\,\mathrm{cm}$.

The area of the curved surface of the cone is $\displaystyle A\,\mathrm{cm}^2$.

By considering how the sector can be folded to exactly form the curved surface of the cone, with $\displaystyle R$ and $\displaystyle L$ suitably chosen,

(a) prove that

Sand is poured onto a horizontal surface at a constant rate of $\displaystyle 1.5\,\mathrm{cm}^3\mathrm{s}^{-1}$. The sand forms a pile in the shape of a right circular cone. Its height is always three times the radius of its base.

Given that

where $\displaystyle k$ is a constant,

(b) find the exact value of $\displaystyle k$.

(c) Calculate the rate, in $\displaystyle \mathrm{cm}^2\mathrm{s}^{-1}$ to 3 significant figures, at which the curved surface area of the pile is increasing when the height of the pile is $\displaystyle 24\,\mathrm{cm}$.

The curve $\displaystyle C$ has equation

(a) Use calculus to find the coordinates of each of the stationary points on $\displaystyle C$.

(b) Determine the nature of each of these stationary points. Justify your answers.

The length of each side of a cube $\displaystyle S_1$ is increasing at a constant rate of $\displaystyle 0.1\,\mathrm{m\,s^{-1}}$.

(a) Find, in $\displaystyle \mathrm{m}^3\mathrm{s}^{-1}$, the rate of increase of its volume when the side length is $\displaystyle 2\,\mathrm{m}$.

The total surface area of a different cube $\displaystyle S_2$ is increasing at a constant rate of $\displaystyle 0.05\,\mathrm{m}^2\mathrm{s}^{-1}$.

(b) Find, in $\displaystyle \mathrm{m}^3\mathrm{s}^{-1}$, the rate of increase of its volume when the side length is $\displaystyle 6\,\mathrm{m}$.

A particle $\displaystyle P$ is moving in a straight line. At time $\displaystyle t$ seconds, $\displaystyle t\geq0$, the displacement, $\displaystyle s$ metres, of $\displaystyle P$ from a fixed point $\displaystyle O$ of the line is given by

Find the distance of $\displaystyle P$ from $\displaystyle O$ when $\displaystyle P$ is instantaneously at rest.

Given that

show that

Figure 2 shows a lawn $\displaystyle ABCDEF$, where $\displaystyle ABDE$ is a rectangle of length $\displaystyle y$ metres and width $\displaystyle 2x$ metres.

Each end of the lawn is in the shape of a semicircle of radius $\displaystyle x$ metres.

The perimeter of the lawn is $\displaystyle 90$ metres and the area of the lawn is $\displaystyle S\,\mathrm{m}^2$.

(a) Show that

where $\displaystyle k$ is a constant. State the value of $\displaystyle k$.

(b) Use calculus to find, to 4 significant figures, the value of $\displaystyle x$ for which $\displaystyle S$ is a maximum.

Justify that your value of $\displaystyle x$ gives the maximum value of $\displaystyle S$.

(c) Find, to the nearest whole number, the maximum value of $\displaystyle S$.

Given that