Further Applications of Differentiation (AQA A Level Maths: Pure): Exam Questions

An ice cube, of side length $x\text{ cm}$, is melting such that its volume is decreasing at a constant rate of $0.8{\text{ cm}}^3{\text{ s}}^{-1}$.

Assuming that the ice cube remains in the shape of a cube whilst it melts, find the rate at which its surface area is decreasing at the instant when $x=2$.

A bowl is in the shape of a hemisphere of radius $8\text{ cm}$.

The volume of liquid in the bowl, $V{\text{ cm}}^3$, is given by the formula

$V=8\pi h^2-\frac{1}{3}\pi h^{3}0⩽h⩽8$

where $h\text{ cm}$ is the depth of the liquid.

Liquid is leaking through a small hole in the bottom of the bowl at a constant rate of $5{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate of change of the depth of the liquid in the bowl at the instant when $h=3$.

The function $\text{f}$ is defined by

$\text{f}\left(x\right)=x^2-{\text{e}}^{2x}+1x\in \mathrm{ℝ}$

Find the set of values of $x$ for which the curve $y=\text{f}\left(x\right)$ is concave.

The curve $C$ has equation

$y=x^3+3x^2-2x\in \mathrm{ℝ}$

Find the coordinates of the point of inflection on $C$.

A cube has side length $x\text{ cm}$, surface area $S{\text{ cm}}^2$ and volume $V{\text{ cm}}^3$.

Show that

$\frac{\text{d}S}{\text{d}V}=\frac{4}{x}$

The volume of the cube is decreasing at a constant rate of $0.6{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate at which the surface area of the cube is decreasing when $x=5$.

The curve $C$ has equation

$y=x^2+\ln x-2xx>0$

Show that

$\frac{{\text{d}}^{2}y}{\text{d}{x}^2}=2-\frac{1}{x^2}$

Solve the inequality

$\frac{{\text{d}}^{2}y}{\text{d}{x}^2}>0$

and hence determine the set of values of $x$ for which $C$ is convex.

In a computer animation, the radius, $r\text{ mm}$, of a circle is increasing at a constant rate of $1{\text{ mm s}}^{-1}$.

Find the rate at which the area of the circle is increasing at the instant when the radius is $8\text{ mm}$.

Give your answer as a multiple of $\pi$.

The curve $C$ has equation

$y=x^3-6x^2+9x-1x\in \mathrm{ℝ}$

Find the $x$ coordinates of the stationary points on $C$.

Determine the nature of the stationary points found in part (a).

Determine the $x$ coordinate of the point of inflection on $C$.

Explain why, in this case, the point of inflection is not a stationary point.

The continuous curve $C$ has equation $y=\text{f}\left(x\right)$, where $x\in \mathrm{ℝ}$.

The graph of $C$ has the following properties:

• $C$ crosses the $x$-axis at the points $(b, 0)$, $(c, 0)$ and $(d, 0)$, where $0<b<c<d$

• $C$ crosses the $y$-axis at $(0, g)$

• The turning points of $C$ have $x$ coordinates $e$ and $f$, where $e<f$

• $C$ is concave for $x<a$

• $C$ is convex for $x>a$

Given also that $\text{f}\left(a\right)>0$, sketch the curve $C$.

On your sketch, label the coordinates of the points where $C$ crosses the coordinate axes, and mark the $x$ coordinates of the turning points and the point of inflection.

The side length, $x\text{ cm}$, of a cube is increasing at a constant rate of $0.1{\text{ cm s}}^{-1}$.

Assuming that the cube remains cubical at all times, find the rate of change of the volume of the cube at the instant when $x=5$.

In the production process of a glass sphere, hot glass is blown such that the radius, $r\text{ cm}$, increases over time, $t$ seconds.

The rate of increase of the radius is directly proportional to the temperature, $T °\text{C}$, of the glass.

Let $V{\text{ cm}}^3$ be the volume of the glass sphere.

Find an expression, in terms of $r$, $T$ and a constant of proportionality $k$, for the rate of change of the volume of the sphere.

When the temperature of the glass is $1200 °\text{C}$, the sphere has a radius of $2\text{ cm}$ and its volume is increasing at a rate of $5{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate of increase of the radius at this instant.

The curve $C$ has equation

$y=x^4+3x^3+2x\in \mathrm{ℝ}$

Determine the number of points of inflection on $C$ and determine their coordinates.

The surface area, $S{\text{ cm}}^2$, of a spherical air bubble is increasing at a constant rate of $4\pi {\text{ cm}}^2{\text{ s}}^{-1}$.

Given that the surface area of a sphere is $S=4\pi r^2$, where $r\text{ cm}$ is the radius of the bubble,

find an expression, in terms of $r$, for the rate at which the radius of the bubble is increasing.

The curve $C$ has equation

$y=4x-x^2-2x^{3}x\in \mathrm{ℝ}$

Find the coordinates of the stationary points on $C$, and determine their nature.

Determine the coordinates of any points of inflection on $C$, and hence state the intervals in which $C$ is convex and concave.

The diagram below shows a sketch of the graph with equation $y=\mathrm{f}\left(x\right).$

Figure 1 shows a sketch of a continuous curve with equation $y=\text{f}\left(x\right)$.

On the sketch in Figure 1,

(i) mark the approximate locations of the intercepts with the coordinate axes using the letter $C$

(ii) mark the approximate locations of the stationary points using the letter $S$

(iii) mark the approximate locations of the points of inflection using the letter $I$

(iv) highlight the sections of the curve where the graph is convex.

A container is in the shape of an inverted right circular cone. The radius of the base of the cone is $3\text{ cm}$ and the height of the cone is $9\text{ cm}$.

Sand is poured into the container at a constant rate of $0.2{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate of change of the depth of the sand in the cone at the instant when the radius of the sand is $1.2\text{ cm}$.

[The volume, $V$, of a cone with radius $r$ and height $h$ is given by $V=\frac{1}{3}\pi r^{2}h$.]

A spherical balloon is being inflated.

At time $t$ seconds, where $t\ge 0$, the radius, $r\text{ cm}$, of the balloon is modelled by the equation

$r=\frac{t}{\pi }+\frac{1}{2}$

The material used to make the balloon is designed such that the balloon will burst if the rate of increase of its volume exceeds $16{\text{ cm}}^3{\text{ s}}^{-1}$.

Show that the maximum time for which the balloon can be inflated without bursting is $\frac{3\pi }{2}$ seconds.

[The volume, $V$, of a sphere with radius $r$ is given by $V=\frac{4}{3}\pi r^3$.]

An ice lolly is in the shape of a cylinder.

At time $t$ seconds, the cylinder has radius $r\text{ cm}$ and length $8r\text{ cm}$.

The ice lolly is melting such that its volume is decreasing at a constant rate of $0.4{\text{ cm}}^3{\text{ s}}^{-1}$.

Assuming that the ice lolly remains mathematically similar to its original shape whilst it melts, find the rate at which its surface area is decreasing at the instant when $r=0.3$.

The volume of liquid, $V$, in a hemispherical bowl is given by the formula

$V=\frac{1}{3}\pi h^2(3R-h)$

where $R$ is the radius of the bowl and $h$ is the depth of the liquid.

Liquid is leaking through a small hole in the bottom of the bowl at a rate directly proportional to the depth of the liquid.

Show that the depth of the liquid in the bowl is decreasing by

$\frac{k}{\pi (2R-h)}$

where $k$ is a constant.

The curve $C$ has equation

$y=x^2{\text{e}}^{0.3x}x\in \mathrm{ℝ}$

Show that there are two points of inflection on $C$.

Find the coordinates of these two points of inflection, giving your answers to 3 significant figures.

The curve $C$ has equation

$y=2x+4\text{sin}3x0<x<\pi$

Show that $C$ is convex in the interval $\left(\frac{\pi }{3}, \frac{2\pi }{3}\right)$.

A plant pot is in the shape of an inverted square-based pyramid. The plant pot has a height of $100\text{ cm}$ ($1\text{ m}$) and a base side length of $40\text{ cm}$.

Soil is added to the plant pot at a constant rate of $72{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate at which the depth of the soil is increasing at the instant when the depth is $60\text{ cm}$.

[The volume, $V$, of a pyramid with base area $A$ and height $h$ is given by $V=\frac{1}{3}Ah$.]

The curve $C$ has equation

$y={\text{e}}^{0.4x}(x^2+3x-4)x\in \mathrm{ℝ}$

Find the coordinates of the stationary points on $C$, and determine their nature.

Give all numerical values in your answer to 3 significant figures.

Find the coordinates of any points of inflection on $C$, and determine whether they coincide with the stationary points.

Hence state the intervals in which $C$ is convex and concave.

Give all numerical values in your answer to 3 significant figures.

The curve $C_1$ has equation $y=3x^3$ and the curve $C_2$ has equation $y=3x^3+2x$.

Explain why $C_1$ has a point of inflection which is also a stationary point, but $C_2$ has a point of inflection that is not a stationary point.

On the same diagram, sketch the curves $C_1$ and $C_2$.

An expanding spherical air bubble has radius $r\text{ cm}$ at time $t$ seconds, where $t\ge 0$.

The radius of the bubble is modelled by the equation

$r=0.3+0.1t^2$

The bubble will burst if the rate of increase of its volume exceeds $4t{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the length of time the bubble expands for, giving your answer to 1 decimal place.

[The volume, $V$, of a sphere with radius $r$ is given by $V=\frac{4}{3}\pi r^3$.]

A container is in the shape of a right circular cone. The container rests on its flat, horizontal circular base, with its vertex pointing vertically upwards. The height of the cone is $6\text{ cm}$ and the radius of its base is $2\text{ cm}$.

Salt is poured into the container through a small hole at its vertex at a constant rate of $0.3{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate of change of the depth of the salt at the instant when the container is half full by volume, giving your answer to 3 significant figures.

[The volume, $V$, of a cone with radius $r$ and height $h$ is given by $V=\frac{1}{3}\pi r^{2}h$.]

A large block of ice, used by sculptors, is in the shape of a cuboid.

At time $t$ seconds, the dimensions of the cuboid are $x\text{ m}$ by $2x\text{ m}$ by $5x\text{ m}$.

The block melts uniformly such that its surface area is decreasing at a constant rate of $k{\text{ m}}^2{\text{ s}}^{-1}$, where $k$ is a positive constant.

You may assume that as the block melts, it remains mathematically similar to its original shape.

Show that the rate of decrease of the volume of the block is given by

$\frac{15kx}{34}{\text{ m}}^3{\text{ s}}^{-1}$

In the case when $k=0.2$, the block of ice remains solid enough to be sculpted as long as the rate of decrease of its volume does not exceed $0.05{\text{ m}}^3{\text{ s}}^{-1}$.

Find the value of $x$ for the largest block of ice that can be used for ice sculpting under these conditions, giving your answer as a fraction in its lowest terms.

The volume of liquid, $V$, in a hemispherical bowl is given by the formula

$V=\frac{1}{3}\pi h^2(3R-h)$

where $R$ is the radius of the bowl and $h$ is the depth of the liquid.

Liquid is leaking through a small hole in the bottom of the bowl at a rate directly proportional to the depth of the liquid.

When the bowl is full, the rate of volume loss is $\pi$.

Show that the rate of change of the depth of the liquid is inversely proportional to

$R(h-2R)$