Further Applications of Differentiation (A Level only) (OCR A Level Maths: Pure)

Exam Questions

4 hours35 questions
1a
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3 marks

A curve has equation y equals x cubed plus 2 x plus 1.

Find expressions for fraction numerator d y over denominator d x end fraction  and  fraction numerator d squared y over denominator d x squared end fraction

1b
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3 marks
(i)
Write down the value of x for which  fraction numerator straight d squared y over denominator straight d x squared end fraction equals 0.

(ii)
By considering the sign of  fraction numerator straight d squared y over denominator straight d x squared end fraction  just either side of the value of x found in part (i), show that this is a point of inflection.

1c
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1 mark

Hence, or otherwise, find the coordinates of the point of inflection on the curve with equation y equals x cubed plus 2 x plus 1.

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2a
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2 marks

Given  y equals x cubed minus 6 x squared show that fraction numerator straight d squared y over denominator straight d x squared end fraction space equals 6 x minus 12.

2b
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2 marks

Solve the inequality  fraction numerator straight d squared y over denominator straight d x squared end fraction space less than 0  and hence determine the set of values for which the graph of  y equals x cubed minus 6 x squared space is concave.

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3a
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3 marks

In a computer animation, the side length, s mm, of a square is increasing at a constant rate of 2 millimetres per second.

(i)
Write down the value of fraction numerator d s over denominator d t end fraction, where t is time and measured in seconds.

(ii)
Write down a formula for the area, A mm2, of the square and hence find  fraction numerator d A over denominator d s end fraction.
3b
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3 marks

Use the chain rule to find an expression for  fraction numerator d A over denominator d t end fraction in terms of s and hence find the rate at which the area is increasing when  s=10.

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4a
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4 marks

Find the value of  fraction numerator d y over denominator d x end fraction  and  fraction numerator straight d squared y over denominator straight d x squared end fraction  at the point where  x equals 2 space for the curve with equation  y equals x cubed minus 6 x squared plus 9 x plus 4.

4b
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2 marks

Show that space x equals 2 space space is a point of inflection.

4c
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1 mark

Explain why  x equals 2  is not a stationary point.

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5
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3 marks

The graph of  y equals straight f left parenthesis x right parenthesis, where space straight f left parenthesis x right parenthesis space spaceis a continuous function has the following properties:

  • The graph intercepts the x-axis at  (-a , 0)  and the y-axis at  (0 , b)
  • It is concave in the interval  (negative infinity , c)
  • It is convex in the interval  (c , infinity)
  • a, b and c are such that  0 less than a less than b less than c

Sketch a graph of the function.

Label the points where the graph intercepts the coordinate axes and label the point of inflection.

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6a
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3 marks

The side length, x cm, of a cube increases at a constant rate of 0.1 cm s-1.

(i)
Write down the value of fraction numerator d x over denominator d t end fraction, where t is time and measured in seconds.

(ii)
Write down a formula for the volume, V cm3, of the cube and hence find  fraction numerator d V over denominator d x end fraction.
6b
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3 marks

Use the chain rule to find an expression for fraction numerator d V over denominator d t end fraction in terms of x and hence find the rate at which the volume is increasing when the side length of the cube is 4 cm.

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7a
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2 marks

The rate at which the radius, r cm, of a sphere increases over time (t seconds) is directly proportional to the temperature (T °C) of its immediate surroundings.
Write down an equation linking  fraction numerator d r over denominator d t end fraction,  T  and the constant of proportionality, k.

7b
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3 marks

When the surrounding temperature is 20 °C, the radius of the sphere is increasing at a rate of 0.4 cm s-1.
Find the value of k.

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8a
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5 marks
(i)
Write down a formula for the volume of a cube, V cm3, and the surface area, S cm2, of a cube, in terms of the side length of a cube, x cm.

(ii)
Show that  fraction numerator d x over denominator d V end fraction equals fraction numerator 1 over denominator 3 x squared end fractionand find an expression for  fraction numerator d S over denominator d x end fraction.
8b
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5 marks

The volume of a cube is decreasing at a constant rate of 0.6 cm3 s-1.

(i)
Explain why fraction numerator d V over denominator d t end fraction, where t is time in seconds, has the value of -0.6.

(ii)
Use the chain rule to find an expression for fraction numerator d S over denominator d t end fraction in terms of fraction numerator d S over denominator d x end fraction comma space fraction numerator d x over denominator d V end fraction and  fraction numerator d V over denominator d t end fraction.

(iii)
Hence write  fraction numerator d S over denominator d t end fraction in terms of x and find the rate at which the area of the cube is decreasing at the instant when its side length is 5 cm.

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1
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5 marks

Find the coordinates of the point of inflection on the curve with equation

y equals x cubed plus 3 x squared minus 2

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2a
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3 marks

Given  y equals x squared plus ln space x space minus 2 x,  where x greater than 0,  show that

fraction numerator d squared y over denominator d x squared end fraction equals 2 minus 1 over x squared

2b
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3 marks

Solve the inequality

fraction numerator d squared y over denominator d x squared end fraction greater than 0

and hence determine the set of values for which the graph of space y equals x squared plus ln space x minus 2 x space  is convex.

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3
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4 marks

In a computer animation, the radius of a circle increases at a constant rate of 1 millimetre per second.  Find the rate, per second, at which the area of the circle is increasing at the time when the radius is 8 millimetres.  Give your answer as a multiple of pi.

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4a
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4 marks

Find the x-coordinates of the stationary points on the graph with equation
y equals x cubed minus 6 x squared plus 9 x minus 1.

4b
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3 marks

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

4c
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3 marks

Determine the x-coordinate of the point of inflection on the graph with equation space y equals x cubed minus 6 x squared plus 9 x minus 1.

4d
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1 mark

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

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5
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4 marks

The graph of a continuous function has the following properties:

The function is concave in the interval  (negative infinity , a).

The function is convex in the interval  (a , infinity).

The graph of the function intercepts the x-axis at the points (b , 0), (c , 0) and (d , 0), where b, c and d are such that d greater than c greater than b greater than 0.

The x-coordinates of the turning points of the function are e and f, which are such that f greater than e.

The graph of the function intercepts the y-axis at  (0 , g)

Given that the value of the function is positive when x equals a, sketch a graph of the function.  Be sure to label the x-axis with the x-coordinates of the stationary points and the point of inflection, and also to label the points where the graph crosses the coordinate axes.

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6
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5 marks

The side length of a cube increases at a rate of  0.1  cm s-1.
Find the rate of change of the volume of the cube at the instant the side length is 5 cm.
You may assume that the cube remains cubical at all times.

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7a
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5 marks

In the production process of a glass sphere, hot glass is blown such that the radius, r cm, increases over time (t seconds) in direct proportion to the temperature (T °C) of the glass.
Find an expression, in terms of r and T, for the rate of change of the volume (V cm3) of a glass sphere.

7b
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3 marks

When the temperature of the glass is 1200 °C, a glass sphere has a radius of 2 cm and its volume is increasing at a rate of 5 cm3s-1.
Find the rate of increase of the radius at this time.

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8
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6 marks

An ice cube, of side length x cm, is melting at a constant rate of 0.8 cm3 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 melting at the point when its side length is 2 cm2.

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9
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5 marks

A bowl is in the shape of a hemisphere of radius 8 cm.

The volume of liquid in the bowl is given by the formula

V equals 8 pi h squared minus 1 third pi h cubed

where h cm is the depth of the liquid (ie the height between the bottom of the bowl and the level of the liquid).

Liquid is leaking through a small hole in the bottom of the bowl at a constant rate of 5 cm3s-1.  Find the rate of change of the depth of liquid in the bowl at the instant the height of liquid is 3 cm.

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1
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5 marks

Determine the number of points of inflection on the curve with equation

y equals x to the power of 4 plus 3 x cubed plus 2

and determine their coordinates.

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2
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4 marks

Find the values for which the curve defined by the function

f x equals x squared minus e to the power of 2 x end exponent plus 1

is concave.

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3
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4 marks

A spherical air bubble’s surface area is increasing at a constant rate of 4π cm2s-1.
Find an expression for the rate at which the radius is increasing per second.
(The surface area of a sphere is 4Ï€r2.)

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4a
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4 marks

Find the coordinates of the stationary points, and their nature, on the graph with equation space y equals 4 x minus x squared minus 2 x cubed.

4b
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4 marks

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

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5
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4 marks

The diagram below shows a sketch of the graph with equation y equals straight f open parentheses x close parentheses.

i_Q65bb0_q8a-7-3-further-differentiation-vh-a-level-maths-pure-screenshots

On the sketch, mark the approximate locations of the following ...

(i)
... intercepts with the coordinate axes using the letter C
(ii)
... stationary points using the letter S
(iii)
... points of inflection using the letter I

Also highlight sections of the curve where the graph is convex.

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6
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6 marks

A cone, stood on its vertex, of radius 3 cm and height 9 cm, is being filled with sand at a constant rate of  0.2 cm3s-1.
Find the rate of change of the depth of sand in the cone at the instant the radius of the sand is 1.2 cm.

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7
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6 marks

The material used to make a spherical balloon is designed so that it can be inflated at a maximum rate of  16 cm3s-1 without bursting.

Given that the radius of the balloon is determined by the function space r open parentheses t close parentheses equals t over pi plus 1 half space comma space t greater or equal than 0 show that the maximum time the balloon can be inflated for, without bursting, is 3 over 2 pi seconds.

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8
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6 marks

An ice lolly which is in the shape of a cylinder of radius r cm and length 8r cm is melting at a constant rate of  0.4 cm3s-1.

Assuming that the ice lolly remains in the shape of a cylinder (mathematically similar to the original cylinder) whilst it melts, find the rate at which its surface area is melting at the point when it’s radius is 0.3 cm. 

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9
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6 marks

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

 

space space V equals 1 third pi h squared left parenthesis 3 R minus h right parenthesis

 

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

(ie the height between the bottom of the bowl and the level of the liquid).

 In a particular case, a bowl is leaking liquid through a small hole in the bottom at a rate directly proportional to the depth of liquid.

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

fraction numerator k over denominator pi open parentheses 2 R minus h close parentheses end fraction

where k is a constant.

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1a
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5 marks

Show that there are two points of inflection on the curve with equation

y equals x squared e to the power of 0.3 end exponent to the power of x

1b
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2 marks

Find the coordinates of the two points of inflection on the curve with equation

y equals x squared e to the power of 0.3 x end exponent

giving your answers to three significant figures.

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2
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5 marks

Show that the shape of the curve with equation

y equals 2 x plus 4 sin space 3 x space space space space space space space space space space space space space space space space space space space space space space space space space space space 0 less than x less than pi

is convex in the interval open parentheses straight pi over 3 space comma fraction numerator 2 pi over denominator 3 end fraction close parentheses.

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3
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6 marks

A plant pot in the shape of square-based pyramid (stood on its vertex) is being filled with soil at a rate of 72 cm3s-1.
The plant pot has a height of 1 m and a base length of 40 cm.
Find the rate at which the depth of soil is increasing at the moment when the depth is 60 cm.

(The volume of a pyramid is a third of the area of the base times the height.)

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4
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7 marks

Find the coordinates of the stationary points, and their nature, on the graph with equation  y equals e to the power of 0.4 x end exponent left parenthesis x squared plus 3 x minus 4 right parenthesis.

Find the coordinates of any points of inflection, determine if these are also stationary points and find the intervals in which the graph is convex and concave.

Give values to three significant figures.

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5a
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4 marks

Explain why the graph ofspace space y equals 3 x cubed has a point of inflection which is also a stationary point but the the graph of space y equals 3 x cubed plus 2 x spacehas a point of inflection that is not a stationary point.

5b
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2 marks

On the same diagram, sketch the graphs of space y equals 3 x cubed spaceand y equals 3 x cubed plus 2 x.

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6
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6 marks

An expanding spherical air bubble has radius, r cm, at a time, t seconds, determined by the function r open parentheses t close parentheses equals 0.3 plus 0.1 t squared.

The bubble will burst if the rate of expansion of its volume exceeds 4 t space cm cubed space straight s to the power of negative 1 end exponent.

Find, to one decimal place, the length of time the bubble expands for.

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7
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7 marks

A small conical pot, stood on its base, is being filled with salt via a small hole at its vertex.  The cone has a height of 6 cm and a radius of 2 cm.

Salt is being poured into the pot at a constant rate of 0.3 cm3s-1.
Find, to three significant figures, the rate of change in depth of the salt at the instant when the pot is half full by volume.

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8a
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5 marks

A large block of ice used by sculptors is in the shape of a cuboid with dimensions x m by 2x m by 5x m.  The block melts uniformly with its surface area decreasing at a constant rate of k m2 s-1. You may assume that as the block melts, the shape remains mathematically similar to the original cuboid.

Show that the rate of melting, by volume, is given by

  fraction numerator 15 k x over denominator 34 end fraction space straight m cubed space straight s to the power of negative 1 end exponent.

8b
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3 marks

In the case whenspace k equals 0.2, the block of ice remains solid enough to be sculpted as long as the rate of melting, by volume, does not exceed 0.05 space straight m cubed space straight s to the power of negative 1 end exponent.

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

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9
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7 marks

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

V equals 1 third pi h squared left parenthesis 3 R minus h right parenthesis

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

(ie the height between the bottom of the bowl and the level of the liquid).

In a particular case, liquid is leaking through a small hole in the bottom of a bowl at a rate directly proportional to the depth of liquid.

When the bowl is full, the rate of volume loss is equal to pi.

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

R left parenthesis h minus 2 R right parenthesis

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