Differentiation & Gradients (Edexcel IGCSE Maths A (Modular): Higher Unit 2): Exam Questions

The curve  is shown on the grid.

By drawing a suitable tangent, find an estimate of the gradient of the curve when x = 1.

A point lies on the curve.
The x  co-ordinate of is negative.
The gradient of the tangent at is 0.

Write down the co-ordinates of .

Ellie runs a race. The graph shows Ellie's speed in the first 10 seconds after the start of the race.

By drawing a suitable tangent, work out the acceleration when = 9. Give the units of your answer.

Describe what happens to Ellie after 7 seconds.

The diagram shows the graph of for 

Use the graph to solve the equation .

By drawing a suitable tangent, find an estimate of the gradient of the graph when x = 1.

Karol runs in a race.

The graph shows her speed, in metres per second, seconds after the start of the race.

Calculate an estimate for the gradient of the graph when  You must show how you get your answer.

Describe fully what your answer to part (a) represents.

Explain why your answer to part (a) is only an estimate.

A curve has equation .

Find: the coordinates where the curve crosses the -axis,

the coordinates where the curve crosses the -axis,

the coordinates of the turning point on the curve,

Sketch the curve showing the points you have found.

A fish bowl is being filled with water.
The graph shows how the diameter of the surface of the water changes with time.

Find an estimate for the gradient at  = 10.

Give an interpretation of the gradient.

The diagram shows the graph of  for 

The point on the curve has  coordinate 2

Use the graph to find an estimate for the gradient of the curve at .

Hence find an equation of the tangent to the curve at . Give your answer in the form  .

A particle is moving along a straight line. The fixed point lies on this line.

The displacement of the particle from  at time  seconds is  metres where

Find an expression for the velocity,  of the particle at time  seconds.

Find the time at which the velocity is instantaneously zero.

Part of the curve with equation  is shown on the grid.

Find an estimate for the gradient of the curve at the point where 
Show your working clearly.

For the curve with equation :
find

find the -coordinates of the two turning points on the curve.

By considering the shape of the curve determine which of your answers to (b) is the -coordinate of a maximum point.

Liquid is leaking out of a container.

The graph shows the depth of the liquid for 60 seconds.

Use the graph to work out an estimate of the rate of decrease of depth at 10 seconds.

You must show your working.

...........................................cm/s

The curve  has equation .

Part of the graph of  is shown below.

Write the coordinates of A.

Points B and C are stationary points on .

Find the coordinates of points B and C, stating the nature of the stationary point in each case. 

For which values of is the gradient of the curve  negative?

For the curve with equation 

find

find the coordinates of the stationary points on the curve.

find the exact distance between the two stationary points.

The graph shows information about the speed of a vehicle during the final 50 seconds of a journey.

At the start of the 50 seconds the speed is k metres per second. The distance travelled during the 50 seconds is 1.35 kilometres.

Work out the value of .

= .......................

A particle is moving along a straight line and passes a fixed point .

The displacement of the particle, from point , at time  seconds is

where  is measured in metres. Initially how far is the particle from ?

Find, in terms of , the velocity of the particle.

Find the time at which the particle’s velocity is at its minimum.

For how long is the particle decelerating?

A homeowner wishes to enclose a rectangular part of their garden by building a fence, using an existing wall as one side of the rectangle as shown in the diagram below.

The width of the enclosed rectangle is  metres and its length metres.

The homeowner has 40 metres of fence to use and would like to use it all in order to maximise the area of the garden to be enclosed. Show that

Show that the area of the garden to be enclosed, , is given by = 40 − 2

Find 

Find the value of that maximises

Find the dimensions of the rectangle that produce the maximum area that can be enclosed using all of the fence.

Also find the maximum area.

The diagram shows a cuboid of volume 

Show that

There is a value of  for which the volume of the cuboid is a maximum.

Find this value of .
Show your working clearly.
Give your answer correct to 3 significant figures.

A particle is moving along a straight line.
The fixed point lies on this line.
At time seconds where , the displacement, metres, of from is given by

Find the displacement of from when is instantaneously at rest.

Give your answer in the form  where and are integers.

A cuboid with a square cross section is to be made from rods as shown in the diagram. The shorter rods making the square are of length xx cm and the longer rods

are of length  cm.

Explain why 12 rods in total will be needed to make the cuboid, and state how many of each length will be required. 

The total length of the rods is to be fixed at 36 cm.

The total length of the rods is to be fixed at 36 cm.

Find  in terms of

Show that the volume of the cuboid,  cm³ is  = 9 − 2.

Find the value of  that maximises the volume.

Find the maximum volume.

A particle is moving along a straight line that passes through the fixed point O.
The displacement, metres, of the particle from O at time seconds is given by

Find the value of when the acceleration of the particle is 5 m/s²

A curve, , has equation  where  is a constant.

Show that when  = 0, the turning point on  has coordinates (0, -3).

Show that when  ≠ 0, the turning point on  must have a negative -coordinate.

When ≠ 0 determine whether or not the -coordinate of the turning point is negative.

Part of the graph with equation is shown below.

The graph has three stationary points, indicated on the graph by points , and.
Find the area of the triangle .

The diagram shows a cuboid with a square cross-section.

The sides of the square face are cm and the length of the cuboid is cm.

The cuboid is to have a fixed surface area, , of 25 cm².

Show that the volume of the cuboid,  cm³ is given by

Show that the value of  that maximises the volume of the cuboid is

Find the maximum volume of the cuboid, correct to 3 significant figures.

A particle moves along a straight line that passes through the fixed point

The displacement, metres, offrom at time t seconds, where , is given by

The direction of motion of  reverses when is at the point on the line.

The acceleration of at the instant when is at is . Find the value of  .

a = ..................................... 

Two particles, and , move along a straight line.

The fixed point lies on this line.

The displacement of from at time seconds is metres, where

The displacement of from at time seconds is metres, where

Find the range of values of where for which both particles are moving in the same direction along the straight line.

The point A is the only stationary point on the curve with equation   where is a constant.

Given that the coordinates of are 

find the value of .
Show your working clearly.

................................................. 

The curve has equation where and are constants.

The point with coordinates (2, –6) lies on .

The gradient of the curve at is 16.

Find the coordinate of the point on the curve whose coordinate is 3.
Show clear algebraic working.

A particle is moving along a straight line.

The fixed point lies on the line.

At time t seconds , the displacement of from is s metres where

Find the minimum speed of .

...................................................... m/s

is a five-sided shape.

is a rectangle.
is an equilateral triangle.

The perimeter of is 100 cm.
The area of is cm²

Show that 

(i) Find the value of for which has its maximum value.

Give your answer in the form where and are integers.

....................................................... [2]

(ii) Explain why the maximum value of is given by this value of .

[1]

A particle moves along a straight line.
The fixed point  lies on this line.
The displacement of the particle from  at time  seconds , , is  metres where

At time  seconds the velocity of  is  where 

Find an expression for  in terms of .

Give your expression in the form  where  and  are integers to be found.

The velocity-time graph shows the first 40 seconds of a car in a race.

Work out the average acceleration for the first 40 seconds.
Give the units of your answer.

Estimate the time during the 40 seconds when the instantaneous acceleration = the average acceleration.

You must show your working on the graph.

The graph gives information about the variation in the temperature of an amount of water that is left to cool from 80° C.  

Work out an estimate for the rate of decrease of temperature at = 300.

Work out the average rate of decrease of the temperature of the water between  = 0 and = 800.

The instantaneous rate of decrease of the temperature of the water at time seconds is equal to the average rate of decrease of the temperature of the water between = 0 and = 800.

Find an estimate for the value of .
You must show how you got your answer.

The diagram shows part of the graph of  

By drawing a suitable straight line, use your graph to find estimates for the solutions of 

is the point on the graph of where 

Calculate an estimate for the gradient of the graph at the point .

Clare emptied a tank and recorded the depth of water each minute.

Plot the graph of depth against time.

Work out the average rate of decrease of the depth of the water in Clare's tank between = 0 and = 10.

Find an estimate for the gradient at = 7.

Give an interpretation of part (c).

Olympic medallist Usain runs in a race.
The graph shows his speed, in metres per second (m/s), during the first 10 seconds of the race.

Use the graph to find how long it took Usain to reach his top speed.

Work out an estimate for Usain's acceleration at 2 seconds. Give the units of your answer.

Calculate the difference in Usain's acceleration between 2 and 6 seconds.

The table shows some values of  .

  Complete the table of values.   

Draw the graph of    for  .

A straight line through (0, –17) is a tangent to the graph of  .

(i) On the grid, draw this tangent.    

[1]

(ii) Find the co-ordinates of the point where the tangent meets your graph.   

(................ , ................) [1]

(iii) Find the equation of the tangent. Give your answer in the form  .  

 = ................................................ [3]