Estimating Gradients & Areas under Graphs (Edexcel GCSE Maths)

Exam Questions

2 hours33 questions
1
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3 marks

The distance-time graph shows information about part of a car journey.

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Use the graph to estimate the speed of the car at time 5 seconds.

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2a
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3 marks

Karol runs in a race.

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

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Calculate an estimate for the gradient of the graph when t space equals space 4
You must show how you get your answer.

2b
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2 marks

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

2c
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1 mark

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

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3a
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2 marks

The graph shows information about the velocity of a parachutist after jumping from a plane.

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By drawing a suitable tangent, find an estimate of the gradient of the curve after 3 seconds.

3b
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1 mark

Interpret the value of the gradient.

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4a
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2 marks

The graph shows the temperature of a fish tank over the first 6 hours after a heater is added.

q4a-medium-2-19-estimating-gradients-edexcel-gcse-maths

By drawing a suitable tangent, find an estimate of the gradient of the curve when h = 3.

4b
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1 mark

Interpret the value of the gradient.

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5a
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2 marks

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

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Find an estimate for the gradient at t = 10.

5b
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1 mark

Give an interpretation of the gradient.

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6a
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1 mark

The diagram shows parts of the graphs of y=f(x) and y=g(x)

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Write down the value of x where the gradient of the curve y=g(x) is zero.

6b
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2 marks

Calculate an estimate for the gradient of the curve y = f(x) at the point on the curve where x = 4.

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7a
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2 marks

The diagram shows part of the graph of y space equals space x to the power of 2 space end exponent long dash space 2 x space plus space 3

q7a-medium-2-19-estimating-gradients-edexcel-gcse-maths

By drawing a suitable straight line, use your graph to find estimates for the solutions of x squared space long dash space 3 x space long dash space 1 space equals space 0

7b
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3 marks

P is the point on the graph of y space equals space x squared space long dash space 2 x space plus space 3 where x space equals space 2

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

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8a
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2 marks

The curve space y space equals space x cubed long dash space 3 x long dash 2 spaceis shown on the grid.

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Write down the co-ordinates of the points where the gradient of the curve is zero.

8b
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1 mark

Write down the range of values of x when the gradient of the curve is negative.

8c
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2 marks

Find an estimate of the gradient of the curve when x = 2.

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9
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3 marks

Part of the curve with equation y space equals space straight f left parenthesis x right parenthesis is shown on the grid.

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Find an estimate for the gradient of the curve at the point where x space equals space 2
Show your working clearly.

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10a
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2 marks

A and B are points on a curve.

A is left parenthesis 2 comma space 7 right parenthesis space   B is left parenthesis 12 comma space 0 right parenthesis

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Work out the instantaneous rate of change of y with respect to x at point A.

10b
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1 mark

The average rate of change of space y spacewith respect to x between points A and B is worked out.

Which statement is correct?
Tick one box.

begin mathsize 26px style square end style It is positive.
begin mathsize 26px style square end style It is zero.
begin mathsize 26px style square end style It is negative.
size 26px square You cannot tell if it is positive or negative.

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11a
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3 marks

The speed-time graph shows 20 seconds of a car journey.

Harry wants to estimate the distance the car travels in this time.

He uses a triangle and a trapezium, as shown, to estimate the area under the graph.

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Complete Harry’s method to estimate the distance the car travels.

.......................m

11b
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1 mark

For this journey, which of these is true for Harry’s method?
Tick one box.

begin mathsize 26px style square end style      It works out an overestimate of the distance

size 26px square        It works out an underestimate of the distance

size 26px square      It could work out an overestimate or an underestimate of the distance

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12a
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1 mark

A container is filled with water in 5 seconds.
The graph shows the depth of water, d cm, at time t seconds.

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The water flows into the container at a constant rate.

Which diagram represents the container?
Circle the correct letter.

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12b
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2 marks

Use the graph to estimate the rate at which the depth of water is increasing at 3 seconds.

You must show your working.

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

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13
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2 marks

A ball is thrown from a point 6 metres above the ground.

The graph shows the height of the ball above the ground, in metres.

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Estimate the speed of the ball, in m/s, after 1 second.

You must show your working.

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

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14
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4 marks

The graph shows the distance travelled by a particle over 8 seconds.

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Estimate the speed of the particle at 5 seconds.

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

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15
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3 marks

The graph shows the speed, v metres per second (m/s), of a car at time t seconds.

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Use the graph to estimate the acceleration at t = 7.

...................................................m/s2

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1a
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3 marks

Here is a speed-time graph for a train.

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Work out an estimate for the distance the train travelled in the first 20 seconds.
Use 4 strips of equal width.

1b
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1 mark

Is your answer to (a) an underestimate or an overestimate of the actual distance the train travelled?
Give a reason for your answer.

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2a
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2 marks

The graph shows information about the velocity, v m/s, of a parachutist t seconds after leaving a plane.

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Work out an estimate for the acceleration of the parachutist at t = 6

2b
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3 marks

Work out an estimate for the distance fallen by the parachutist in the first 12 seconds after leaving the plane.
Use 3 strips of equal width.

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3a
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3 marks

The graph shows the speed of a car, in metres per second, during the first 20 seconds of a journey.

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Work out an estimate for the distance the car travelled in the first 20 seconds.
Use 4 strips of equal width.

3b
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1 mark

Is your answer to part (a) an underestimate or an overestimate of the actual distance the car travelled in the first 20 seconds?
Give a reason for your answer.

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4a
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3 marks

Here is a speed-time graph for a car.

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Work out an estimate for the distance the car travelled in the first 10 seconds.
Use 5 strips of equal width.

4b
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1 mark

Is your answer to (a) an underestimate or an overestimate of the actual distance?
Give a reason for your answer.

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5a
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2 marks

The curve y space equals space x cubed space plus 2 x squared space long dash 4 x is shown on the grid.

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

5b
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1 mark

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

Write down the co-ordinates of D.

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6a
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3 marks

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

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

6b
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1 mark

Describe what happens to Ellie after 7 seconds.

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7a
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1 mark

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.

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

7b
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3 marks

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

7c
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3 marks

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

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8a
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2 marks

The diagram shows the graph of y space equals space x over 2 space plus space 2 over x spacefor 0 less than straight x less or equal than 8.

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Use the graph to solve the equation x over 2 space plus space 2 over x space equals 3
.

8b
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3 marks

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

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9a
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2 marks

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

  Time (t minutes) 0 1 2 3 4 5 6 7 8 9 10
  Depth (m metres) 30 29.5 29 28 27 26 24.5 22.5 19.5 15 9

Plot the graph of depth against time.

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9b
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2 marks

Work out the average rate of decrease of the depth of the water in Clare's tank between t = 0 and t = 10.

9c
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2 marks

Find an estimate for the gradient at t = 7.

9d
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1 mark

Give an interpretation of part (c).

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10
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3 marks

Part of the curve with equation y space equals space h left parenthesis x right parenthesis is shown on the grid.

q18c-paper2hr-june2019-edexcel-igcse-maths

Find an estimate for the gradient of the curve at the point where x space equals space minus 0.5
Show your working clearly.

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11
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3 marks

Liquid is leaking out of a container.

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

q25-paper3h-nov2017-aqa-gcse-maths

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

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12
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2 marks

The graph shows the speed, in metres per second, of a particle over the first four seconds of motion.

q20-paper4-june2017-ocr-gcse-maths

Use the graph to estimate the distance travelled by the particle in the four seconds.

............................................... metres

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1a
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3 marks

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

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

1b
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2 marks

Estimate the time during the 40 seconds when the instantaneous acceleration = the average acceleration.
You must show your working on the graph.

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2a
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2 marks

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

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

2b
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2 marks

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

2c
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2 marks

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

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

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3a
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2 marks

The diagram shows part of the graph of  y space equals space x squared space long dash space 2 x space plus space 3

q3-vhard-2-19-estimating-gradients-edexcel-gcse-maths

By drawing a suitable straight line, use your graph to find estimates for the solutions of x squared space long dash space 3 x space long dash space 1 space equals space 0

3b
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3 marks

P is the point on the graph of space y space equals space x squared space long dash space 2 x space plus space 3 spacewhere x space equals space 2

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

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4a
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4 marks

Leo runs for 12 seconds.

The graph shows his speed.

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Show that the distance he runs is less than 67.5 metres.

4b
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2 marks

Work out his average acceleration for the first 9 seconds.

State the units of your answer.

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5a
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4 marks

A car moves from rest.

The graph gives information about the speed, v metres per second, of the car t seconds after it starts to move.

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i)
Calculate an estimate of the gradient of the graph at t= 15
[3]
ii)
Describe what your answer to part (i) represents.
[1]
5b
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3 marks

Work out an estimate for the distance the car travels in the first 20 seconds of its journey.
Use 4 strips of equal width.

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6a
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3 marks

The diagram shows the graph of y space equals space straight f space left parenthesis x right parenthesis for negative 4 space less-than or slanted equal to space x space less-than or slanted equal to space 12

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The point space P spaceon the curve has x coordinate 2

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

6b
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2 marks

Hence find an equation of the tangent to the curve at P.
Give your answer in the form y space equals space m x space plus space c .

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