A train leaves a station. Its speed, v m/s, is measured at different times, t seconds.
t
0
2
6
10
v
0
5
9
10
Use three trapezia to estimate the distance the train travels in the first 10 seconds.
Work out an estimate of the mean speed over the 10 seconds.
4 marks
Show answer
a) 71 mb) 7.1 m/s
2
The temperature, T °C, of a cup of tea is plotted against time, t minutes, and the points form a smooth curve going down.
At t = 10 a tangent is drawn to the curve. The tangent passes through the points (2, 80) and (14, 56).
Work out the gradient of the tangent.
What does this gradient tell you about the tea at t = 10?
3 marks
Show answer
a) -2b) the temperature is falling at 2 °C per minute at t = 10
3
A runner's velocity-time graph is a smooth curve. The table gives her velocity v m/s at time t seconds.
t: 0, 10, 20, 30 v: 0, 4.2, 6.6, 7.8
Use the trapezium rule with 3 strips of equal width to estimate the distance the runner travels in the first 30 seconds.
3 marks
Show answer
147 m
4
A cyclist accelerates away from traffic lights. Her speed-time graph is a smooth curve that gets less steep as time increases. The table gives her speed v m/s at time t seconds.
t: 0, 5, 10, 15, 20 v: 0, 3.6, 5.4, 6.4, 7.0
Use the trapezium rule with 4 strips of equal width to estimate the distance travelled in the first 20 seconds.
State, with a reason, whether your answer to part (a) is an underestimate or an overestimate of the actual distance.
4 marks
Show answer
a) 94.5 mb) underestimate, because the curve bends above the chords so each trapezium misses some area under the curve
5
The speed-time graph for a car is a smooth curve for 0 ≤ t ≤ 12, where t is the time in seconds and the speed is in m/s.
A tangent drawn by eye to the curve at t = 6 passes through the points (2, 8) and (10, 32).
Use the tangent to estimate the acceleration of the car at t = 6.
Explain why your answer is only an estimate.
4 marks
Show answer
a) 3 m/s2b) the acceleration is the gradient of the curve, and a tangent drawn by eye only approximates that gradient
6
A ball rolls down a slope. Its velocity-time graph is a smooth curve. The table gives the velocity v m/s at time t seconds.
t: 0, 2, 4, 6 v: 0, 2.2, 5.2, 9.0
A tangent drawn to the curve at t = 4 passes through the points (1, 0.1) and (6, 8.6). Use the tangent to estimate the acceleration of the ball at t = 4.
Use the trapezium rule with 3 strips of equal width to estimate the distance the ball travels in the 6 seconds.
5 marks
Show answer
a) 1.7 m/s2 (accept about 1.7 m/s2)b) 23.8 m
7
Water flows out of a tank. The rate of flow, r litres per minute, is measured every 2 minutes. The graph of r against time t is a smooth curve that falls quickly at first and then flattens out.
t (minutes)
0
2
4
6
8
r (litres/min)
12
8.5
6
4.5
4
Use the trapezium rule with 4 strips to estimate the area under the graph.
What does this area tell you about the tank?
Is your answer to part (a) an over-estimate or an under-estimate? Give a reason.
4 marks
Show answer
a) 54b) the total volume of water (litres) that has flowed out in 8 minutesc) over-estimate: the curve is concave up, so the straight tops of the trapezia lie above the curve
8
A drone accelerates from rest. Its velocity-time graph is a smooth curve that gets less steep throughout the first 12 seconds. The table gives the velocity v m/s at time t seconds.
t: 0, 3, 6, 9, 12 v: 0, 4.5, 7.8, 9.9, 10.8
Use the trapezium rule with 4 strips of equal width to estimate the distance travelled by the drone in the 12 seconds.
A tangent drawn to the curve at t = 6 passes through the points (0, 2.4) and (12, 13.2). Use the tangent to estimate the acceleration of the drone at t = 6.
Is your answer to part (a) an underestimate or an overestimate of the actual distance? Give a reason.
7 marks
Show answer
a) 82.8 mb) 0.9 m/s2c) underestimate: the velocity increases at a decreasing rate, so the curve bows above each chord and the trapezia miss some area
Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.