Printable · GCSE Higher · ages 14-16

Gradients and areas under graphs worksheet — GCSE Higher

Fifteen questions on "gradients and areas under graphs" — DfE statement A15. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.

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Gradients and areas under graphs worksheet — GCSE Higher

Maths · GCSE Higher · A15 · Gradients and areas under graphs · 15 questions

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1.A car's journey is represented by a velocity-time graph. The total distance travelled, found from the area under the graph, is 350 metres, and the total time for the journey is 25 seconds. Work out the car's average speed for the whole journey, in m/s.
(a)375 m/s
(b)14 m/s
(c)8750 m/s
(d)0.07 m/s
2.A car's speed, in m/s, during a journey is: 0 at t = 0 s, 8 at t = 4 s, and 8 (constant) from t = 4 s to t = 10 s, increasing at a constant rate between t = 0 and t = 4. Estimate the total distance the car travels, using the areas of a triangle and a rectangle.
(a)48 m
(b)40 m
(c)64 m
(d)80 m
3.The table shows the speed, in m/s, of a car at times, in seconds, all 2 seconds apart: 0 at t = 0, 3 at t = 2, 8 at t = 4, 15 at t = 6, 24 at t = 8. A student uses the trapezium rule with these five readings to estimate the distance the car travels. By finding the second differences of the speed values, decide whether this trapezium estimate is an overestimate or an underestimate of the true distance.
(a)Underestimate — second differences are constant.
(b)Overestimate — the speed values are increasing.
(c)Underestimate — the curve bends upward (convex).
(d)Overestimate — the curve bends upward (convex).
4.The tangent to a curve at the point (5, 2) is the line y = mx + c. This tangent crosses the y-axis at (0, −8). Work out the gradient, m, of the tangent.
(a)0.5
(b)−1.2
(c)−2
(d)2
5.The table shows the speed, in m/s, of a cyclist at times, in seconds, 1 second apart: 0 at t = 0, 7 at t = 1, 12 at t = 2, 15 at t = 3, 16 at t = 4, 14 at t = 5, 9 at t = 6. Using the trapezium rule with strip width 1 second, estimate the distance the cyclist travels between t = 2 and t = 5 only.
(a)68.5 m
(b)39 m
(c)28.5 m
(d)44 m
6.The table shows the velocity, in m/s, of a train at various times, in seconds: 0 at t = 0, 20 at t = 4, 35 at t = 10, 27 at t = 14. Assuming the velocity changes at a constant rate between each pair of readings, estimate the distance the train travels between t = 0 and t = 12, using the trapezium rule.
(a)271 m
(b)275 m
(c)282 m
(d)267 m
7.A car's velocity–time graph is a straight line from (0 s, 4 m/s) rising to (6 s, V m/s), followed by a straight line falling from (6 s, V m/s) to (9 s, 0 m/s). The gradient of the second line is −8 m/s². Work out the total distance travelled between t = 0 and t = 9 seconds.
(a)120 m
(b)156 m
(c)48 m
(d)204 m
8.The trapezium rule is used to estimate the area under a curve between x = 2 and x = 6. The curve is concave up (it curves upwards, like the inside of a bowl) throughout this interval. Which statement about the estimate is correct?
(a)Overestimates — trapezium edges lie above the curve.
(b)Underestimates — trapezium edges lie below the curve.
(c)Exact — straight lines model any curve with no error.
(d)Overestimates, because more strips always increase it.
9.A car accelerates uniformly from rest at 2.5 m/s² until it reaches a speed of 20 m/s, then travels at this constant speed for a further 30 seconds. Work out the total distance travelled.
(a)900 m
(b)760 m
(c)240 m
(d)680 m
10.The tangent to a curve at the point where x = 4 passes through the points (2, 5) and (6, 21). Use these two points to estimate the gradient of the curve at x = 4.
(a)4
(b)16
(c)0.25
(d)8
11.A tap's flow rate, in litres per minute, is plotted against time, in minutes, on a graph. What are the units of the area under this graph?
(a)Litres
(b)Litres per minute squared
(c)Litres per minute
(d)Minutes
12.The table shows the rate at which water flows from a tank, in litres per minute, over a 6-minute period: 12 at t = 0 min, 12 at t = 2 min, 3 at t = 5 min, and 0 at t = 6 min. Between t = 0 and t = 2 the rate is constant. Between t = 2 and t = 5, and between t = 5 and t = 6, the rate decreases at a constant rate in each section. Estimate the total volume of water that has flowed out, using the areas of a rectangle and two trapeziums.
(a)48 litres
(b)24 litres
(c)49.5 litres
(d)36 litres
13.A cyclist's speed rises from rest to a maximum — quickly at first, then more and more slowly — so her speed-time graph is a curve that is concave down (its gradient decreases as time goes on). A trapezium estimate is made for the distance travelled during this phase, using two points on the curve. Is the trapezium estimate an overestimate or an underestimate of the true distance, and why?
(a)Underestimate: the chord lies below the curve
(b)Overestimate: the trapezium rule always overestimates
(c)Exact: a trapezium and a curve enclose equal areas
(d)Overestimate: the chord lies above the curve
14.A tram sets off from a stop. Its velocity-time graph rises in a straight line from 0 m/s to 12 m/s over the first 8 seconds, then stays constant at 12 m/s for a further 10 seconds. Work out the total distance the tram travels in these 18 seconds, using the area under the graph.
(a)120 m
(b)216 m
(c)108 m
(d)168 m
15.A cup of tea cools after being made. Its temperature-time graph is a curve. Three tangents are drawn on the graph: at t = 2 minutes the tangent has gradient −8; at t = 10 minutes the tangent has gradient −3; at t = 20 minutes the tangent has gradient −0.5. Which of these statements about the tea's cooling is correct?
(a)Fastest at t = 20 min — largest gradient.
(b)Cools at the same rate throughout.
(c)Fastest at t = 2 min — steepest gradient.
(d)Fastest at t = 10 min — the middle reading.

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