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.The tangent to a curve at the point (6, 1) has gradient −3. Work out the y-coordinate of the point where this tangent crosses the y-axis.
(a)−2
(b)19
(c)−17
(d)18
2.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).
3.The table shows the height, in metres, of a firework rocket at various times, in seconds, during its flight: 40 at t = 2, 54 at t = 3, and 60 at t = 4. Use the chord between t = 2 and t = 4 to estimate the gradient of the height-time graph at t = 3, stating the correct units.
(a)14 m/s
(b)10 m
(c)10 m/s
(d)0.1 s/m
4.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
5.Two students each use the trapezium rule to estimate the area under the same curve between x = 0 and x = 8. Student A uses 4 strips of width 2. Student B uses 8 strips of width 1. Which estimate is more likely to be closer to the true area, and why?
(a)Student A's — fewer strips means fewer rounding errors.
(b)Student B's — narrower strips fit the curve more closely.
(c)They are always the same; the trapezium rule is exact.
(d)Student A's — wider strips give a better average height.
6.A speed-time graph shows a constant speed of 15 m/s for 20 seconds. Work out the distance travelled, using the area under the graph.
(a)7.5 m
(b)1.33 m
(c)35 m
(d)300 m
7.The area under a speed-time graph between t = 0 and t = 6 seconds is estimated using three strips of equal width, using the speeds, in m/s, at t = 0, 2, 4 and 6: 0, 5, 9 and 12. Using the trapezium rule with these three trapezia, estimate the distance travelled.
(a)28 m
(b)80 m
(c)52 m
(d)40 m
8.A car is decelerating. The tangent to its velocity-time graph at t = 12 seconds passes through the points (8, 22) and (16, 6), where velocity is in m/s and time is in seconds. Work out the gradient of this tangent, in m/s².
(a)−2 m/s²
(b)−16 m/s²
(c)2 m/s²
(d)−0.5 m/s²
9.A cyclist's velocity increases from 3 m/s to 11 m/s over 5 seconds, at a constant rate. Work out the cyclist's acceleration, in m/s².
(a)0.625 m/s²
(b)1.6 m/s²
(c)8 m/s²
(d)2.2 m/s²
10.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
11.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
12.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
13.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
14.The curve y = x² − 3x. Use the chord between x = 1 and x = 4 to estimate the gradient of the curve at x = 2.5.
y = x² − 3x
(a)2/3
(b)1/2
(c)2
(d)6
15.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

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