Gradients and areas under graphs worksheet — GCSE Higher

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.

2.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.

3.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.

4.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?

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?

6.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.

7.A tram travels between two stops. Its velocity-time graph consists of straight line segments joining the points (0, 0), (5, 20), (12, 20), (16, 4) and (20, 4), where time is in seconds and velocity is in m/s. Work out the average speed of the tram over the whole 20 seconds. Give your answer to 1 decimal place.

8.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.

9.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.

10.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.

11.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.

12.A company's cost, in £, for producing x items is shown on a graph. The tangent to the curve at x = 50 passes through (30, 400) and (70, 800). Interpret the gradient of this tangent in the context of the company's costs.

13.A cyclist accelerates uniformly from rest to 6 m/s in 4 seconds, travels at a constant 6 m/s for 10 seconds, then decelerates uniformly to rest in 3 seconds. Work out the total distance the cyclist travels.

14.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.

15.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.