1. © Nuffield Foundation 2011 Nuffield Mathematics Activity Mean values

2. The speed of a car varies with time. How can you work out the average speed of the car? Can you work it out from a speed-time graph?

3. Speed–time graph Think about Can you describe what happens? Can you find the average speed? This activity is about finding mean values.

4. Mean speed Car travelling at 15 m s –1 Area = The distance travelled in 4 seconds is 60 metres 4 × 15 = 60 v m s –1 t seconds 0 4 15 where distance travelled = area under the speed–time graph

5. t seconds v m s –1 0 24 6 8 10 2 4 6 8 12 14 Car travelling along road in town Think about… The areas under this graph are triangles and trapezia. Can you recall the formulae for these areas? b h

6. A B C D E t seconds v m s –1 0 24 6 8 10 2 4 6 8 12 14 Area of A = Area of C == 29.295 Area of B = = 4.8 = 11.2 Area of E == 11.135 Area of D = = 26.8 Car travelling along road in town = 83.23 Total area Mean speed = 8.5 m s –1 (to 1dp)

7. Example Area = 0.025 t 4 – 0.002 t 5 – 200 = 250 Distance = 50 metres Mean speed Integration allows you to find the area under a curve. The speed of a cyclist along a road can be modelled by the function v = 0.1 t 3 – 0.01 t 4 = 5 m s –1 v t 0 v = 0.1 t 3 - 0.01 t 4 10

8. Using Integration Integration gives the area under a graph A = (6.1, 13.7) (8.1, 13.1) v t 0 D For section D: Gradient, m = –0.3 Using (6.1, 13.7) in y = mx + c 13.7 = - 0.3 × 6.1 + c c = 15.53 Distance = = 115.9515– 89.1515 = 26.8 For a linear graph this method takes longer.

9. Reflect on your work Can you summarise the method for finding the mean value of a quantity from its graph? In cases where the graph consists of straight lines, you can use either geometrical formulae to find the area or integration. Which of these methods do you prefer? Why? Mean values