Rates of Change may be determined graphically, using a chord or tangent to a graph; or numerically, by using known numbers. Other uses for determining rates of change include calculating flow of water through pipes, train scheduling, fuel consumption in engines and factory production.
1. A racing car accelerates from stationary and covers 260m in the first six seconds. In the following six seconds, it covers a further 460m. What is the average velocity of the car over the first 12 seconds?
Total distance covered is 260 + 460 = 720m
Total time = 12 seconds
Average velocity = `frac(text(distance))(text(time))` = 720 ÷ 12 = 60m/s
The graph shows distance against time for a new locomotive. What is the speed of the locomotive at 5 seconds?
Draw a tangent to the graph at 5 seconds.
The gradient of the tangent gives the velocity.
gradient = `frac(text(up))(text(along))` = `frac(47 - 18)(6 - 4)` = `frac(29)(2)` = 14.5 m/s