Main Concept

The odometer on a car measures how far the car travels. The speedometer measures how fast it travels. These two measurements are intimately connected: the faster the car goes, the farther it goes in a given time period, and vice-versa. This tight connection means that if you know how one of the two measurements is changing over time then you can generally figure out the other one. In particular, if the speed is increasing, then the graph of the distance traveled by the car (as measured by the odometer) will bend upwards, becoming steeper and steeper. Similarly, if the graph of the distance traveled is becoming less steep, then the car must be slowing down, so the graph of its speed must be decreasing.

Note also that while the speedometer on a car can go up and down, the odometer can only go up—it represents an increasing function of time.

This relationship between distance and speed traveled is illustrated on these graphs, which represent a car going on a short trip. Click either graph to show the distance traveled and speed corresponding to a certain time. The car's odometer is displayed above the distance graph, and its speedometer is above the speed graph.

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