In the first post of this series I mentioned that a car's odometer and speedometer perform related functions :), and these functions are a good way to think about the fundamental operations of calculus. Suppose a car is going in a straight line (say, on the Bonneville salt flats). We need to deal only with the positive x-axis.

Distance (often labeled s) is a function of time:

If the velocity is constant and we start from s = 0 then:

Distance equals velocity times time. (60 miles per hour for 2 hours = 120 miles).

Now, velocity

*might*be changing, it might be a function of time. Let's say we'll figure out how it changes later, and just write in a general way:

If the velocity is changing, we call the rate of change in velocity the acceleration. (It is the slope of the curve for v as a function of t). If the acceleration is constant (like gravity), then:

For a car, imagine increasing the pressure on the gas pedal steadily so that after 1 second you are going 10 mph, after 2 seconds 20 mph, after 3 seconds 30 mph. If we continue at the same rate, we'll accelerate from 0 to 60 mph in 6 seconds.

As I said before, there is a special trick symbolized with a prime mark placed next to the s (the physicists put a dot on top). The trick is called differentiation:

What it is depends on the exact form of s as a function of time. If the acceleration is constant:

Do you remember the rule from last time?

Or, using the time and distance symbols:

So that is where the factor of 1/2 comes from in the equation involving acceleration. It is needed to cancel the the 2 that comes out of the exponent when we differentiate. One way to think about this is to say that the velocity after a period of constant acceleration is the

*average*of the initial acceleration and the final acceleration, times the time. But this differentiation mumbo jumbo will work even if the velocity and acceleration are more complicated functions of time.

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