Derivatives as rates of change

A rock dropped from 64 feet has height −16t² + 64. Differentiate once and you get its velocity, −32t; it hits the ground at t = 2 moving at 64 feet per second.

Position, velocity, acceleration

If s(t) is the position of an object moving along a line at time t, its velocity is v(t) = s′(t), the instantaneous rate of change of position. Velocity has a sign: positive when the object moves in the positive direction, negative when it moves the other way. Speed is the absolute value |v(t)|, which ignores direction. The acceleration is a(t) = v′(t) = s″(t), the rate at which velocity changes. OpenStax collects these in section 3.4 and uses them for falling and thrown objects.

Take the rock from the limits course, s(t) = −16t² + 64 feet. Its velocity is v(t) = −32t feet per second. At t = 0.5 that is −16, the number the secant slopes pointed to. The rock reaches the ground when s(t) = 0: 16t² = 64, t = 2 seconds. Its velocity then is v(2) = −64 feet per second, so its speed is 64 feet per second. The acceleration is a(t) = −32 feet per second squared at every moment, the constant pull of gravity in these units. The negative signs all mean the same thing: downward.

Something thrown up

An OpenStax exercise tosses a stone up from the ground at 15 meters per second, with height h(t) = 15t − 4.9t² meters. Its velocity is v(t) = 15 − 9.8t. At t = 1 the stone is still rising at 5.2 meters per second. It stops rising when v(t) = 0, at t = 15/9.8 ≈ 1.53 seconds, and that moment is its highest point: h ≈ 11.48 meters. After that v is negative and the stone falls. The acceleration is −9.8 meters per second squared throughout, the value of g used in that exercise.

Notice the method in that last calculation: the top of the flight is where the velocity, a derivative, equals zero. That is a first taste of unit 6. Notice too that average and instantaneous rates are different questions. The rock's average velocity over its whole fall is (0 − 64)/2 = −32 feet per second, while its instantaneous velocity runs from 0 at the start to −64 at the end.

Beyond motion

Any quantity that changes with another has a derivative that means something. If P(t) is a population, P′(t) is its growth rate in individuals per year. If C(x) is the cost of making x items, C′(x), the marginal cost, approximates the cost of making one more. If V is the volume of water in a tank, dV/dt is the flow rate. In each case the units of the derivative tell you how to read it, just as feet per second did for the rock.