Mathematics · Ch 7 — Applications of Differential Calculus

Derivative as Rate of Change

7.2.2

Derivative as Rate of Change

The derivative also measures the rate of change of one variable with respect to another — population growth rates, production rates, water-flow rates, velocity and acceleration are everyday instances.

Motion along a line. Fix an origin and a positive direction on the line of motion. If $s=f(t)$ is the position function (signed distance from the origin at time $t$ ), then the velocity and acceleration at time $t$ are

v(t)=\frac{ds}{dt},\qquad a(t)=\frac{dv}{dt}=\frac{d^2s}{dt^2}.

Note

Working facts used throughout the exercises:

Speed is the absolute value of velocity, regardless of direction: $\text{Speed}=|v(t)|=\left|\dfrac{ds}{dt}\right|$ .
The particle is momentarily at rest when $v(t)=0$ ; it moves forward when $v(t)>0$ and backward when $v(t)<0$ ; it changes direction exactly where $v(t)$ changes sign.
If the particle changes direction at a time $t_c\in(t_1,t_2)$ , the total distance travelled from $t_1$ to $t_2$ is $|s(t_c)-s(t_1)|+|s(t_2)-s(t_c)|$ — the two "legs" of the trip are added as positive distances, not simply subtracted as a net displacement.
Near Earth's surface, a body in free fall (gravity only, no air resistance) has constant acceleration $a=-g$ (taking "up" as positive), so $v=-gt+v_0$ and $s=-\tfrac12gt^2+v_0t+s_0$ , where $g\approx9.8\ \text{m/s}^2\ (32\ \text{ft/s}^2)$ . …

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