Rates of Change: Velocity and Marginals

Definition: The average rate of change of a function $y = f(x)$ over the interval $[a, b]$ is given by the formula: $\text{Average rate of change} = \frac{f(b) - f(a)}{b - a} = \frac{\Delta y}{\Delta x}$
This represents the slope of the secant line passing through the points $(a, f(a))$ and $(b, f(b))$ on a graph, as seen in Figure 2.18.
Example 2 (Falling Object): For a height function $s = -16t^2 + 100$ , the average velocity over different intervals is:
- $[1, 2]$ : $-48\,\text{ft/sec}$
- $[1, 1.5]$ : $-40\,\text{ft/sec}$
- $[1, 1.1]$ : $-33.6\,\text{ft/sec}$

The instantaneous rate of change at a specific point is the derivative of the function at that point.
Velocity Function: The derivative of the position function $s(t)$ is the velocity function $v(t) = s'(t)$ .
Position Function (Free-falling object): $s(t) = -16t^2 + v_{0}t + s_{0}$ , where $v_{0}$ is initial velocity and $s_{0}$ is initial height.
Speed: Defined as the absolute value of velocity, $|v(t)|$ .
Example 4 (Diver): A diver jumps from $32\,\text{feet}$ with initial velocity $16\,\text{ft/sec}$ .
- Position: $s(t) = -16t^2 + 16t + 32$
- Impact Time: Hits water when $s = 0$ at $t = 2\,\text{seconds}$ .
- Impact Velocity: $v(2) = s'(2) = -32(2) + 16 = -48\,\text{ft/sec}$ .

Marginals refer to the rates of change of profit ( $P$ ), revenue ( $R$ ), and cost ( $C$ ) with respect to the number of units ( $x$ ) produced or sold.
Basic Relationship: $P = R - C$ .
Marginal Definitions:
- Marginal Profit: $\frac{dP}{dx}$
- Marginal Revenue: $\frac{dR}{dx}$
- Marginal Cost: $\frac{dC}{dx}$
Discrete variables (like individual product units) are treated as continuous variables in calculus to find optimal values, then rounded to the nearest sensible unit.
Example 5 (Profit): For $P = 0.0002x^3 + 10x$ , the marginal profit at $x = 50$ is $\$11.50$ per unit. This closely approximates the actual gain for the $51\text{st}$ unit ( $\$11.53$ ).