3.4 - Velocity and Other Rates of Change

Motion along a straight line

This is implied instantaneous velocity.
The first derivative of the position function is the velocity function – $s (t)$ is $s^{'} (t) = v (t)$ .
The particle stops when $v = 0$ .
The particle is moving right (or up) if the velocity is positive. The particle is moving left (or down) if the velocity is negative.
A particle is speeding up if the signs of velocity and acceleration are the same sign. A particle is slowing down if the signs of velocity and acceleration are opposite signs.

This is implied instantaneous acceleration.
It’s the second derivation of the position function, or the first derivative of velocity – $s^{″} (t) = v^{'} (t) = a (t)$ .

Average Cost: $\frac{C (x)}{x}$
Average Increase in Cost: $\frac{C (x + h) - C (x)}{h}$
Marginal Cost: Estimates the cost of producing one unit beyond the present level of production – $lim_{h \to 0} (\frac{C (x + h) - C (x)}{h})$
“actual additional cost”: $C (x + h) - C (x)$
Demand Function: Expresses the relationship between unit price and the quantity demanded – $p = f (x)$
Revenue: $R (x) = p x$
Marginal Revenue: $R^{'} (x)$
- Estimates the increase in revenue that will result from selling one additional unit
- “actual additional revenue made” $R (x + h) - R (x)$
Profit: $P (x) = R (x) - C (x)$
Marginal Profit: Measures the rate of change of the profit function P and provides us with a good approximation of the actual profit or loss realized from the sale of 1 more unit of an item sold – $P x^{'} ()$
- “actual gain in profit”: $P (x + h) - P (x)$