1. The Instantaneous Rate of Change.
Archetype: A ball thrown straight up in the air is -16t2+75t + 6 feet high after t seconds. How fast is it traveling 1 second after thrown?
Multi-Variable Version: A ball rolls down a spiral-shaped slide. Assume the friction force is negligible and estimate how fast the ball is traveling when it is half-way down the slide. How far away from the end of the slide will the ball land? Where does it land?
Comments: We are being asked to discuss how fast the position of the
ball is changing, not on average, but at the exact instant when t = 1 second.
This is complicated by the fact that the velocity changes ... the ball
is not traveling as quickly at t = 1 second as it was when it left the
hand. If the ball had constant velocity, V = D/T, where V is velocity,
D is distance, and T is time; so we could simply divide.
Examples of this type of problem abound in many fields, including biology,
economics
For further discussion of this problem, see Homework Solutions, Example #1.
2. The Total Change Accumulated.
Archetype: If velocity of a car t seconds after applying brakes is -2.5 t2+40 (in feet per second), find the total distance traveled by the car before it comes to a complete stop.
Multi-Variable Version: For the ball on the slide from Question 1, what total distance does the ball travel from the time it is released until it lands?
Comments: We are being asked to find how much the position of the car
changes during some time interval. This is complicated again by the fact
that the velocity changes ... the car is slowing down. if the car had constant
velocity, V T = D, so we could simply multiply.