1. Denver & Rio Grande Railroad Gunnison River, Colorado www.apcalculusbc.com

2. Consider a graph of displacement (distance traveled) vs. time. time (hours) distance (miles) Average velocity can be found by taking: A B The speedometer in your car does not measure average velocity, but instantaneous velocity. (The velocity at one moment in time.)

3. Velocity is the first derivative of position. www.apcalculusbc.com

4. Example: Free Fall Equation Gravitational Constants: Speed is the absolute value of velocity.

5. Acceleration is the derivative of velocity. example: If distance is in: Velocity would be in: Acceleration would be in:

6. time distance acc pos vel pos & increasing acc zero vel pos & constant acc neg vel pos & decreasing velocity zero acc neg vel neg & decreasing acc zero vel neg & constant acc pos vel neg & increasing acc zero, velocity zero It is important to understand the relationship between a position graph, velocity and acceleration:

7. Rates of Change: Average rate of change = Instantaneous rate of change = These definitions are true for any function. ( x does not have to represent time. )

8. Example 1: For a circle: Instantaneous rate of change of the area with respect to the radius. For tree ring growth, if the change in area is constant then dr must get smaller as r gets larger.

9. from Economics: Marginal cost is the first derivative of the cost function, and represents an approximation of the cost of producing one more unit. www.apcalculusbc.com

10. Example 13: Suppose it costs: to produce x stoves. If you are currently producing 10 stoves, the 11 th stove will cost approximately: marginal cost The actual cost is: actual cost Note that this is not a great approximation – Don’t let that bother you.

11. Marginal cost is a linear approximation of a curved function. For large values it gives a good approximation of the cost of producing the next item.  www.apcalculusbc.com