3.4 Velocity, Speed, and Rates of Change

1. 3.4 Velocity, Speed, and Rates of Change Greg Kelly, Hanford High School, Richland, Washington

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

3. Velocity is the first derivative of position.

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

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

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

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

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

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

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

11. Note that this is not a great approximation – Don’t let that bother you. 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. p