Denver & Rio Grande Railroad Gunnison River, Colorado Interpretations of the Derivative (Rates of Change) Denver & Rio Grande Railroad Gunnison River, Colorado Greg Kelly, Hanford High School, Richland, Washington Revised by: Jon Bannon, Siena College Photo by Vickie Kelly, 2008

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

Velocity is the first derivative of position.

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

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

It is important to understand the relationship between a position graph, velocity and acceleration: time distance 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

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. )

Instantaneous rate of change of the area with respect to the radius. 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.

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

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

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