1. 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

2. 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.)

3. Velocity is the first derivative of position.

4. Speed is the absolute value of velocity.
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. 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

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. 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.

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.

10. 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

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