1. 3.4 Velocity and Rates of Change

2. 3.4 Velocity and other Rates of Change
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. 3.4 Velocity and other Rates of Change
Velocity is the first derivative of position.
Acceleration is the second derivative
of position.

4. 3.4 Velocity and other Rates of Change
Gravitational
Constants:
Example:
Free Fall Equation
Speed is the absolute value of velocity.

5. 3.4 Velocity and other Rates of Change
Acceleration is the derivative of velocity.
example:
If distance is in:
Velocity would be in:
Acceleration would be in:

6. 3.4 Velocity and other Rates of Change
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. 3.4 Velocity and other 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. 3.4 Velocity and other Rates of Change
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. Evaluate the rate of change of the area of a circle A at r = 5 and r = 10.

10. EXAMPLE: An object moves along a linear path according to the equation
where s is measured in feet and t in seconds.
Determine its velocity when t = 4 and when t = 2.
When is the velocity zero?

11. EXAMPLE: An object moves along a linear path according to the equation
where s is measured in feet and t in seconds.
Determine its position, velocity, and acceleration when t = 0 and when t = 3 seconds.

12. EXAMPLE: An object moves along a linear path according to the equation
where s is measured in feet and t in seconds.
When is the velocity zero?
On what intervals is the object moving to the right? To the left?
We consider the intervals determined by the times when the velocity is zero t = 0, t = 2, and t = 4 sec.
For 0 < t < 2 and for t > 4, velocity is positive, so the object is moving to the right.
For 2 < t < 4, velocity is negative, so the object is moving to the left.

13. EXAMPLE: A dynamite blast propels a heavy rock straight up with a launch
velocity of 160 ft/sec. It reaches a height of feet after t seconds.
How high does the rock go?

14. EXAMPLE: A dynamite blast propels a heavy rock straight up with a launch
velocity of 160 ft/sec. It reaches a height of feet after t seconds.
What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? on the way down?

15. EXAMPLE: A dynamite blast propels a heavy rock straight up with a launch
velocity of 160 ft/sec. It reaches a height of feet after t seconds.
What is the acceleration of the rock at any time t during its flight (after the blast)?
When does the rock hit the ground?

16. 3.4 Velocity and other Rates of Change
from Economics:
Marginal cost is the first derivative of the cost function, and represents an approximation of the cost of producing one more unit.

17. 3.4 Velocity and other Rates of Change
Example 13:
Suppose it costs:
to produce x stoves.
If you are currently producing 10 stoves, the 11th stove will cost approximately:
The actual cost is:
marginal cost
actual cost

18. 3.4 Velocity and other Rates of Change
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.

19. HOMEWORK P
#1-4, 9-20