1. 3.4 Velocity and Other Rates of Change
AP Calculus AB
3.4 Velocity and Other Rates of Change
Greg Kelly, Hanford High School, Richland, Washington

2. Instantaneous Rate of Change
Definition –
Instantaneous Rate of Change
The (instantaneous) rate of change of f with respect to x at a is the derivative
provided the limit exists.

3. Example 1
(a) Write the volume V of a cube as a function of the side length s.
(b) Write the (instantaneous) rate of change of the volume V with respect to a side s.
(c) Evaluate the rate of change of V at s = 1 and s = 5.
(d) If s is measured in inches and V is measured in cubic inches, what units would be appropriate for dV/ds?
= 3
= 75

4. Average velocity can be found by taking:
Consider a graph of displacement (distance traveled) vs. time. The position as a function of time is s = f(t).
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.)

5. Example 2
The position of a particle moving along a straight line at any time t ≥0 is s(t) = t2 – 3t + 2.
Find the displacement during the first 5 seconds.
Find the average velocity during the first 5 seconds. And,
Find the instantaneous velocity when t = 4.
= 10 m

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

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

8. Example 3
The position of a particle moving along a straight line at any time t ≥0 is s(t) = t2 – 3t + 2.
Find the acceleration of the particle when t = 4.
At what values of t does the particle change direction?
Where is the particle when s is a minimum?
(b) Change in direction happens when the velocity is zero. So,
(c) s(t) is at a minimum also when v(t) = 0 which is at

9. 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

10. 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. )

11. Example 4
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.

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

13. Example 5
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

14. 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