1. 3.4 Velocity, Speed, and Rates of Change

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. A fly ball is hit vertically upward (a pop-fly)
A fly ball is hit vertically upward (a pop-fly). It’s position is given by the equation s(t) = -5t2+30t, where the origin is at ground level and the positive direction is up.
a) Find the max height of the baseball.
Grade 10: Complete the Square
s(t)= -5(t2 – 6t)
= -5(t2 – 6t +9 – 9)
= -5[(t – 3)2 – 9]
= -5(t – 3)2 +45
Calculus: Use derivatives
Find where v(t)= s’(t) = 0
v(t)= -10t + 30
Set v(t) = 0 and solve
-10t + 30 = 0
t = 3
Sub t = 3 into s(t) to find height:
s(3) = -5(3)2 + 30(3)
= 45
The max height of the ball is 45m

6. b) When will the ball hit the ground?
Set s(t) = 0 and solve for t
-5t2+30t = 0
-5t(t - 6) = 0
t = 0 or t = 6
c) What is the velocity when it hits the ground?
v(6) = -10(6) + 30
= -30
It was travelling 30m/s when it hit the ground.
It took 6 seconds to return to the ground.

7. The position function of an object moving in a straight line is
s(t) = 3t2 – 0.25t4. At time t = 3, is the object moving towards
the origin or away from the origin?
Solution:
The velocity of the object is: v(t) = s’(t)
= 6t – t3
At time t = 3,
s(3) = and v(3) = - 9
In the diagram, let the positive direction be to the right.
Thus, s(3) >0 implies the object is to the right of the origin
and v(3) < 0 implies the object is moving to the left
(in this case, to the origin) when t = 3. s
Direction of motion at t = 3

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

9. Starting at time t = 0, a dragster accelerates down a strip and then brakes and comes to a stop. Its position function s(t) is s(t) = 6t2 – 0.2t3.
After how many seconds does the dragster stop?
What distance does the dragster travel?
At what time does the braking commence?
Solutions:
v(t) = 12t – 0.6t2. To stop v(t) = 0
12t – 0.6t2=0 so that t = 0 or t = 20. It stops after 20s.
It travels from s(0) = 0 to s(20)= Since it did not reverse direction, the distance travelled was 800m.
a(t) = v’(t) = 12 – 1.2t = 1.2(10 – t)
Note: when t<10 then a(t) >0 and that if t>10 then a(t) <0. Thus the dragster accelerates from 0s to 10s and brakes from 10s to 20s. The braking begins after 10s.

10. A ball thrown in the air slows as it rises and speeds up as it falls
Negative acceleration means that the ball slows down as it rises.
It indicates that the velocity is decreasing.
Positive acceleration means that velocity is increasing.
An object is accelerating if a(t) x v(t) > 0
decelerating if a(t) x v(t) < 0

11. 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

12. 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. )

13. 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.

14. from Economics:
The instantaneous rate of change of cost with respect to the
number of items produced is called the marginal cost.
Marginal cost is the first derivative of the cost function, and represents an approximation of the cost of producing one more unit.
Marginal revenue is the first derivative of the revenue function,
sometimes given as R(x) = xp(x) where p(x) is the price per
unit (p(x) is sometimes called the price function or demand
function.) and x is the number of items.

15. 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 have a marginal cost of approximately:
Note that this is not a great approximation – Don’t let that bother you.
The actual cost is:
marginal cost
actual cost

16. Marginal cost is a linear approximation of a curved function
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

17. Guidelines for Successful Problem Solving with Related Rates
Make a sketch and label known quantities
Introduce variables to rep. quantities that change (let x rep…)
Identify what you are to find
Find an equation (area, volume, pythag…)
Find the derivative
Substitute and solve
Conclude in the context of the problem