Transcript Calculus 3.4 - Vista Unified School District
2.4 Velocity, Speed, and Rates of Change
Photo by Vickie Kelly, 2008
Denver & Rio Grande Railroad Gunnison River, Colorado
Vista High, AB Calculus. Book Larson, V9 2010
Consider a graph of displacement (distance traveled) vs. time.
distance (miles)
A B
s
Average velocity can be found by taking: change in position change in time
s
t
t
time (hours)
V
ave
s
t
t
The speedometer in your car does not measure average velocity, but instantaneous velocity.
ds dt
lim
t
0
t
(The velocity at one moment in time.)
Velocity is the first derivative of position.
Example: Free Fall Equation
s
1 2 2
s
2 32
t
2
s
16
t
2
V
ds dt
32
t
Speed is the absolute value of velocity.
Gravitational Constants:
g
32 ft sec 2
g
9.8 m sec 2
g
980 cm sec 2
Acceleration is the derivative of velocity.
a
dv dt
dt
2 example:
v
32
t a
32
If distance is in:
feet
Velocity would be in: feet sec Acceleration would be in:
ft sec sec
ft sec 2
It is important to understand the relationship between a position graph, velocity and acceleration: distance acc pos vel pos & increasing acc zero vel pos & constant acc neg vel pos & decreasing velocity zero acc neg vel neg & decreasing acc zero vel neg & constant acc pos vel neg & increasing acc zero, velocity zero time
Rates of Change: Average rate of change =
h
h
Instantaneous rate of change =
f
lim
h
0
h
h
These definitions are true for any function.
( x does not have to represent time. )
Example 1: For a circle:
A
r
2
dA
dr d dr
r
2
dA
2
r dr
Instantaneous rate of change of the area with respect to the radius.
dA
2
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.
Example 13: Suppose it costs: to produce
x
stoves.
x
3
6
x
2
15
x
3
x
2
12
x
15
If you are currently producing 10 stoves, the 11 th stove will cost approximately:
c
2 Note that this is not a great approximation – Don’t let that bother you.
$195
The actual cost is: 11 3 2
C
10 3
C
2 marginal cost
$220
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.