3.4 Velocity and Rates of Change

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3.4 Velocity and Rates of Change
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3.4 Velocity and other Rates of Change
Consider a graph of displacement (distance
traveled) vs. time.
Average velocity can be found by taking
The speedometer in your car does not measure
average velocity, but instantaneous velocity.
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3.4 Velocity and other Rates of Change
Velocity is the first derivative of position.
Acceleration is the second derivative of
position.
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3.4 Velocity and other Rates of Change
Example
Free Fall Equation
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3.4 Velocity and other Rates of Change
example
If distance is in
Velocity would be in
Acceleration would be in
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3.4 Velocity and other Rates of Change
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
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3.4 Velocity and other Rates of Change
Rates of Change
These definitions are true for any function.
( x does not have to represent time. )
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3.4 Velocity and other Rates of Change
For a circle
Instantaneous rate of change of the area
with respect to the radius.
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Evaluate the rate of change of the area of a
circle A at r 5 and r 10.
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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?
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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.
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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 lt t lt 2 and for t gt 4, velocity is
positive, so the object is moving to the right.
For 2 lt t lt 4, velocity is negative, so the
object is moving to the left.
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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?
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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?
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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?
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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.
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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
marginal cost
actual cost
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3.4 Velocity and other Rates of Change
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.
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HOMEWORK

• P. 135-137
• 1-4, 9-20