Calculus 3.4

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Interpretations of the Derivative
(Rates of Change)
Denver Rio Grande Railroad Gunnison River,
Colorado
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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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Velocity is the first derivative of position.
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Example
Free Fall Equation
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example
If distance is in
Velocity would be in
Acceleration would be in
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It is important to understand the relationship
between a position graph, velocity and
acceleration
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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Rates of Change
These definitions are true for any function.
( x does not have to represent time. )
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Example 1
For a circle
Instantaneous rate of change of the area
with respect to the radius.
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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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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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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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