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3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

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Consider a graph of position vs. time. time (hours) distance from starting place (miles) Average velocity can be found by comparing: A B The speedometer in your car measures instantaneous velocity (but without direction information!)

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Velocity, the change in position as time goes by, is the first derivative of position.

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Example: Free Fall Position Equation Gravitational Constants: Speed is the absolute value of velocity. s = - 4.9 t 2 m velocity = s = speed =

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Acceleration is the derivative of velocity, and the second derivative of position. example: If distance is in: velocity would be in: acceleration would be in: meters second

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Jerk, the change in acceleration as time goes by, is the third derivative of position. Snap, Crackle and Pop are the fourth, fifth and sixth derivatives of position. (Honestly!)

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time position p increases => p pos It is important to understand the relationship between a position graph, velocity (p) and acceleration (p): p horizontal => p´ zero p constant => p zero p decreases => p neg p horizontal => p zero p increases => p pos p decreases=> p neg p decreases => p neg p constant => p zero p increases => p pos p decreases => p neg p increases => p pos p constant => p zero p decreases => p neg

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Rates of Change: Average rate of change in f Instantaneous rate of change in f These definitions are true for any function ( and x does not have to represent time! )

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Example 1: For a circle: Instantaneous rate of change in surface area as the radius changes. For tree ring growth, if the change in area is constant, then dr must get smaller as r gets larger.

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From economics: Marginal cost is the first derivative of the cost function, and represents the change in cost as the number of manufactured items changes. The marginal cost is also thought of as the increase in cost for manufacturing one additional item.

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Marginal cost is a linear approximation of a curved function. For large values of x, it gives a good approximation of the cost of producing the next item.

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Example 13: Suppose it costs: to produce x stoves. If you are currently producing 10 stoves, the next stove will cost roughly: marginal cost after the 10 th stove The actual cost is: Note that for small values of x, this is not a perfect approximation– its much better as x grows large! then c (x) = 3x 2 - 12x + 15

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