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Velocity, Speed, and Rates of Change - Understanding the Relationship

Explore the concept of velocity, speed, and rates of change through examples from physics, mathematics, and economics. Understand how graphs, derivatives, and instantaneous rates of change play a role in these concepts.

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Velocity, Speed, and Rates of Change - Understanding the Relationship

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  1. Photo by Vickie Kelly, 2008 Greg Kelly, Hanford High School, Richland, Washington 3.4 Velocity, Speed, and Rates of Change Denver & Rio Grande Railroad Gunnison River, Colorado

  2. B distance (miles) A time (hours) (The velocity at one moment in time.) 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.

  3. Velocity is the first derivative of position.

  4. Gravitational Constants: Speed is the absolute value of velocity. Example: Free Fall Equation

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

  6. distance time 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

  7. Average rate of change = Instantaneous rate of change = Rates of Change: These definitions are true for any function. ( x does not have to represent time. )

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

  9. from Economics: Marginal cost is the first derivative of the cost function, and represents an approximation of the cost of producing one more unit.

  10. Note that this is not a great approximation – Don’t let that bother you. The actual cost is: 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

  11. 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. p

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