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Velocity and Other Rates of Change

Velocity and Other Rates of Change. 3.4. 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:.

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Velocity and Other Rates of Change

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  1. Velocity and Other Rates of Change 3.4

  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. Instantaneous Rates of Change

  4. Example Instantaneous Rates of Change

  5. Motion Along a Line

  6. Instantaneous Velocity Velocity is the first derivative of position.

  7. Gravitational Constants: Example: Free Fall Equation

  8. Example Finding Velocity

  9. Speed

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

  11. Acceleration

  12. distance time It is important to understand the relationship between a position graph, velocity and acceleration:

  13. Speed • Speed is increasing if a(t) and v(t) have the same sign. • Speed is decreasing if a(t) and v(t) have a different sign.

  14. 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. )

  15. Example a. Find the of change of the area A of a circle with respect to its radius r. • Evaluate the rate of change of at r = 5 and at r = 10. • If r is measured in inches and A is measured in square inches, what units would be appropriate for dA/dr.

  16. Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 f/s. It reaches a height of s = 160t-16t2 ft after t seconds. • How high does the rock go? • What is the velocity and speed of the rock when it is 256 ft above the ground on the way up and on the way down? • What is the acceleration of the rock at at any time t during its flight? • When does the rock hit the ground?

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

  18. Derivatives in Economics

  19. Example Derivatives in Economics

  20. 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 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. actual cost

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