1 / 22

3.4 Velocity, Speed, and Rates of Change

3.4 Velocity, Speed, and Rates of Change. downward. -256. 2, 8. X=3, 7. 64. -32. 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:.

eaton-hebert
Télécharger la présentation

3.4 Velocity, Speed, and Rates of Change

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 3.4 Velocity, Speed, and Rates of Change

  2. downward -256 2, 8

  3. X=3, 7 64 -32

  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: The speedometer in your car does not measure average velocity, but instantaneous velocity.

  5. Velocity is the first derivative of position.

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

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

  8. 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

  9. 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. )

  10. 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.

  11. 4 2 4 6 -4 Particle Motion • A particle P moves back and forth on the number line. The graph below shows the position of P as a function of time. • Describe the motion of the particle over time. • Graph the particle’s velocity and speed (where defined).

  12. 4 2 2 4 6 -2 -4 Particle Motion Particle is moving right when P‘(t) > 0 or (0,1) Particle is moving left when P’(t) < 0 or (2,3), (5,6) Particle is standing still when P’(t) = 0 or (1,2), (3,5)

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

  14. 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

  15. 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

  16. 3.5 Derivatives of Trig Functions

  17. 0 y = 12x - 35 12

  18. slope Consider the function We could make a graph of the slope: Now we connect the dots! The resulting curve is a cosine curve.

  19. slope We can do the same thing for The resulting curve is a sine curve that has been reflected about the x-axis.

  20. We can find the derivative of tangent x by using the quotient rule.

  21. Derivatives of the remaining trig functions can be determined the same way. p

More Related