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Velocity and Rates of Change in Calculus

Learn about instantaneous rate of change, velocity, acceleration, and how to find average and instantaneous rates of change. Explore examples and applications in different areas.

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Velocity and Rates of Change in Calculus

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  1. AP Calculus AB/BC 3.4 Velocity and Other Rates of Change, p. 127 Greg Kelly, Hanford High School, Richland, Washington

  2. Definition – Instantaneous Rate of Change, p. 127 The (instantaneous) rate of change of f with respect to x at a is the derivative provided the limit exists.

  3. Example 1 (see #1, p. 135) (a) Write the volume V of a cube as a function of the side length s. (b) Write the (instantaneous) rate of change of the volume V with respect to a side s. (c) Evaluate the rate of change of V at s = 1 and s = 5. (d) If s is measured in inches and V is measured in cubic inches, what units would be appropriate for dV/ds? = 3 = 75

  4. B distance (miles) A time (hours) (The velocity at one moment in time.) Consider a graph of displacement (distance traveled) vs. time. The position as a function of time is s = f(t). Average velocity can be found by taking: The speedometer in your car does not measure average velocity, but instantaneous velocity.

  5. Example 3 • The position of a particle moving along a straight line at any time t≥0 is s(t) = t2 – 3t + 2. • Find the displacement during the first 5 seconds. • Find the average velocity during the first 5 seconds. And, • Find the instantaneous velocity when t = 4. = 10 m

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

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

  8. Example 4 • The position of a particle moving along a straight line at any time t≥0 is s(t) = t2 – 3t + 2. • Find the acceleration of the particle when t = 4. • At what values of t does the particle change direction? • Where is the particle when s is a minimum? (b) Change in direction happens when the velocity is zero. So, (c) s(t) is at a minimum also when v(t) = 0 which is at

  9. 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

  10. 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. )

  11. For tree ring growth, if the change in area is constant then dr must get smaller as r gets larger. Example 5 For a circle: Instantaneous rate of change of the area with respect to the radius.

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

  13. Note that this is not a great approximation – Don’t let that bother you. The actual cost is: Example 6 Suppose it costs: to produce x stoves. If you are currently producing 10 stoves, the 11th stove will cost approximately: marginal cost actual cost

  14. 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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