Velocity and Rates of Change in Calculus

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