1. 3.4 Velocity, Speed, and Rates of Change

2. downward -256 2, 8

3. X=3, 7 64 -32

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

5. Velocity is the first derivative of position.

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

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

8. time distance acc pos vel pos & increasing acc zero vel pos & constant acc neg vel pos & decreasing velocity zero acc neg vel neg & decreasing acc zero vel neg & constant acc pos vel neg & increasing acc zero, velocity zero It is important to understand the relationship between a position graph, velocity and acceleration:

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

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

11. A particle P moves back and forth on the number line. The graph below shows the position of P as a function of time. a)Describe the motion of the particle over time. b)Graph the particle’s velocity and speed (where defined). 4 -4 246 Particle Motion

12. 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) 4 -4 246 2 -2 Particle Motion

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

15. 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. 

16. 3.5 Derivatives of Trig Functions

18. 0 y = 12x - 35 12

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

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

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

22. Derivatives of the remaining trig functions can be determined the same way. 