3.4 Velocity and Rates of Change

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

3.4 Velocity and other Rates of Change Velocity is the first derivative of position. Acceleration is the second derivative of position.

3.4 Velocity and other Rates of Change Gravitational Constants: Example: Free Fall Equation Speed is the absolute value of velocity.

3.4 Velocity and other Rates of Change Acceleration is the derivative of velocity. example: If distance is in: Velocity would be in: Acceleration would be in:

3.4 Velocity and other Rates of Change time distance 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

3.4 Velocity and other 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. )

3.4 Velocity and other Rates of Change 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.

Evaluate the rate of change of the area of a circle A at r = 5 and r = 10.

EXAMPLE: An object moves along a linear path according to the equation where s is measured in feet and t in seconds. Determine its velocity when t = 4 and when t = 2. When is the velocity zero?

EXAMPLE: An object moves along a linear path according to the equation where s is measured in feet and t in seconds. Determine its position, velocity, and acceleration when t = 0 and when t = 3 seconds.

EXAMPLE: An object moves along a linear path according to the equation where s is measured in feet and t in seconds. When is the velocity zero? On what intervals is the object moving to the right? To the left? We consider the intervals determined by the times when the velocity is zero - t = 0, t = 2, and t = 4 sec. For 0 < t < 2 and for t > 4, velocity is positive, so the object is moving to the right. For 2 < t < 4, velocity is negative, so the object is moving to the left.

EXAMPLE: A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of feet after t seconds. How high does the rock go?

EXAMPLE: A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of feet after t seconds. What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? on the way down?

EXAMPLE: A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of feet after t seconds. What is the acceleration of the rock at any time t during its flight (after the blast)? When does the rock hit the ground?

3.4 Velocity and other Rates of Change from Economics: Marginal cost is the first derivative of the cost function, and represents an approximation of the cost of producing one more unit.

3.4 Velocity and other Rates of Change Example 13: Suppose it costs: to produce x stoves. If you are currently producing 10 stoves, the 11th stove will cost approximately: The actual cost is: marginal cost actual cost

3.4 Velocity and other Rates of Change 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.

HOMEWORK P. 135-137 #1-4, 9-20