Lesson 3-4: Velocity, Speed, and Rates of Change AP Calculus Mrs. Mongold

Definitions Instantaneous Rate of Change: the instantaneous rate of change of f with respect to x at a is the derivative Motion along a line –Position Function –Displacement –Average Velocity

Instantaneous Velocity Speed Acceleration

Free Fall Constants (Earth) Sensitivity to Change Derivatives in Economics

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.)

Velocity is the first derivative of position.

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

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

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:

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. )

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.

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

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.

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.

Things to Remember Velocity is the first derivative (how fast an object is moving as well as the direction of motion) Speed is the absolute value of the first derivative (tells how fast regardless of direction) Acceleration is the second derivative

Examples Find the rate of change of the area of a circle with respect to its radius A=πr 2 Evaluate the rate of change at r = 5 and r = 10 What units would be appropriate if r is in inches

Examples Find the rate of change of volume of a cube with respect to s V=s 3 Evaluate rate of change at s=1, s=3, and s=5 What would units be if measured in inches

Example Suppose a ball is dropped from the upper observation deck, 450 m above ground. What is the velocity of the ball after 5 seconds?

Example When does the particle move… Forward Backward Speed up Slow down

Example Cont… When is acceleration Positive Negative Zero

Example Cont… When is it at its greatest speed When is it still for more than an instant

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t 2 ft. after t seconds. How high does it go 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? What is the acceleration How long does it take for the rock to return to the ground?

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