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

h

afhafaf

)()()('

• 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:

change in position

change in time

s

t

t

sA

B

ave

f t t f tsV

t t

The speedometer in your car does not measure average velocity, but instantaneous velocity.

0

limt

f t t f tdsV t

dt t

(The velocity at one moment in time.)

Velocity is the first derivative of position.

Example: Free Fall Equation

21

2s g t

GravitationalConstants:

2

ft32

secg

2

m9.8

secg

2

cm980

secg

2132

2s t

216 s t

32 ds

V tdt

Speed is the absolute value of velocity.

Acceleration is the derivative of velocity.

dva

dt

2

2

d s

dt example: 32v t

32a

If distance is in: feet

Velocity would be in:feet

sec

Acceleration would be in:ft

sec sec

2

ft

sec

time

distance

acc posvel pos &increasing

acc zerovel pos &constant

acc negvel pos &decreasing

velocityzero

acc negvel neg &decreasing acc zero

vel neg &constant

acc posvel 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 = f x h f x

h

Instantaneous rate of change = 0

limh

f x h f xf x

h

These definitions are true for any function.

( x does not have to represent time. )

Example 1:

For a circle:

2A r

2dA dr

dr dr

2dA

rdr

Instantaneous rate of change of the area withrespect to the radius.

For tree ring growth, if the change in area is constant then dr must get smaller as r gets larger.

2 dA r dr

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: 3 26 15c x x x x

to produce x stoves. 23 12 15c x x x

If you are currently producing 10 stoves, the 11th stove will cost approximately:

210 3 10 12 10 15c

300 120 15

$195

marginal costThe actual cost is: 11 10C C

3 2 3 211 6 11 15 11 10 6 10 15 10

770 550 $220 actual cost

Note that this is not a great approximation – Don’t let that bother you.

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 radiusA=πr2

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=s3

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 – 16t2 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?

Homework

• Pages 129-130/ 1-12, 13, 23 and 24