3.4 Velocity and Rates of Change. Consider a graph of displacement (distance traveled) vs. time. time (hours) distance (miles) Average velocity can be.

3.4 Velocity and Rates of Change

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.)

3.4 Velocity and other Rates of Change


Velocity is the first derivative of position.

Acceleration is the second 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:

ftsec 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


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. )


For a circle: 2A r2dA 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


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?



Determine its position, velocity, and acceleration when t = 0 and when t = 3 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 launchvelocity of 160 ft/sec. It reaches a height of feet after t seconds.

How high does the rock 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 of the rock at any time t during its flight (after the blast)?

When does the rock hit the ground?

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

$195marginal cost

The 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.

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

3.4 Velocity and Rates of Change. Consider a graph of displacement (distance traveled) vs. time. time (hours) distance (miles) Average velocity can be.

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