3.4 Velocity, Speed, and Rates of Change

Created by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

Consider a graph of position vs. time.

time (hours)

distance from

starting place

(miles)

Average velocity can be found by comparing:

change in position

change in time

s

t

t

sA

B

2 1avg

2 1( ) ( )

f t f tfV

t t t

The speedometer in your car measures instantaneous velocity (but without direction information!)

0

limt

f t t f tdfv t

dt t

Velocity, the change in position as time goes by, is the first derivative of position.

Example:Free Fall Position Equation

21

2s g t

GravitationalConstants:

2

ft2

c3

seg

2

m.8

c9

seg

2

cm980

secg

Speed is the absolute value of velocity.

s = - 4.9 t2 m

velocity =

s = 2 m9.1

( 8)2

t

speed = ds

dtm/s9.8t

ds

dt4.9 2t m/s9.8t

Acceleration is the derivative of velocity, and the second derivative of position.

dva

dt

2

2

d s

dt

example: 32ft

v ts

232ft

as

If distance is in:

velocity would be in:

acceleration would be in:

meters

meters

second

msecsec 2

mor

sec

Jerk, the change in acceleration as time goes by, is the third derivative of position.

Snap, Crackle and Pop are the fourth, fifth and sixth derivatives of position. (Honestly!)

time

positionp increases => p′ pos

It is important to understand the relationship between a position graph, velocity (p′) and acceleration (p′′):

p horizontal => p´ zero

p′ constant => p′′ zero

p decreases => p′ neg

p horizontal => p′ zero

p increases => p′ pos

p increases => p′ pos p decreases=> p′ neg

p decreases

=> p′ neg

p′ constant => p′′ zero

p′ increases => p′′ pos

p′ decreases => p′′ negp′ decreases => p′′ neg

p′ increases

=> p′′ pos

p′ constant => p′′ zero

p′ decreases => p′′ neg

Rates of Change:

Average rate of change in f f x h f x

h

Instantaneous rate of change in f 0

limh

f x h f xf x

h

These definitions are true for any function

( and x does not have to represent time! )

Example 1:

For a circle:

2A r

2dA dr

dr dr

2dA

rdr

Instantaneous rate of change in surface area as the radius changes.

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 the change in cost as the number of manufactured items changes.

The marginal cost is also thought of as the increase in cost for manufacturing one additional item.

Marginal cost is a linear approximation of a curved function. For large values of x, it gives a good approximation of the cost of producing “the next item.”

Example 13: Suppose it costs: 3 26 15c x x x x to produce x stoves.

If you are currently producing 10 stoves, the next stove will cost roughly:

23 10 20 01 1 1 15c 300 120 15

$195

marginal cost after the 10th stove

The actual cost is: 11 10C C 3 2 3 211 6 11 15 11 10 6 10 15 10

770 550 $220

Note that for small values of x, this is not a perfect approximation– it’s much better as x grows large!

then c′ (x) = 3x2 - 12x + 15