2.4 Velocity, Speed, and Rates of Change

Photo by Vickie Kelly, 2008

Denver & Rio Grande Railroad Gunnison River, Colorado

Vista High, AB Calculus. Book Larson, V9 2010

Consider a graph of displacement (distance traveled) vs. time.

distance (miles)

A B



s

Average velocity can be found by taking: change in position change in time  

s



t



t

time (hours)

V

ave  

s



t

 

t

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



ds dt

 lim

t

0 

t

 (The velocity at one moment in time.) 

Velocity is the first derivative of position.



Example: Free Fall Equation

s

 1 2 2

s

2 32

t

2

s

 16

t

2

V



ds dt

 32

t

Speed is the absolute value of velocity.

Gravitational Constants:

g

 32 ft sec 2

g

 9.8 m sec 2

g

 980 cm sec 2 

Acceleration is the derivative of velocity.

a



dv dt



dt

2 example:

v



32

t a



32

If distance is in:

feet

Velocity would be in: feet sec Acceleration would be in:

ft sec sec

 ft sec 2 

It is important to understand the relationship between a position graph, velocity and acceleration: 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 time 

Rates of Change: Average rate of change = 

h

 

h

Instantaneous rate of change =

f

  lim

h

 0 

h

 

h

These definitions are true for any function.

( x does not have to represent time. ) 

Example 1: For a circle:

A

 

r

2

dA



dr d dr



r

2

dA

 2 

r dr

Instantaneous rate of change of the area with respect to the radius.

dA



2

 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.



x

3 

6

x

2 

15

x

  

3

x

2 

12

x



15

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

c

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

 

$195

The actual cost is:   11 3   2

C

10 3 

C

2  marginal cost 

$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.

