3.4 Velocity, Speed, and Rates of Change

downward-256

2, 8

−∞,144( ⎤⎦

5,144( )

X=3, 7

x=158−∞,5⎛

⎝⎜

⎞

⎠⎟

64

-32

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

Δ=Δ

tΔ

sΔA

B

( ) ( )ave

f t t f tsV

t t

+ Δ −Δ= =

Δ Δ

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

( ) ( ) ( )0

limt

f t t f tdsV t

dt tΔ →

+ Δ −= =

Δ

(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 rπ=

2dA dr

dr drπ=

2dA

rdr

π=

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 drπ=

→

A particle P moves back and forth on the number line. The graph below shows the position of P as a function of time.

a) Describe the motion of the particle over time.

b) Graph the particle’s velocity and speed (where defined).

4

-4

2 4 6

Particle Motion

Particle is moving right when P‘(t) > 0 or (0,1)

Particle is moving left when P’(t) < 0 or (2,3), (5,6)

Particle is standing still when P’(t) = 0 or (1,2), (3,5)

4

-4

2 4 6

2

-2

Particle Motion

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.

π

3.5 Derivatives of Trig Functions

3π4

97.403°

32

Domain: all realsRange: −1, 1⎡

⎣⎢⎢

⎤

⎦⎥⎥

Domain: x≠kπ2 (k odd integer)

Range: all reals

±12

0

Multiply by 1+cos h1+cos h and use

identity 1-cos2h=sin2h

y = 12x - 35

12

π

2

π0

2

π−

π−

Consider the function ( )siny θ=

We could make a graph of the slope: θ slope

1−

0

1

0

1−Now we connect the dots!

The resulting curve is a cosine curve.

( )sin cosd

x xdx

=

→

π

2

π0

2

π−

π−

We can do the same thing for ( )cosy θ= θ slope

0

1

0

1−

0The resulting curve is a sine curve that has been reflected about the x-axis.

( )cos sind

x xdx

= −

→

We can find the derivative of tangent x by using the quotient rule.

tand

xdx

sin

cos

d x

dx x

( )2

cos cos sin sin

cos

x x x x

x

⋅ − ⋅ −

2 2

2

cos sin

cos

x x

x

+

2

1

cos x

2sec x

( ) 2tan secd

x xdx

=

→

Derivatives of the remaining trig functions can be determined the same way.

sin cosd

x xdx

=

cos sind

x xdx

=−

2tan secd

x xdx

=

2cot cscd

x xdx

=−

sec sec tand

x x xdx

= ⋅

csc csc cotd

x x xdx

=− ⋅

π