ClassMeeting

fl6

n4ore

on

Derivatives

-ff1*{x},

then units

for

dyldx

will

be

y

units

over

x

units.

For

example,

distance-time

functions have

S=f(t),

where

S is distance

&

t

is time. d$/dt will

be rneasured in ff/sec

&

gives

velocity at

any t.

-IfP=f(t),

where P is

a

given

population

size

at

any

time

t,

dP/dt will

meazure how

this

population

is

changing

instantateously

with

time,

thus

gvins

a

rale

of

grol?vgE

or

possibly,

a

rate

gf

dec+v.

if

the

given population

decreases.

-If

C(x) represents a total cost

function

for

producing

x

items,

then

C

'

(x) grves

the

rate

at which

cost

per

itenr

is

changing at

x.

This is better

known

as

Marsiqal

Cost.

-Exge$b

lf a derivative

is l0

cubic feet

ofwater

per

second

what could this

describe?

-All

derivatives

give

in$taqtanfouE

rflte$

of ctmge

ofthe dependent

variable

with respect

to

(urt)

the

independent

variable.

*kludg"

Let

T={t) repregent

the temperature

of a

cup of

coffee

placed

on

the

kitchen

table

where

t

i$

in

minutes

&

gives

the time

elapsed

alter

placement,

What

would

dT/dt

represent?

What

are the units?

What

would

the

expression f '

(20)

give?

Would f

'

(t)

be

positive

or negative? Why?

(assume

the

kitchen

temp is

a

constant

72"

E).

ApBroximating

the

value

oJa

functipn

usine

the

d$rivative

Let's

say

we know

y{x)

and the value

at

>ra,

t{a)

and

Would like

an

estirnate of the

value of t{x),

where x

is

close

to

a.

Let's

a$sume we

have dy/d:r=1"

(x),

which

gives

us the

slope of the

tangert line at

x*a

(see

diagram

on

the left).

Let

d:r=x-a

(change

in

x)

(called

the diftbrential

of

x),

then

we define

dy

=

f

'

(x)

dx,

which

mea$ures

the

y

change

along

the tangent

line for

the

giverr

dx. This is

called

the differential

ofy.

-Then,

ifx

is close to

q

{x)

can be

approximated

by: f(x)

*

f(a)

+

f

'

(a}dx

(the

smaller

the

dx,

the

better the

approximation)

(tangent

line

is

closer to the curve at x).

Example:

If f(10F7.53

and

f

'

(lOF

-0.2,

estimate a)

{11)

b}

fl15)