,/

Class

Meeting

#5

6*"'\

The

Derivative

1) locate

a

pt

x

on

the

x-axis.

Then

express

this

position on

the curve.

Call

it Pr(x,

f(x))

2)

adda small

distance

h

(Ax

is also used)

to

x &

locate that

position

on

the curve.

Call

it

Pz(x+[

f(x+h))

3)

connect

the

two

points

&

express

the

slope

of the

secant line.

The

slope of

PrPz:

[f(x+h)-f(x)]/h

4)

Let

h---+0.

This

forces

Pz

to slide

towards

Pr.

The

secant

lines for

each

position

get

closer

to the

tangent

line

at

Pr.

5) So,

the slope

expression

for the

secant

line

gets

closer

to

the

slope

slope

expression

for the

tangent

tine

(which

is our

goal).

6)

In symbols,

we

write

it this

waY:

lim

[f(x+h)-f(x)/h

:

f

'

(x)

(Call

the

derivative

of f at

x)

x-*0

7)

This

is Newtonrs

difference

quotient

definition

of a derivatve.

8)

View the

video

on the

first

recommended

web site

(lesson

1, ch 2)

If we set

up

the diagram

using

a as the

first

position

on

the

x-axis

and

x as another

position

to the right of a,

wewouldhavethederivativeinadifferentform.

Itwouldlooklike:

lim

[f(x)-f(a)](x-a)

:f

'(a)

x-+a

The

result

in this

case

is the

numerical

value of

the derivative

at x:a,

not a

general

expression

for the derivative

in terms

of x.

This

form

is

popular in more

theoretical

courses.

See

the

gaph

in

fig.

2.

Example:

Given

f(x) =

x2,

calculate

f

'(x)

using

the

definition

of

the

derivative.

Using

the

TI-83

to

gaph

the

derivative

of a

function,

f(x):

1)

go

to

the

y:

menu

(leave

blank)

2) click

the

Math

menu,

then

go

to 8

3i

code

in

(f(x), x, x),

then

press

zoom

fit or

set up

your own friendly

viewing

window.

Note:

It

is

much

more

difficult to

graph

the

derivative

by

just

looking

at the

gaph

of

f(x). You have to

visualize

each

y

value

on the

graph

of the

derivative

as the

slope of the

tangent

to the

curve

(x)

at that

point.

Consider

the

graPh

below

R